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Physics Part I Chapter 4 Motion In A Plane

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    NCERT Solution For Class 11 Physics Physics Part I

    Motion In A Plane Here is the CBSE Physics Chapter 4 for Class 11 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 11 Physics Motion In A Plane Chapter 4 NCERT Solutions for Class 11 Physics Motion In A Plane Chapter 4 The following is a summary in Hindi and English for the academic year 2021-2022. You can save these solutions to your computer or use the Class 11 Physics.

    Question 1
    CBSEENBI11019949
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    A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone. 
    Solution

    Given, 
    Length of the string, l = 80 cm
    Number of revolutions = 14
    Time taken = 25 s
    Frequency comma space straight nu space equals space fraction numerator No. space of space revolutions over denominator Time space taken end fraction space equals space 14 over 25 Angular space frequency comma space straight omega space equals space 2 πν space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 cross times 22 over 7 cross times 14 over 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 88 over 25 space rad divided by sec space Centripetal space acceleration comma space straight a subscript straight c space equals space straight omega squared straight r space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 88 over 25 close parentheses squared cross times 0.85 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 9.91 space straight m divided by straight s squared space

    The direction of centripetal acceleration is always directed along the string, toward the centre, at all points.

    Question 2
    CBSEENPH110024934
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    Explain and derive parallelogram law of vector additor.
    Solution

    The parallelogram law of vector addition states that if two vectorsare represented by the two adjacent sides of a parallelogram by both magnitude and direction from a point, then their resultant can be represented by the diagonal of the parallelogramfrom the same point.

    Question 3
    CBSEENPH110024935
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    A projectile is thrown at an angle θ from the horizontal with velocity u under the gravitation field of the earth. Derive expressions for its:
    a)Time of its flight
    b)Height
    c)Horizontal Range
    Solution


    Let us consider a ball projected at an angle θ with respect to horizontal x-axis with the initial velocity u .

    The point O is called the point of projection, θ is the angle of projection and OB = horizontal range. The total time taken by the particle from reaching O to B is called the time of flight.

    Now,

    (a). The total time of flight is

    Resultant displacement is zero in Vertical direction.

    Therefore, by using equation of motion

    s = ut-12gt2gt=2 sin θt = 2 sin θg

    (b). The  horizontal range is

    Horizontal range OA = horizontal component of velocity × total flight time

    R =u cos θ × 2u sin θgR = u2 sin 2θg

    (c). The maximum height is

    It is the highest point of the trajectory point A. When the ball is at point A, the vertical component of the velocity will be zero.

    By using equation of motion

    v2=u2 -2as0 = u2sin2θ-2gHH= u2sin2θ2g

    Hence, this is the required solution

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    Question 4
    CBSEENPH11017861
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    What is a scalar quantity?

    Solution
    A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
    Question 5
    CBSEENPH11017862
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    What is a vector quantity?

    Solution
    A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
    Question 6
    CBSEENPH11017863
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    Give three examples of scalar quantities.

    Solution
    Mass, temperature and energy
    Question 7
    CBSEENPH11017864
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    Give three examples of vector quantities.

    Solution
    Force, impulse and momentum.
    Question 8
    CBSEENPH11017865
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    What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

    Solution
    A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 

    Question 9
    CBSEENPH11017866
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    Is a physical quantity, having a direction, necessarily a vector?

    Solution
    No, all physical quantities having  direction is not a vector. 
    Question 10
    CBSEENPH11017867
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    Name a quantity which is both scalar as well as vector.

    Solution
    Surface.
    Question 11
    CBSEENPH11017868
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    Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

    Solution
    Acceleration and momentum are vector quantities. 
    Question 12
    CBSEENPH11017869
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    Pick up the odd one from the following: force, linear momentum, mass, impulse.

    Solution
    Mass, because mass is a scalar and all others are vector quantities.
    Question 15
    CBSEENPH11017877
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    What are polar vectors?

    Solution
    The vectors having linear effect are called polar vectors. A vector which reverses sign when the cordinate axis is reversed are called as polar vectors. 
    Question 16
    CBSEENPH11017878
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    Give two examples of polar vectors.

    Solution
    Force and velocity are polar vectors. 
    Question 17
    CBSEENPH11017885
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    What are axial vectors?

    Solution
    A vector that does not reverse its sign when the coordinate system is changed to a new system by a reflection in the origin.
    Axial vectors act along axis of rotation.

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    Question 18
    CBSEENPH11017886
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    Give two examples of axial vectors.

    Solution
    Angular momentum and torque are two examples of axial vectors. 
    Question 19
    CBSEENPH11017887
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    What are tensors?

    Solution
    Tensors are the physical quantities having different values in different direction but not having specified direction. 
    Question 20
    CBSEENPH11017889
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    Give one example of tensor.

    Solution
    Moment of inertia.
    Question 21
    CBSEENPH11017893
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    What is module of vector?

    Solution
    The measure of magnitude of vector is known as the module or modulus of a vector. 
    Question 22
    CBSEENPH11017894
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    What are equal vectors?

    Solution
    Equal vectors are the vectors having same magnitude and same direction. 
    Question 23
    CBSEENPH11017895
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    What are negative vectors?

    Solution
    Two vectors are said to be negative, if they have same magnitude and opposite direction. 
    Question 24
    CBSEENPH11017899
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    What are co-initial vectors?

    Solution
    The vectors having their initial points same are called as co-initial vectors. 
    Question 25
    CBSEENPH11017900
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    Define unit vector.

    Solution
    The vector whose magnitude is unity is called unit vector. 
    Question 26
    CBSEENPH11017901
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    Define collinear vectors.

    Solution
    Two or more vectors lying along the same line or parallel lines are known as collinear vectors. 
    Question 27
    CBSEENPH11017902
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    When two or more vectors are said to be coplanar vectors?

    Solution
    When three or more vectors lie on the same plane, they are known as coplanar vectors.
    Question 28
    CBSEENPH11017903
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    Define parallel vectors.

    Solution
    When the cross-product of two vectors is zero, they are known as parallel vectors. 
    For example, the zero vector is parallel to all other vectors. 
    Question 29
    CBSEENPH11017904
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    Is it necessary for two parallel vectors to have equal magnitude?

    Solution
    No, two parallel vectors need not have equal magnitude.
    Question 30
    CBSEENPH11017905
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    Are negative vectors unlike parallel vectors?

    Solution
    Yes, negative vectors are unlike parallel vectors.
    Parallel vectors may or may not be of the same magnitude. But, a negative vector is of the same magnitude. 
    Question 31
    CBSEENPH11017906
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    Are unlike parallel vectors negative vectors?

    Solution
    No, unlike vectors and negative vectors are different vectors. Unlike parallel vectors need not have equal magnitude. 
    Question 32
    CBSEENPH11017907
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    Can two non-coplanar vectors be parallel vectors?

    Solution
    No, coplanar vectors cannot be paralle vectors as they are intersecting vactors. 
    Question 33
    CBSEENPH11017908
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    Comment on the statement: Like parallel vectors are equal vectors.

    Solution
    The statement is wrong because like parallel vectors need not have equal magnitude while equal vectors have equal magnitude. 
    Question 34
    CBSEENPH11017909
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    What changes take place in the vector when it is replaced parallel to itself without changing its magnitude and direction?

    Solution
    When the vector is replaced parallel to itself without changing it's magnitude and direction, the vector remains unchanged.
    Question 35
    CBSEENPH11017910
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    What is free vector?

    Solution
    When a vector is replaced parallel to itself, and the effect of the vector remains unchanged, the vector is called a free vector.  
    Question 36
    CBSEENPH11017911
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    Give two examples of free vector.

    Solution
    Force, momentum, impulse are some of the examples of free vector. 
    Question 37
    CBSEENPH11017912
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    What is fixed vector?

    Solution
    A vector whose initial point (tail) is fixed is called fixed vector.
    Question 38
    CBSEENPH11017913
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    Give one example of fixed vector.

    Solution
    Displacement vector is an example of a fixed vector. 
    Question 39
    CBSEENPH11017915
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    If <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> then what is the relation between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.?
    Solution

    The relation between A and B is,
     
                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

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    Question 40
    CBSEENPH11017918
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    The magnitude of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is equal and equals to one. Can we write straight i with hat on top space equals space straight j with hat on top?
    Solution

    Since the direction of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is different, straight i with hat on top space not equal to space j with hat on top

    Question 41
    CBSEENPH11017921
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    How can we change the vector into negative vector?
    Solution

    When a given vector is multiplied by -1, the direction of the vector is reversed. 

    Question 42
    CBSEENPH11017925
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    If space straight A with rightwards arrow on top equals 3 straight i with hat on top space space and space straight B with rightwards arrow on top space equals space 2 straight i with hat on top then what is the relation between straight A with hat on top space and space straight B with hat on top ?
    Solution

    The vectors is along the same direction.
    That is, space straight A with hat on top parallel to straight B with hat on top 

    Question 43
    CBSEENPH11017926
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    Can we multiply a vector by scalar?

    Solution
    Yes, a vector can be multiplied by a scalar quantity. 
    For example, S is any scalar and is multiplied to vector A, then the vector becomes Sstraight A with rightwards harpoon with barb upwards on top, whose magnitude is s times the magnitude of the vector A. 
    Question 44
    CBSEENPH11017927
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    Can we divide a vector by scalar?

    Solution
    Yes, we can divide a vector by scalar. The magnitude of the vector is reduced by the scalar quantity, when divided. 
    Question 45
    CBSEENPH11017928
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    Can we divide a scalar by vector?

    Solution
    No, we cannot divide a scalar by vector.
    Question 46
    CBSEENPH11017933
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    What is the composition of vectors?

    Solution
    The process of compounding two or more vectors into a single vector is called composition of vectors.
    Composition of vectors determines the resultant of two or more vectors.
    Question 47
    CBSEENPH11017934
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    What types of quantities are added geometrically?

    Solution
    Vector quantities are added geometrically. Vector quantities cannot be added by simple laws of algebra. 
    Question 48
    CBSEENPH11017935
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    Two vectors of magnitude A and B are acting at angle straight theta. What is the magnitude of resultant?

    Solution

    The magnitude of the resultant vector is given by, 
                          square root of straight A squared plus straight B squared plus 2 AB space cosθ. end root

    Question 49
    CBSEENPH11017937
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    Two vectors of magnitude A and B are acting at angle straight theta. What is the angle that the resultant of two vectors makes with vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> ?
    Solution

    Given, two vectors A and B. 
    That is, the angle between the resultant of two vectors is, 
     
                         tan to the power of negative 1 end exponent open parentheses fraction numerator straight B space sinθ over denominator straight A plus straight B space cosθ end fraction close parentheses

    Question 50
    CBSEENPH11017939
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    Can we add two or more vectors not lying in the same plane?

    Solution
    Yes. Two or more vectors which are not lying in the same plane can be added. 
    Question 51
    CBSEENPH11017940
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    Three vectors are represented by three sides of a triangle in same order. What is their resultant? 

    Solution
    Given the vectors are of the same order. Hence, their resultant vector is equal to zero. 
    Question 52
    CBSEENPH11017941
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    State two properties of vector addition.

    Solution

    Two properties of vector addition are:
    (i) Addition of vectors is commutative.
    That is, (straight A with rightwards harpoon with barb upwards on top space plus stack space B space with rightwards harpoon with barb upwards on top equals space B with rightwards harpoon with barb upwards on top space plus space A with rightwards harpoon with barb upwards on top 
    (ii) Addition of vectors is associative.  
    Mathematically, left parenthesis straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top right parenthesis space plus space straight C with rightwards harpoon with barb upwards on top space equals space straight A with rightwards harpoon with barb upwards on top space plus space left parenthesis straight B with rightwards harpoon with barb upwards on top plus straight C with rightwards harpoon with barb upwards on top right parenthesis

    Question 53
    CBSEENPH11017942
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    Is vector addition commutative?

    Solution

    Yes, vector addition is commutative.
                 space straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top plus straight A with rightwards arrow on top

    Question 54
    CBSEENPH11017943
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    Comment on the statement that vector subtraction is commutative.

    Solution
    The given statement is wrong.
    Vector subtraction does not follow commutative law because, 
                         straight A with rightwards arrow on top minus stack straight B space with rightwards arrow on top not equal to straight B with rightwards arrow on top space minus space straight A with rightwards arrow on top
    Question 55
    CBSEENPH11017944
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    Is vector addition associative?

    Solution
    Yes, vector addition is associative. 
    Question 56
    CBSEENPH11017945
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    Is vector subtraction associative? 

    Solution
    No, vector subtraction is not associative. 

    Question 57
    CBSEENPH11017946
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    Can commutative law and associative law be applied to vector subtraction?

    Solution
    Commutative law and associative law cannot be applied to vector subtraction. 
    Question 58
    CBSEENPH11017947
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    When the magnitude of the resultant of two vectors is equal to the sum of the magnitude of two vectors?

    Solution
    The resultant of two vectors is equal to the sum of the magnitude of two vectors, if two vectors are like parallel vectors. The nature of resultant vectors is same as that of the given vectors. 
    Question 59
    CBSEENPH11017948
    Wired Faculty App

    What is the magnitude of difference of the two unit vectors?

    Solution
    Magnitude of difference of two unit vectors lies between 0 and 2. 
    Question 60
    CBSEENPH11017949
    Wired Faculty App

    Do you think that magnitude straight i with hat on top minus straight j with hat on top is zero?
    Solution

    Magnitude = square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals square root of 2  

    Question 61
    CBSEENPH11017950
    Wired Faculty App

    What is the resultant of vectors 2 straight i with hat on top space minus space 3 straight j with hat on top space minus space 4 straight k with hat on top and negative 2 straight i with hat on top plus 3 straight j with hat on top plus 4 straight k with hat on top?
    Solution

    The resultant of the vector is zero vector.

    Question 62
    CBSEENPH11017951
    Wired Faculty App

    Can straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top minus straight B with rightwards arrow on top?
    Solution

    Ifspace straight B with rightwards arrow on top is a zero vector, then straight A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top minus B with rightwards harpoon with barb upwards on top. 

    Question 63
    CBSEENPH11017952
    Wired Faculty App

    What is the resultant of two perpendicular vectors of magnitude 12 and 5?

    Solution

    The resultant of two vectors is given by,
     straight R space equals space square root of straight A squared plus straight B squared end root = square root of 12 squared plus 5 squared end root space equals space square root of 169 space equals space 13

    Question 64
    CBSEENPH11017953
    Wired Faculty App

    If magnitude of two vectors is 5 and 2 units respectively, then what can be the maximum possible magnitude of the resultant of these two vectors?

    Solution
    Given, 
    Magnitude of first vector = 5
    Magnitude of second vector = 2
    Resultant of two vectors = square root of 5 squared plus 2 squared end root  
    Therefore, the maximum possible magnitude of the resultant of the two vectors is 7 units. 
    Question 68
    CBSEENPH11017957
    Wired Faculty App

    What is the minimum number of vectors of equal magnitude required to produce zero resultant?

    Solution

    Minimum number of vectors of equal magnitude to produce a zero resultant is 2. 

    Question 69
    CBSEENPH11017958
    Wired Faculty App

    Can ever two vectors of different magnitude be added to get null vector?

    Solution

    Two vectors of different magnitude cannot be added to get a null vector. 

    Question 70
    CBSEENPH11017959
    Wired Faculty App

    Can ever three vectors of different magnitude be added to get null vector?

    Solution
    Yes. If all the three vectors lie in the same plane, three vectors of different magnitude can be added. 
    Question 71
    CBSEENPH11017960
    Wired Faculty App

    If space space straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals straight C with rightwards arrow on top and straight A squared plus straight B squared equals straight C squared then what is the angle between vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top?
    Solution

    The two vectors are perpendicular to each other. The angle between the vectors is 90 degree.

    Question 72
    CBSEENPH11017961
    Wired Faculty App

    The two equal velocities have got a resultant equal to either. What is the angle between them?

    Solution

    The angle between the vectors is 120 degree. 

    Question 73
    CBSEENPH11017962
    Wired Faculty App

    Is open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space open vertical bar straight A minus straight B close vertical bar?
    Solution

    Yes, this is possible if straight A with rightwards arrow on top space and space straight B with rightwards arrow on top are unlike parallel vectors. 

    Question 74
    CBSEENPH11017963
    Wired Faculty App

    When does the direction of resultant of two vectors coincide with each of them?

    Solution
    When the two vectors are like parallel vectors, then the direction of the resultant of two vectors coincides with each of them. 
    Like vectors are the vectors in the same direction. 
    Question 75
    CBSEENPH11017964
    Wired Faculty App

    When does the direction of resultant of two vectors coincide with each of them?

    Solution
    When the two vectors are like parallel vectors, then the direction of the resultant of two vectors coincides with each of them.
    Like vectors are vectors in the same direction. 
    Question 76
    CBSEENPH11017965
    Wired Faculty App

    If straight R equals square root of straight A squared plus straight B squared end root comma then what is the angle between them?
    Solution

    The angle between the vectors A and B is given by, 
                                space tan to the power of negative 1 end exponent open parentheses straight B over straight A close parentheses.

    Question 78
    CBSEENPH11017967
    Wired Faculty App

    Is the magnitude of straight A with rightwards arrow on top minus straight B with rightwards arrow on top and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> equal?
    Solution

    Yes, the magnitude is equal. 

    Question 79
    CBSEENPH11017968
    Wired Faculty App

    What is the magnitude of difference of two vectors of magnitude A and B acting at angle 90°?

    Solution

    The magnitude of two vectors A and B which are at angle 900 is given by,  square root of straight A squared plus straight B squared end root. 

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    Question 80
    CBSEENPH11017969
    Wired Faculty App

    What is magnitude of the difference of two vectors of magnitude A and B acting in opposite direction?

    Solution

    The magnitude of the difference of vectors acting in opposite direction is A+B. 

    Question 81
    CBSEENPH11017970
    Wired Faculty App

    If the magnitude of the two forces are 2 and 3 and the magnitude of their scalar product is 3square root of 2, then find the angle between the vectors. 
    Solution

    Here, A = 2; B = 3 and A. B = 3square root of 2,
    We know that, 
    A. B = AB cos straight theta 
    So, cosstraight theta = fraction numerator straight A with rightwards harpoon with barb upwards on top space. space straight B with rightwards harpoon with barb upwards on top over denominator AB end fraction space equals space fraction numerator 3 square root of 2 over denominator 2 space cross times space 3 end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction = cos 45o. 
    That is  straight theta  = 45o, is the angle between the vectors. 

    Question 82
    CBSEENPH11017971
    Wired Faculty App

    When does the resultant of two vectors equally inclined to both the component vectors?

    Solution
    When component vectors are equal in magnitude, the resultant of two vectors are equally inclined to both the component vectors. 
    Question 83
    CBSEENPH11017972
    Wired Faculty App

    What is the greatest and least value of space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar?
    Solution

    The greatest value of space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar is A + B and least value is A - B.

    Question 84
    CBSEENPH11017973
    Wired Faculty App

    What is resolution of vectors?

    Solution
    Resolution of vectors is the process of splitting a given vector into two or more vectors in given direction which together produce the same effect as is produced by the single vector alone. 
    Question 85
    CBSEENPH11017974
    Wired Faculty App

    When the resolved components are said to be rectangular component?

    Solution
    When  resolved components are perpendicular to each other, the components are said to be rectangular components. 
    Question 86
    CBSEENPH11017975
    Wired Faculty App

    Is it necessary that magnitude of the component of vector less than its magnitude of vector itself?

    Solution
    The magnitude of the component vector will be greater than the magnitude of vector itself.
    Question 87
    CBSEENPH11017976
    Wired Faculty App

    Can a vector have many sets of components?

    Solution
    Yes.
    Question 88
    CBSEENPH11017977
    Wired Faculty App

    How many sets of two resolved components of vector are there in two given directions?

    Solution
    There is one and only one way in which a vector can be resolved into two component vectors along the direction of two given vectors. 
    Question 89
    CBSEENPH11017978
    Wired Faculty App

    Comment upon the statement:

    A vector in XY plane can be split into only two components.

    Solution
    The given statement is wrong. A vector in XY plane can be split into more than two components. 
    Question 90
    CBSEENPH11017979
    Wired Faculty App

    What is the X-component of vector 3 straight i with hat on top space minus space 4 straight j with hat on top?

    Solution

    The x-component of the vector is 3. 

    Question 95
    CBSEENPH11017984
    Wired Faculty App

    If l, m and n are the rectangular components of a vector, then what is the magnitude of vector?

    Solution

    Magnitude of the vector is given by, square root of straight l squared plus straight m squared plus straight n squared end root

    Question 96
    CBSEENPH11017985
    Wired Faculty App

    Can a null vector have rectangular components of non-zero magnitude?

    Solution

    No. The magnitude of a null vector will always be zero.

    Question 97
    CBSEENPH11017986
    Wired Faculty App

    Can a null vector have resolved components of non-zero magnitude?

    Solution
    Yes, the magnitude of resolved components of a null vector can be non-zero. 
    Question 98
    CBSEENPH11017987
    Wired Faculty App

    Can the magnitude of rectangular components of a vector be greater than the magnitude of that vector?

    Solution
    No, the magnitude of a reactangular component of a vector will not be greater than the magnitude of the component of the vector. 
    Question 99
    CBSEENPH11017988
    Wired Faculty App

    Can the magnitude of resolved components of a vector be greater than the magnitude of that vector?

    Solution
    Yes, the magnitude of resolved component of a vector will be greater than the magnitude of that vector. 
    Question 100
    CBSEENPH11017989
    Wired Faculty App

    Do you think, “all the rectangular components of a unit vector are unit vectors”?

    Solution
    No. It is not necessary that all the rectangular components of a unit vector are unit vectors. 
    Question 101
    CBSEENPH11017990
    Wired Faculty App

    What is X-component of the resultant of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and space space minus 3 straight j with hat on top plus straight k with hat on top ?

    Solution

    The X component of the resultant vector is 3 units.

    Question 102
    CBSEENPH11017991
    Wired Faculty App

    What is unit vector in direction of vector 5 straight j with hat on top ?
    Solution

    The unit vector is in the direction of straight j with hat on top.

    Question 103
    CBSEENPH11017992
    Wired Faculty App

    What is the rectangular component of vector left parenthesis negative 3 straight i with hat on top space minus space 5 straight j with hat on top right parenthesis along Z-axis? 
    Solution
    The rectangular component of the vector along z-axis is zero. 
    Question 104
    CBSEENPH11017993
    Wired Faculty App

    State True or False: 

    The magnitude of a vector is always a scalar.

    Solution
    A. TRUE
    Question 105
    CBSEENPH11017994
    Wired Faculty App

    State True or False:

    Each component of a vector is always a scalar.

    Solution
    A. FALSE
    Question 108
    CBSEENPH11017997
    Wired Faculty App

    What is the angle subtended by the resultant of left parenthesis negative 2 straight i with hat on top minus 3 straight j with hat on top plus straight k with hat on top right parenthesis and left parenthesis negative 3 straight i with hat on top plus 3 straight j with hat on top minus straight k with hat on top right parenthesis with X-axis?
    Solution

    Here, 
    straight A with rightwards harpoon with barb upwards on top space equals space left parenthesis negative 2 i with hat on top minus 3 j with hat on top space plus k with hat on top right parenthesis B with rightwards harpoon with barb upwards on top space equals space left parenthesis negative 3 i with hat on top space plus space 3 j with hat on top space minus space k with hat on top right parenthesis 
    Therefore, the angle subtended by the resultant of the vectors with X-axis is zero. 

    Question 109
    CBSEENPH11017998
    Wired Faculty App

    One of the rectangular components of velocity of a particle is 12 m/s and the other is 16 m/s. What is the speed of the particle?

    Solution

    The rectangular components of velocity of a particle is 12 m/s and 16 m/s. 
    Therefore, the velocity of the particle, v = square root of 12 squared plus 16 squared end root space equals space square root of 400 space equals space 20 space m divided by s 

    Question 110
    CBSEENPH11017999
    Wired Faculty App

    What is the position vector of a point having coordinates (2, –1, 3)?

    Solution

    The position vector is given by 2 straight i with hat on top minus straight j with hat on top plus 3 straight k with hat on top. 

    Question 111
    CBSEENPH11018000
    Wired Faculty App

    If the coordinates of points P and Q are (3, 1, -2) and (1, -2, 2) respectively, then find the vector PQ with rightwards arrow on top ?
    Solution

    The vector PQ is [(3-1)straight i with hat on top + (1-(-2)straight j with hat on top + (-2-2)straight k with hat on top 
    So, 
                    PQ with rightwards arrow on top space equals space minus 2 straight i with hat on top minus 3 straight j with hat on top plus 4 straight k with hat on top.

    Question 112
    CBSEENPH11018001
    Wired Faculty App

    What is the component of vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> perpendicular to XY plane?
    Solution

    The component of vector is perpendicular in the XY plane in the Z direction. 
    So, it is 2.9 straight k with hat on top. 

    Question 113
    CBSEENPH11018002
    Wired Faculty App

    What is the direction of null vector?
    Solution

    Null vector cantake any arbitrary direction.

    Question 114
    CBSEENPH11018003
    Wired Faculty App

    Which component of vector 2.4 straight j with hat on top space minus space 9 straight i with hat on top is null vector?
    Solution
    Y-component of the given vector is null vector.  
    Question 115
    CBSEENPH11018004
    Wired Faculty App

    Give one example of resolution of a vector.

    Solution
    A person walking on road is an example of resolution of a vector.
    Question 116
    CBSEENPH11018005
    Wired Faculty App

    If one of the rectangular components of a vector of magnitude 50 has magnitude 30, then what is the magnitude of other rectangular component of the vector?

    Solution

    Given, magnitude of vectors are 50 and 30.
    Therefore, magnitude of the rectangular component = square root of 50 squared minus 30 squared end root space equals space square root of 2500 minus 900 end root space equals space square root of 1600 space equals space 40 

    Question 117
    CBSEENPH11018006
    Wired Faculty App

    What is null or zero vector?

    Solution
    A vector of zero magnitude is called null vector.
    Question 118
    CBSEENPH11018007
    Wired Faculty App

    Does the direction of null vector matter?

    Solution
    No, the direction of null vector is immaterial.
    Question 119
    CBSEENPH11018008
    Wired Faculty App

    Give one example of zero vector.

    Solution
    The difference of two equal vectors is a null vector.

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    Question 120
    CBSEENPH11018009
    Wired Faculty App

    What is position vector of origin?

    Solution
    The position vector of origin is called as the zero vector. The magnitude of this vector will be zero. 
    Question 121
    CBSEENPH11018010
    Wired Faculty App

    Can we subtract the vector from zero vector?

    Solution
    Yes, when any vector is subtracted from the zero vector, negative vector is obtained. 
    Question 122
    CBSEENPH11018011
    Wired Faculty App

    With what scalar, a vector must be multiplied to get null vector? 

    Solution

    When a vector is multiplied with zero, we get a null vector. 

    Question 123
    CBSEENPH11018012
    Wired Faculty App

    Scalar product of two vectors is scalar. Can scalar product of two vectors be negative?

    Solution
    When two vectors are anti-parallel then straight theta space equals space 180 to the power of straight o
    Then, cos 1800 = -1 
    Here, the product of two scalar vectors is negative. 
    Question 124
    CBSEENPH11018013
    Wired Faculty App

    If two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top represent the two adjacent sides of a parallelogram, then what is the area of the parallelogram?
    Solution

    Area of the parallelogram formed by two adjacent vectors, straight A with rightwards arrow on top space and space straight B with rightwards arrow on top, is straight A with rightwards arrow on top cross times straight B with rightwards arrow on top. 

    Question 125
    CBSEENPH11018014
    Wired Faculty App

    Can the product of two vectors be scalar?

    Solution
    Yes, dot product of two vectors is a scalar quantity. 
    Question 126
    CBSEENPH11018015
    Wired Faculty App

    Does the cross product of two vectors commute?

    Solution
    No, the cross product of two vectors do not commute. 
    Question 127
    CBSEENPH11018016
    Wired Faculty App

    Does the scalar product of two vectors commute?

    Solution
    Yes, the scalar products of the two vectors commute. 
    Question 128
    CBSEENPH11018017
    Wired Faculty App

    State distributive law.

    Solution
    The distributive law states that,
                 A × (B + C)=A × B + A × C
    Question 129
    CBSEENPH11018018
    Wired Faculty App

    Does the scalar product follow distributive law?

    Solution
    Yes, scalar product follows distributive law. 
    Question 130
    CBSEENPH11018019
    Wired Faculty App

    Does the vector product follow distributive law?

    Solution
    Yes, vector product follows distributive law. 
    Question 131
    CBSEENPH11018020
    Wired Faculty App

    What is the ratio of magnitude of dot product to that of cross product?

    Solution

    The ratio of magnitude of dot product to that of cross product is open vertical bar tan space straight theta close vertical bar

    Question 132
    CBSEENPH11018021
    Wired Faculty App

    Give one physical quantity in which we make the use of dot product of vectors.

    Solution
    Work is dot product of force and displacement. 
                                  W = stack straight F space with rightwards harpoon with barb upwards on top. space d with rightwards harpoon with barb upwards on top 
    Force and displacement both are vector quantities and work is a scalar quantity. 
    Question 133
    CBSEENPH11018022
    Wired Faculty App

    Give one physical relation which makes the use of cross product of vectors.

    Solution
    Torque. It is a cross product of position vector and force vector.
    Question 134
    CBSEENPH11018023
    Wired Faculty App

    What is dot product of two perpendicular vectors?

    Solution

    When angle between two vectors is 900, their dot product is zero. 

    Question 135
    CBSEENPH11018024
    Wired Faculty App

    What does <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> signify?
    Solution

    It gives the magnitude of vector straight A with rightwards arrow on top.

    Question 136
    CBSEENPH11018025
    Wired Faculty App

    What is cross product of two parallel vectors?

    Solution
    When two vectors are parallel to each other angle between them is 0 and sin 0 = 0o. Therefore, their cross product of two parallel vectors is 0. 
    Question 137
    CBSEENPH11018026
    Wired Faculty App

    Can straight A with rightwards arrow on top. straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top cross times straight B with rightwards arrow on top?
    Solution

    No, both cannot be equal to each other. This is because space straight A with rightwards arrow on top. straight B with rightwards arrow on top is a scalar and straight A with rightwards arrow on top cross times straight B with rightwards arrow on top is a vector. A scalar quantity and a vector quantity cannot be equal. 

    Question 138
    CBSEENPH11018027
    Wired Faculty App

    Can <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> then how are the two vectors related?
    Solution
    The two vectors are unlike parallel vectors.
    Question 139
    CBSEENPH11018028
    Wired Faculty App

    What is the dot product of unit vector with itself?

    Solution
    The dot product of unit vector with itself is one. 
    Question 140
    CBSEENPH11018029
    Wired Faculty App

    What is the value of straight A with rightwards arrow on top. straight B with rightwards arrow on top cross times straight A with rightwards arrow on top?
    Solution

    Zero.

    Question 141
    CBSEENPH11018030
    Wired Faculty App

    What is the unit vector in direction of vector straight P with rightwards arrow on top space equals space minus 3 straight i with hat on top plus 2 straight j with hat on top plus 5 straight k with hat on top?
    Solution

    Unit vector in direction of straight P with rightwards arrow on top = fraction numerator straight P with rightwards harpoon with barb upwards on top over denominator vertical line straight P with rightwards harpoon with barb upwards on top vertical line end fraction 
    Therefore, we get, 
    P with hat on top space equals space fraction numerator 1 over denominator square root of 38 end fraction open parentheses negative 3 straight i with hat on top plus 2 straight j with hat on top plus 5 straight k with hat on top close parentheses 

    Question 142
    CBSEENPH11018031
    Wired Faculty App

    A vectors <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> points north and vector space straight B with rightwards arrow on top points east. What is the direction of straight B with rightwards arrow on top cross times straight A with rightwards arrow on top?
    Solution

    The direction of straight B with rightwards harpoon with barb upwards on top space cross times space A with rightwards harpoon with barb upwards on top is vertically upwards.

    Question 143
    CBSEENPH11018032
    Wired Faculty App

    What type of operation on the vectors is not commutative?

    Solution
    Cross product and subtraction of two vectors is not commutative.
    Question 144
    CBSEENPH11018033
    Wired Faculty App

    What is the angle between straight A with rightwards arrow on top cross times straight B with rightwards arrow on top and straight B with rightwards arrow on top cross times straight A with rightwards arrow on top ?
    Solution

    The given two vectors are anti-parallel to each other. 
    Therefore, angle between them is 180 degree

    Question 145
    CBSEENPH11018034
    Wired Faculty App

    What is the angle between straight A with rightwards arrow on top cross times straight B with rightwards arrow on top and straight A with rightwards arrow on top plus straight B with rightwards arrow on top?
    Solution

    Angle between the vector is 90 degree. 

    Question 146
    CBSEENPH11018035
    Wired Faculty App

    It is found that for two vectors <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  straight A with rightwards arrow on top. space straight B with rightwards arrow on top space equals space minus AB. What is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top?
    Solution

    The vectors are unlike parallel vectors. 
    Therefore, angle between them is 180 degree.

    Question 147
    CBSEENPH11018036
    Wired Faculty App

    What is equilibrium?

    Solution
    When the net force acting on a body is zero, the body is in equilibrium. The body is either at rest or moves with uniform velocity. 
    Question 148
    CBSEENPH11018037
    Wired Faculty App

    What is equilibrant force?

    Solution
    Equilibrant force is the force, which is needed to bring the body in equilibrium. The equilibrant force is equal and opposite to the resultant of all the forces acting on the body. 
    Question 149
    CBSEENPH11018038
    Wired Faculty App

    A particle is moving under the action of three forces acting simultaneously in a plane. Can the particle be in equilibrium?

    Solution
    Yes, the particles acn be in equilibrium if the resultant of two forces is equal and opposite to the third force. 
    Question 150
    CBSEENPH11018039
    Wired Faculty App

    If straight R with rightwards arrow on top is the resultant of two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top then what is the equilibrant vector of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top?
    Solution

    The equilibrant vector is opposite in direction to the resultant vector.
    So, equilibrant vector of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 

    Question 151
    CBSEENPH11018040
    Wired Faculty App

    How will you represent a vector quantity?

    Solution
    A vector quantity can be represented by drawing a line parallel to the vector quantity having length equal to magnitude of quantity. Arrow-head is drawn in the direction of vector quantity. 
    Question 152
    CBSEENPH11018041
    Wired Faculty App

    Do you think that if any quantity has direction, necessarily be a vector? Give example in support of your answer.

    Solution
    No, direction is not the only condition for a quantity to be a vector. In addition, the vector axioms should be followed by the physical quantity. 
    For example, current possesses the direction even then it is a scalar because current does not follow the axiom vector addition. 
    Question 153
    CBSEENPH11018042
    Wired Faculty App

    Pick up the scalar and vector quantities out of the following quantities: mass, force, temperature, displacement, frequency, surface energy, current, electric field, gravitational field. 

    Solution

    Vector quantities are force, displacement, electric field, gravitational field.
    Scalar quanties are mass, temperature, frequency, surface energy, current.

    Question 154
    CBSEENPH11018043
    Wired Faculty App

    State for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, and angular velocity. 

    Solution

    Volume - scalar quantity
    Speed - scalar quantity
    Acceleration - vector quantity
    Density - scalar quantity
    Number of moles - scalar quantity
    Velocity - vector quantity
    Angular frequency - scalar quantity
    Angular velocity - vector quantity

    Question 155
    CBSEENPH11018044
    Wired Faculty App

    Define unit vector and write the unit vector in direction of straight A with rightwards arrow on top.
    Solution

    A unit vector is whose magnitude is unity.
    Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be any vector whose magnitude is A.
    Unit vector in direction of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is represented by straight A with hat on top and is  given by, 
     
                             straight A with hat on top space equals space fraction numerator straight A with rightwards arrow on top over denominator open vertical bar straight A close vertical bar end fraction

    Question 156
    CBSEENPH11018045
    Wired Faculty App

    Define the following:
    (i) Equal vector               (ii) Negative vector
    (iii) Parallel vector           (iv) Co-planer vector
    (v) Co-initial vector          (vi) Co-linear vector
    Solution
    (i) Equal vector: Two vectors are said to be equal vectors, if they have equal magnitude and same direction.
    Figure (i) shows the two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top having same magnitude and same direction, therefore, they are equal vectors. 
                   
    (ii) Negative vector: A vector is said to be negative of another vector if they have same magnitude but are in opposite direction.
    (iii) Parallel vector: If the lines of action of the two vectors are same or parallel to each other, then the vectors are parallel. The magnitude of two vectors need not to be equal for parallel vectors. 
    The parallel vectors are of two types: 
    (a) Like parallel vector: Two vectors are said to be like parallel vectors if they have same direction.  
    (b) Unlike parallel vectors: Two vectors in the opposite direction are called unlike parallel vectors. 
    (iv) Co-planer vectors: The vectors lying in one plane are called co-planer vectors. 
    (v) Co-initial vectors: The vectors having common initial point are called co-initial vectors.
    (vi) Co-linear vectors: The vectors having same line of action are called co-linear vectors. 
    Question 157
    CBSEENPH11018046
    Wired Faculty App

    Can we multiply a vector by a scalar?

    Solution
    On multiplying a vector by a scalar, we get another vector whose magnitude is equal to the modulus of scalar times the magnitude of vector. The direction of the resultant vector is either same or opposite as that of vector depending upon whether scalar is positive or negative, respectively. 
    Question 158
    CBSEENPH11018047
    Wired Faculty App

    If we multiply the vector by +2, then what will be the new vector?

    Solution
    On multiplying the vector by +2, we get the new vector whose magnitude is twice the magnitude of original vector. Direction is same as that of the original vector because the sign of the scalar is positive. 
    Question 159
    CBSEENPH11018048
    Wired Faculty App

    If we multiply the vector by -2, then what will be the new vector?

    Solution
    When we multiply the vector by -2, we get a new vector of magnitude, twice the magnitude of original vector. The direction is opposite to that of the original vector as the sacalar has a negative sign. 
    Question 160
    CBSEENPH11018049
    Wired Faculty App

    A vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is multiplied by scalar k. Is the magnitude of new vector straight k space open vertical bar straight A with rightwards arrow on top close vertical bar?
    Solution

    No, because magnitude of a vector is positive quantity.
    If k is a negative scalar then k <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> will be negative which is not possible.
    The magnitude of new vector will be <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 161
    CBSEENPH11018050
    Wired Faculty App

    Which of the following is true or false? Give examples: 

    (i) A scalar quantity can never take negative values.

    (ii) It is necessary for a scalar quantity to be dimensionless. 
    Solution

    (i) False, work is a scalar quantity and can have negative values. 
    (ii) False, work is a scalar quantity and have dimensions.
    The dimensional formula of work is M1 L2 T-2]

    Question 162
    CBSEENPH11018051
    Wired Faculty App

    State triangular law and parallelogram law of vector addition.

    Solution

    Triangle law of vector addition:
    If two vectors acting on the particle at the same time are represented in magnitude and direction by the two sides of a triangle taken in one order, their resultant vector is represented in magnitude and directed by the third side of the triangle taken in opposite order.



    Parallelogram law of vector addition:
    If two vectors acting simultaneously at a point are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, then their resultant is given by the diagonal of parallelogram passing through same point, both in magnitude and direction.

    Question 163
    CBSEENPH11018052
    Wired Faculty App

    State polygon law of vector addition.

    Solution
    Polygon law of vector addition states that if two or more vectors are represented by adjacent sides of a polygon, taken in same order both in magnitude and direction, then the resultant is given by closing side of the polygon taken in opposite order both in magnitude and direction. 
    Question 164
    CBSEENPH11018053
    Wired Faculty App

    Can we always add two vectors? Explain your answer.

    Solution
    If both the vectors are of the same type, then we can add those quantities. We cannot add two different types of vector. 
    For example, velocity vector and displacement vector cannot be added. 
    Question 165
    CBSEENPH11018054
    Wired Faculty App

    Find the magnitude and direction of resultant of two vectors analytically. Discuss the special cases for angle between the vectors: (i) 0° (ii) 180° (iii) 90°.

    Solution

    Let two vectors space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top be represented by two adjacent sides OA and OB of parallelogram OACB.
        From C draw CD perpendicular to OA produced.
                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>       [corresponding angle]
    From increment space CAD comma
                  AD space equals space AC space cos space straight theta space equals space straight Q space cosθ
                  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Special cases:


                     

    Question 166
    CBSEENPH11018055
    Wired Faculty App

    State the properties of vector addition.

    Solution

    The properties of vector addition are: 
    (i) A vector can be added only to a vector. 
    (ii) Closure property of addition: The sum of two vectors is also a vector. Hence, vectors are closed under addition.
    (iii) Vector addition is commutative.
     
    i.e. space straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top plus straight A with rightwards arrow on top

    (iv)Vector addition is associative. 
     
    i.e. left parenthesis straight A with rightwards arrow on top plus straight B with rightwards arrow on top right parenthesis plus straight C with rightwards arrow on top space equals space straight A with rightwards arrow on top plus left parenthesis straight B with rightwards arrow on top plus straight C with rightwards arrow on top right parenthesis 
    (v) Vector addition is distributive.
    i.e. space straight m left parenthesis straight A with rightwards arrow on top plus straight B with rightwards arrow on top right parenthesis space equals space straight m straight A with rightwards arrow on top plus straight m straight B with rightwards arrow on top 
    (vi) Magnitude of the resultant of two vectors is less or equal to the sum of the magnitude of two vectors and greater or equal to the magnitude of the difference of the magnitude of two vectors. 
    i.e. space space space space space space space parallel to right enclose straight A with rightwards arrow on top end enclose space minus space parallel to right enclose straight B with rightwards arrow on top end enclose space less-than or slanted equal to space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space less-than or slanted equal to space open vertical bar straight A with rightwards arrow on top close vertical bar space plus space open vertical bar straight B with rightwards arrow on top close vertical bar

    Question 167
    CBSEENPH11018056
    Wired Faculty App

    Show that the vector addition follows commutative law.

    Solution

    Let straight A with rightwards arrow on top space and space straight B with rightwards arrow on top be represented by <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and OQ with rightwards arrow on top comma the two adjacent sides of parallelogram OPSQ.
    In the figure, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is the diagonal of parallelogram.

    Since <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is equal and parallel to <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴             OP with rightwards arrow on top space equals space QS with rightwards arrow on top space equals space straight A with rightwards arrow on top
    Also, space OQ with rightwards arrow on top is equal and parallel to space PS with rightwards arrow on top
    ∴           OQ with rightwards arrow on top space equals space PS with rightwards arrow on top space equals space straight B with rightwards arrow on top
    Now applying triangle law of vector addition to increment OPS comma space we get
                 OS with rightwards arrow on top space equals space OP with rightwards arrow on top space plus space PS with rightwards arrow on top space equals space straight A with rightwards arrow on top plus straight B with rightwards arrow on top                ...(1)
    Similarly applying triangle law of vector addition to <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> we get
                  OS with rightwards arrow on top space equals space OQ with rightwards arrow on top plus QS with rightwards arrow on top space equals space straight B with rightwards arrow on top plus straight A with rightwards arrow on top                ...(2)
    From equation (1) and (2),
                   space straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top plus straight A with rightwards arrow on top
    Hence the vector addition follows commutative law.

    Question 168
    CBSEENPH11018057
    Wired Faculty App

    State and prove associative law for vector addition.

    Solution
    The associative law of vector states that the vector addition is same, in whatever grouping they are added.
    left parenthesis A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis plus C with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top plus left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis
    Proof: 

    Let there be three vectors straight A with rightwards harpoon with barb upwards on top comma space B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top. These three vectors, both magnitude and direction are represented by three vectors stack OL comma with rightwards harpoon with barb upwards on top space stack L K with rightwards harpoon with barb upwards on top space a n d space stack K G with rightwards harpoon with barb upwards on top space of polygon OLKG. 
    Now, using polygon law of vector addition, the resultant of the vector will be represented by the closing side of the polygon OG with rightwards harpoon with barb upwards on top. 
    Join O with K, L and G. 
    Now, applying triangle law to triangle OLK, we have
                   OK with rightwards harpoon with barb upwards on top space equals space stack O L with rightwards harpoon with barb upwards on top plus stack L K with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top 
    Now, in increment OKG, we have
    OG with rightwards harpoon with barb upwards on top space equals space stack O K with rightwards harpoon with barb upwards on top plus stack K G with rightwards harpoon with barb upwards on top space equals space left parenthesis A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis space plus space C with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis
    Now, applying the triangle law of vector addition to increment LKG comma spacewe have
    LG with rightwards harpoon with barb upwards on top space equals space stack L K with rightwards harpoon with barb upwards on top plus stack K G with rightwards harpoon with barb upwards on top space equals space B with rightwards harpoon with barb upwards on top space plus space C with rightwards harpoon with barb upwards on top
    For triangle OLG, we get
    OG with rightwards harpoon with barb upwards on top space equals space stack O L with rightwards harpoon with barb upwards on top plus stack L G with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top space plus space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis space space space space space space space space space space space... space left parenthesis 2 right parenthesis
    Now, from equations (1) and (2), we get
    left parenthesis straight A with rightwards harpoon with barb upwards on top plus B with rightwards harpoon with barb upwards on top right parenthesis space plus space C with rightwards harpoon with barb upwards on top space equals space A with rightwards harpoon with barb upwards on top space plus space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top right parenthesis
    Therefore, the vector addition is assosciative. 
                   
    Question 169
    CBSEENPH11018058
    Wired Faculty App

    Show that:
    open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar less-than or slanted equal to open vertical bar straight A with rightwards arrow on top close vertical bar space plus space open vertical bar straight B with rightwards arrow on top close vertical bar greater-than or slanted equal to space open vertical bar open vertical bar straight A with rightwards arrow on top close vertical bar minus open vertical bar straight B with rightwards arrow on top close vertical bar close vertical bar

    Solution

    To show: 
    open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals square root of straight A squared plus straight B squared plus 2 ABcosθ end root

    Since <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>,
    open vertical bar straight A with rightwards harpoon with barb upwards on top plus straight B with rightwards harpoon with barb upwards on top close vertical bar space less-than or slanted equal to space square root of straight A squared plus straight B squared plus 2 AB end root space space space space space space space space space space space space space space equals space square root of left parenthesis straight A plus straight B right parenthesis squared end root space space space space space space space space space space space space space space equals space straight A space plus space straight B Therefore comma space open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar less-than or slanted equal to open vertical bar straight A with rightwards harpoon with barb upwards on top space close vertical bar space plus space open vertical bar space straight B with rightwards harpoon with barb upwards on top close vertical bar space Now comma space open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar space equals space square root of straight A squared plus straight B squared plus 2 ABcosθ end root Since comma space cos space straight theta space greater-than or slanted equal to space minus space 1 therefore space open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar greater-than or slanted equal to space square root of straight A squared plus straight B squared minus 2 AB end root space space space space space space space space space space space space space space space space space space space space equals square root of left parenthesis straight A minus straight B right parenthesis squared end root space space space space space space space space space space space space space space space space space space space equals open vertical bar straight A with rightwards harpoon with barb upwards on top space minus space straight B with rightwards harpoon with barb upwards on top close vertical bar space space space space space space space space space space space space space space space space space space space space equals space open vertical bar open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar plus open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar close vertical bar
    Hence proved. 

    Question 170
    CBSEENPH11018059
    Wired Faculty App

    Show that:

    Solution

    We have the inequality, 
                     negative 1 less-than or slanted equal to cosθ less-than or slanted equal to 1  
    i.e., negative 1 space less-than or slanted equal to space minus cos space straight theta space less-than or slanted equal to space 1 space

    Now, multiplying the inequality by 2AB, 
    negative 2 A B space less-than or slanted equal to space minus 2 A B space cos theta space less-than or slanted equal to space 2 A B space
    Now adding A2 + B2, we get 
    straight A squared space plus space straight B squared space minus space 2 AB space less-than or slanted equal to space straight A squared plus space straight B squared space minus space 2 ABcosθ space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space less-than or slanted equal to space straight A squared plus space straight B squared space plus space 2 AB open parentheses open vertical bar open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar minus open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar close vertical bar close parentheses squared space less-than or slanted equal to space open parentheses open vertical bar straight A with rightwards harpoon with barb upwards on top minus straight B with rightwards harpoon with barb upwards on top close vertical bar close parentheses squared space less-than or slanted equal to space open parentheses open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar plus open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar close parentheses squared space
    Taking square root, we get
    open vertical bar open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar minus open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar close vertical bar space less-than or slanted equal to space open vertical bar A with rightwards harpoon with barb upwards on top minus B with rightwards harpoon with barb upwards on top close vertical bar space less-than or slanted equal to space open vertical bar A with rightwards harpoon with barb upwards on top close vertical bar space plus space open vertical bar B with rightwards harpoon with barb upwards on top close vertical bar
    Hence, proved. 
            

    Question 171
    CBSEENPH11018060
    Wired Faculty App

    Explain how a vector space straight B with rightwards arrow on top can be subtracted from vector straight A with rightwards arrow on top ?
    Solution

    To subtract vector straight B with space rightwards arrow on top from vector space straight A with rightwards arrow on top comma first change the vector space straight B with rightwards arrow on top into negative stack straight B comma with rightwards arrow on top and then add it to vector straight A with rightwards arrow on top.

    Then, we have
                             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    This way, the vectors can be subtracted. 

    Question 172
    CBSEENPH11018061
    Wired Faculty App

    Two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top are represented by the two sides of a parallelogram drawn from point O. Show that the diagonal not passing through O is given by plus-or-minus left parenthesis straight A with rightwards arrow on top minus straight B with rightwards arrow on top right parenthesis.
    Solution

    Let two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top be represented by two adjacent sides of parallelogram OP and OQ drawn from O as shown in the figure. to complete the parallelogram OPSQ.

    That space is comma space OQ with rightwards harpoon with barb upwards on top space equals space straight B with rightwards harpoon with barb upwards on top space and space OP with rightwards harpoon with barb upwards on top space equals space straight A with rightwards harpoon with barb upwards on top Through space straight O space draw space straight a space vector space OM with rightwards harpoon with barb upwards on top space equal space and space opposite space to space space to space OQ with rightwards harpoon with barb upwards on top. therefore space OM with rightwards harpoon with barb upwards on top space equals space minus straight B with rightwards harpoon with barb upwards on top space OPNM space parallelogram space is space completed. space Now comma space according space to space the space parallelogram space vector space of space addition comma space ON with rightwards harpoon with barb upwards on top space equals space straight A with rightwards harpoon with barb upwards on top space plus space left parenthesis negative straight B with rightwards harpoon with barb upwards on top right parenthesis space equals space straight A with rightwards harpoon with barb upwards on top space minus space straight B with rightwards harpoon with barb upwards on top Since space the space sides space of space the space parallelogram space OPSQ space and space OPNM space are space equal space and space parallel comma space therefore comma space their space diagonals space are space also space equal space and space parallel. space straight i. straight e. comma space ON with rightwards harpoon with barb upwards on top space equals space QP with rightwards harpoon with barb upwards on top therefore space ON with rightwards harpoon with barb upwards on top space equals space straight A with rightwards harpoon with barb upwards on top space minus space straight B with rightwards harpoon with barb upwards on top and space NO with rightwards harpoon with barb upwards on top space equals space minus space left parenthesis straight A with rightwards harpoon with barb upwards on top space minus space straight B with rightwards harpoon with barb upwards on top right parenthesis Hence comma space the space diagonals space ON with rightwards harpoon with barb upwards on top space and space NO with rightwards harpoon with barb upwards on top space not space passing space through space straight O space represent space the space difference space of space two space vectors. space

    Question 173
    CBSEENPH11018062
    Wired Faculty App

    Two vectors of magnitude A and B are acting at angle straight theta. Find the manitude of the difference of two vectors.
    Solution

    Let two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top of magnitude A and B be acting at angle straight theta.
    We will find the magnitude of straight A with rightwards arrow on top minus straight B with rightwards arrow on top. 
    The vector straight A with rightwards arrow on top minus straight B with rightwards arrow on top can be written as left parenthesis straight A with rightwards arrow on top right parenthesis plus left parenthesis negative straight B with rightwards arrow on top right parenthesis comma.
    Thus, straight A with rightwards arrow on top minus straight B with rightwards arrow on top is equal to resultant of straight A with rightwards arrow on top space and space minus straight B with rightwards arrow on top.
     As the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is straight theta, therefore angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is left parenthesis 180 minus straight theta right parenthesis.
    Thus,
                open vertical bar straight A with rightwards arrow on top minus straight B with rightwards arrow on top close vertical bar space equals space open vertical bar straight A with rightwards arrow on top plus left parenthesis negative stack straight B right parenthesis with rightwards arrow on top close vertical bar
                          equals square root of straight A squared plus straight B squared plus 2 ABcos left parenthesis 180 minus straight theta right parenthesis end root
                          <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 174
    CBSEENPH11018063
    Wired Faculty App

    When the vector sum of the two forces is perpendicular to their vector difference?
    Solution
    The two diagonals of a parallelogram formed by two vectors represent the vector sum and vector difference of two vectors. 
    When the parallelogram is a square, the diagonals are perpendicular. Then, the two force vectors are represented by the adjacent sides of the square. Hence, the two force vectors which are equal in magnitude and perpendicular to each other have their vector sum perpendicular to their vector difference.
    Question 175
    CBSEENPH11018064
    Wired Faculty App

    Can three vectors not in one plane give a zero resultant? Can four vectors also do? 
    Solution

    The resultant of the two vectors lie in the same plane. Hence, three vectors in single plane cannot give the resultant zero. For the resultant of three vectors to be zero, resultant of two should be equal and opposite to the third. 
    Here, since the three vectors do not lie in the same plane, the resultant of the two cannot be in opposite direction of the third, hence resultant can not be zero. 

    Question 176
    CBSEENPH11018065
    Wired Faculty App

    A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?

    Solution
    For a quantity to be a vector, magnitude and direction are not the only sufficient conditions. Rotation of a body is specified by direction. But, it is not necessary that it is always a vector quantity. Finite rotation of a body about an axis is not a vector because finite rotation does not obey the laws of vector additions. Only infinitesimally small rotation is a vector quantity.
    Question 177
    CBSEENPH11018066
    Wired Faculty App

    Does the scalar quantity remain conserved in a process? Give example in support of your answer.

    Solution
    A scalar quantity need not remain conserved in the process.
    For example, during the adiabatic compression, temperature, which is a scalar quantity does not remain conserve while mass, a scalar quantity, remains conserved. 
    Question 178
    CBSEENPH11018067
    Wired Faculty App

    A particle in motion is under the action of four forces such that these forces are represented by the four sides of quadrilateral taken in order. What type of motion the particle undergoes? 
    Solution
    The resultant force of the four forces, represented by four sides of a quadrilateral, taken in order is zero. Thus, the particle does not experience any force and the particle moves with constant velocity. 
    Question 179
    CBSEENPH11018068
    Wired Faculty App

    How can you explain that the flying of a bird is an example of resultant of two vectors? 

    Solution

    Flying of a bird is an example of resultant of two vectors.
    i) When the bird flies, it strikes the air with wings A and B towards O along AO with rightwards arrow on top space and space stack BO. with rightwards arrow on top 
    ii) Action and reaction are opposite. Therefore, the air strikes the wings in opposite direction i.e. along space OA with rightwards arrow on top space and space OB with rightwards arrow on top.
    iii) The resultant of these vectors is in the upward direction which balances the weight of the bird.

    Question 180
    CBSEENPH11018069
    Wired Faculty App

    Can three vectors whose magnitudes are 10, 20 and 25 units lying in plane produce zero resultant?

    Solution
    If the magnitude of any vector lies between sum and difference of the magnitude of other two vectors, and if all the three vectors lay in the same plane, then their resultant can be zero.
    Question 181
    CBSEENPH11018070
    Wired Faculty App

    If straight A with rightwards arrow on top space and space straight B with rightwards arrow on top are orthogonal , then prove that 
                      open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space square root of straight A squared plus straight B squared end root
    Solution

    Magnitude of resultant of two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top acting at angle straight theta is given by, 
                  space space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space square root of straight A squared plus straight B squared plus 2 ABcosθ end root 
    Here, straight A with rightwards arrow on top space and space straight B with rightwards arrow on top are orthogonal, therefore straight theta equals 90 degree 
    Therefore, 
                open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space square root of straight A squared plus straight B squared plus 2 ABcos 90 degree end root 
                          equals square root of straight A squared plus straight B squared end root 

    Question 182
    CBSEENPH11018071
    Wired Faculty App

    The resultant of two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and perpendicular to stack straight A. with rightwards arrow on top Show that straight B with rightwards arrow on top space equals space square root of 2 open vertical bar straight A with rightwards arrow on top close vertical bar.
    Solution

    Let the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and space straight B with rightwards arrow on top be straight theta
    Resultant of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is A.
    Therefore,
                      straight A equals square root of straight A squared plus straight B squared plus 2 AB space cosθ end root
    rightwards double arrow            space straight A squared equals straight A squared plus straight B squared plus 2 AB space cosθ
    rightwards double arrow             straight B squared plus 2 ABcosθ space equals space 0
    rightwards double arrow             straight B squared equals negative 2 straight A space straight B space cosθ              ...(1)
    Also resultant is perpendicular to straight A with rightwards arrow on top.
    ∴              space tan 90 degree equals fraction numerator Bsinθ over denominator straight A plus Bcosθ end fraction
                    straight A plus straight B space cosθ space equals space 0
    rightwards double arrow                <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                ...(2)
    From (1) and (2)
                     straight B squared equals 2 straight A squared
    rightwards double arrow            space space space straight B equals left parenthesis square root of 2 right parenthesis straight A

    Question 183
    CBSEENPH11018072
    Wired Faculty App

    If space space straight R with rightwards arrow on top equals straight A with rightwards arrow on top plus straight B with rightwards arrow on top  and open vertical bar straight R with rightwards arrow on top close vertical bar space equals space open vertical bar straight A with rightwards arrow on top close vertical bar space equals space open vertical bar straight B with rightwards arrow on top close vertical bar then what is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top ?
    Solution

    Let straight theta be the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top. 
    ∴ Resultant is given by,
    open vertical bar straight R with rightwards arrow on top close vertical bar space equals space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space square root of straight A squared plus straight B squared plus 2 AB thin space cosθ end root 
    straight R squared space equals space straight A squared plus straight B squared plus 2 AB space cosθ 
    Hence, R = A = B
    Thus, straight A squared equals straight A squared plus straight A squared plus 2 straight A squared cosθ 
         
    rightwards double arrow  cosθ equals negative 1 half
    rightwards double arrow      straight theta equals 120 degree, is the required angle between them. 

    Question 184
    CBSEENPH11018073
    Wired Faculty App

    At what angle do the two vectors of magnitude A + B and A - B act so that their resultant is <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>?
    Solution

    The magnitude of the resultant of two vectors space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top acting at angle straight theta is given by, 
                          space space straight R equals square root of straight P squared plus straight Q squared plus 2 PQcosθ end root
    Here,       P = A + B,        Q = A - B
    R e s u l t a n t space comma space straight R space equals space square root of 3 straight A squared plus straight B squared end root therefore space P Q space equals space straight A squared space minus space straight B squared P u t t i n g space t h e space v a l u e s space o f space straight P comma space straight Q comma space straight R space a n d space P Q space i n space a b o v e space e q u a t i o n comma space square root of 3 straight A squared plus straight B squared end root space equals space square root of left parenthesis straight A plus straight B right parenthesis squared plus left parenthesis straight A minus straight B right parenthesis squared plus 2 left parenthesis straight A squared minus straight B squared right parenthesis space c o s theta end root T a k i n g space s q u a r e space o n space b o t h space s i d e s comma space space space space space space space space 3 straight A squared space plus space straight B squared space equals space left parenthesis straight A plus straight B right parenthesis squared space plus space left parenthesis straight A minus straight B right parenthesis squared space plus 2 space left parenthesis straight A squared minus straight B squared right parenthesis c o s theta rightwards double arrow space space straight A squared space minus space straight B squared space equals 2 space left parenthesis straight A squared minus straight B squared right parenthesis c o s theta T h a t space i s comma space c o s space straight theta space equals space bevelled 1 half rightwards double arrow space straight theta space equals space 60 to the power of 0
                                      
                        

    Question 185
    CBSEENPH11018074
    Wired Faculty App

    The magnitude of the resultant of stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top is P; If direction of stack straight F subscript 2 with rightwards arrow on top is reversed, the resultant is Q.
    Show that
                  space straight P squared plus straight Q squared space equals space 2 left parenthesis straight F subscript 1 squared plus straight F subscript 2 squared right parenthesis
    Solution

    Let straight theta be the  angle between two vectors space stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top.
    ∴        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>            ...(1)
    If direction of stack straight F subscript 2 with rightwards arrow on top is reversed then, then angle between stack straight F subscript 1 with rightwards arrow on top space and space minus stack straight F subscript 2 with rightwards arrow on top becomes 180 degree minus straight theta.
    ∴           straight Q squared equals straight F subscript 1 squared plus straight F subscript 2 squared plus 2 straight F subscript 1 straight F subscript 2 cos left parenthesis 180 degree minus straight theta right parenthesis.
                     equals straight F subscript 1 squared plus straight F subscript 2 squared minus 2 straight F subscript 1 straight F subscript 2 cosθ       ...(2)
    Adding (1) and (2), we get
                  space straight P squared plus straight Q squared equals 2 left parenthesis straight F subscript 1 squared plus straight F subscript 2 squared right parenthesis

    Question 186
    CBSEENPH11018075
    Wired Faculty App

    Maximum and minimum magnitudes of the resultant of two vectors of magnitude A and B are in the ratio 4:3. Find the relation between A and B.

    Solution
    The maximum magnitude of the resultant of two vectors = A + B.
    Minimum magnitude of the resultant of two vectors = A – B
    Here,       fraction numerator straight A plus straight B over denominator straight A minus straight B end fraction equals 4 over 3
    rightwards double arrow         3 straight A plus 3 straight B equals 4 straight A minus 4 straight B
     
    rightwards double arrow                  straight A equals 7 straight B
    Question 187
    CBSEENPH11018207
    Wired Faculty App

    A motor boat is racing towards north at 25 km/hr and the water current in the river is 10 km/hr in the direction 60° east of south. Find the resultant velocity of the boat.

    Solution

    Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the velocities of boat and river water respectively.
    Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the resultant velocity of boat.
    Boat heads towards north and water current is in the direction <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> east of south.
    Therefore, the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is 120 degree.
    Now using parllelogram law of vector addition, the magnitude of resultant velocity <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is ,
       open vertical bar straight v with rightwards arrow on top close vertical bar space equals space square root of straight v subscript straight b superscript 2 plus straight v subscript straight r superscript 2 plus 2 straight v subscript straight b straight v subscript straight r cosθ end root
           equals square root of left parenthesis 25 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus 2 cross times 25 cross times 10 space cos space 120 end root
           equals square root of 625 plus 100 minus 250 end root space equals space square root of 475 space equals space 21.8 space km divided by hr 
    Let the resultant velocity vector subtend an angle straight alpha with the north.
    Then,
               tan space straight alpha space equals space fraction numerator straight v subscript straight r space sinθ over denominator straight v subscript straight b plus straight v subscript straight r cosθ end fraction space space space space space space space space equals fraction numerator 10 space sin 120 degree over denominator 25 plus 10 space cos space 120 degree end fraction
                      equals fraction numerator 5 square root of 3 over denominator 20 end fraction
    This implies,
                         straight alpha equals 23.4 degree

    Question 188
    CBSEENPH11018219
    Wired Faculty App

    Two forces have magnitude in the ratio 5:12. If the angle between them is 90°, they give the resultant of magnitude 26N. Find the magnitude of resultant force if the angle between them is changed to 120°.

    Solution
    Let magnitude of two forces be F1and F2.
    Here,      straight F subscript 1 over straight F subscript 2 equals 5 over 12 
    rightwards double arrow      space space space straight F subscript 2 equals 12 over 5 straight F subscript 1 equals 2.4 straight F subscript 1     ...(1)
    The resultant of stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top is 26N when angle between them is .
    Therefore,
                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                          equals straight F subscript 1 squared plus straight F subscript 2 squared              ...(2)
    Substituting, straight F subscript 2 equals 2.4 space straight F subscript 1 in equation (2),
          left parenthesis 26 right parenthesis squared equals straight F subscript 1 squared plus straight F subscript 2 squared equals straight F subscript 1 squared plus left parenthesis 2.4 right parenthesis squared straight F subscript 1 squared space space space space space space space space space equals 6.76 straight F subscript 1 squared
     
         left parenthesis 26 right parenthesis squared equals left parenthesis 2.6 right parenthesis squared straight F subscript 1 squared
           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴  straight F subscript 2 equals 2.4 space space space space space straight F subscript 1 equals 24 straight N
    Let R be the magnitude of the resultant when the angle between the two forces is 120 degree comma
    ∴                    straight R squared equals straight F subscript 1 squared plus straight F subscript 2 squared plus 2 straight F subscript 1 straight F subscript 2 space cos space 120 degree
                              equals straight F subscript 1 squared plus straight F subscript 2 squared minus straight F subscript 1 straight F subscript 2
                       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    i.e.,                   straight R equals 20.88, is the magnitude of the resultant between the vectors. 
    Question 189
    CBSEENPH11018220
    Wired Faculty App

    A particle is moving in a circle of radius r centred at O with constant speed v. Find the magnitude of change in velocity moving from A to B left parenthesis angle AOB space equals space 40 degree right parenthesis.
    Solution

    Let stack straight v subscript 1 with rightwards arrow on top space and space stack straight v subscript 2 with rightwards arrow on top be the velocities of particle at A and B.
    Since the speed of the particle (straight v with rightwards harpoon with barb upwards on top) is constant.
    Therefore,
              space space open vertical bar stack straight v subscript 1 with rightwards arrow on top close vertical bar space equals space open vertical bar stack straight v subscript 2 with rightwards arrow on top close vertical bar space equals space straight v
    As angle space space angle AOB space equals space 40 degree comma therefore angle between the velocity vectors is also <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Now,      Δv space equals space open vertical bar stack straight v subscript 2 with rightwards arrow on top minus stack straight v subscript 1 with rightwards arrow on top close vertical bar
                      equals square root of straight v subscript 2 superscript 2 plus straight v subscript 1 superscript 2 minus 2 straight v subscript 2 straight v subscript 1 cos 40 to the power of degree end root 
                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                        = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                        equals square root of 4 straight v squared sin squared 20 degree end root equals space 2 vsin 20 degree
    This is the required change in magnitude of the velocity on moving from A to B. 
                        
                           

    Question 190
    CBSEENPH11018225
    Wired Faculty App

    Can <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> ?
    Solution

    Let straight theta be the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
    Now,             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow       straight A squared plus straight B squared plus 2 AB space cosθ space equals space straight A squared plus straight B squared minus 2 AB space cosθ
    rightwards double arrow                 4 AB space cosθ space equals space 0
    rightwards double arrow                        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow                             straight theta space equals space 90 degree
    Thus,space space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar  can be equal to open vertical bar straight A with rightwards arrow on top minus straight B with rightwards arrow on top close vertical bar if angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is 90 degree. 

    Question 191
    CBSEENPH11018228
    Wired Faculty App

    If straight theta is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top, then show that open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar space equals space 2 sin straight theta over 2.
    Solution

    On evaluating open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar 
    Therefore,
     open vertical bar straight A with hat on top minus straight B with hat on top close vertical bar space equals space square root of open vertical bar straight A with hat on top close vertical bar squared plus open vertical bar straight B with hat on top close vertical bar squared minus 2 open vertical bar straight A with hat on top close vertical bar open vertical bar straight B with hat on top close vertical bar cosθ. end root 
              equals square root of 1 plus 1 minus 2 space cosθ end root 
              equals square root of 2 left parenthesis 1 minus cosθ right parenthesis end root equals space square root of 2.2 sin squared straight theta over 2 end root 
              equals 2 sin straight theta over 2      

    Question 192
    CBSEENPH11018230
    Wired Faculty App

    Show that if <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, then angle between the two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is greater than 90 degree.
    Solution

    Let straight theta be the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
    Here, we have
                space space open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space space less than space space open vertical bar straight A with rightwards arrow on top minus straight B with rightwards arrow on top close vertical bar
    ∴           open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar squared space less than space open vertical bar straight A with rightwards arrow on top minus straight B with rightwards arrow on top close vertical bar squared
    That is,
                 straight A squared plus straight B squared plus 2 ABcosθ less than straight A squared plus straight B squared minus 2 ABcosθ
    rightwards double arrow            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow              cos space straight theta space space less than space 0
    Since cos space straight theta is negative, therefore
                 space space 90 degree less than straight theta less than 180 degree, is the angle between the two vectors. 

    Question 193
    CBSEENPH11018232
    Wired Faculty App

    Two vectors acting in opposite direction have resultant 7 units and if they act at angle 90° to each other, the resultant is 13 units. Calculate the magnitude of the two vectors.

    Solution

    Let A and B be the magnitude of two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top and let A>B.
    When straight A with rightwards arrow on top space and space straight B with rightwards arrow on top act in opposite direction, the magnitude of resultant is A - B
    Now,
                   A - B  = 7 [Given]              ...(1)
    When straight A with rightwards arrow on top space and space straight B with rightwards arrow on top act perpendicular to each other, then their resultant is,
                             square root of straight A squared plus straight B squared end root space equals space 13
    rightwards double arrow              straight A squared plus straight B squared equals 169                   ...(2)
    Now,             left parenthesis straight A squared plus straight B squared right parenthesis minus left parenthesis straight A minus straight B right parenthesis squared equals 2 thin space straight A thin space straight B
    ∴                    2 AB space equals space 169 minus left parenthesis 7 right parenthesis squared equals 120
    and            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                 equals 169 plus 120 equals 289
    or                 space space straight A plus straight B space equals space 17                    ...(3)
    On solving equation (1) and (3), we get
                    A = 12 and B = 5

    Question 194
    CBSEENPH11018234
    Wired Faculty App

    What do you mean by resolution of a vector? What are the resolved parts called? Discuss a method to resolve a vector into two vectors in given direction?

    Solution
    The process of splitting up the vector into two or more vectors so that combined effect of the resultant vector is same as that of the given vectors. 
    The vectors into which a given vector is split is called component of vector.
    The components of a vector are also vectors. 

    Let us resolve vector straight R with rightwards harpoon with barb upwards on top represented by PQ with rightwards harpoon with barb upwards on top into two vectors in the direction of A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top.
    Draw a line from P parallel to vector straight A with rightwards harpoon with barb upwards on top and another line from Q, paralle to vector B.
    We can see that the two lines intersect at point S. 
    Here, PS with rightwards harpoon with barb upwards on top space parallel to A with rightwards harpoon with barb upwards on top space a n d space stack S Q with rightwards harpoon with barb upwards on top space parallel to B with rightwards harpoon with barb upwards on top 
    So, we can write, 
    PS with rightwards harpoon with barb upwards on top space equals space lambda A with rightwards harpoon with barb upwards on top space a n d space stack S Q with rightwards harpoon with barb upwards on top space equals beta B with rightwards harpoon with barb upwards on top 
    Therefore, using triangle law of vector addition to vectors PQ with rightwards harpoon with barb upwards on top comma space stack P S with rightwards harpoon with barb upwards on top space a n d space stack S Q with rightwards harpoon with barb upwards on top, we get 
    PQ with rightwards harpoon with barb upwards on top space equals space stack P S with rightwards harpoon with barb upwards on top space plus space stack S Q with rightwards harpoon with barb upwards on top space space space space space space equals space lambda A with rightwards harpoon with barb upwards on top space plus space beta B with rightwards harpoon with barb upwards on top 
    Therefore, these are the resolved components of vector straight R with rightwards harpoon with barb upwards on top in the direction of straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top.  

    Question 195
    CBSEENPH11018237
    Wired Faculty App

    Show that there is one and only one set of resolved components in a particular direction.

    Solution

    Let a given vector straight R with rightwards arrow on top be resolved in the direction of two non-parallel vectors straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top. 
    Let there be two sets of resolved components of resultant straight R with rightwards harpoon with barb upwards on top in the direction of straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top. 
    That is, left parenthesis straight lambda subscript 1 straight A with rightwards harpoon with barb upwards on top comma space straight beta subscript 1 straight B with rightwards harpoon with barb upwards on top space right parenthesis space and space left parenthesis straight lambda subscript 2 straight A with rightwards harpoon with barb upwards on top comma space straight beta subscript 2 straight B with rightwards harpoon with barb upwards on top space right parenthesis 
    Since, straight lambda subscript 1 straight A with rightwards harpoon with barb upwards on top space and space straight beta subscript 1 straight B with rightwards harpoon with barb upwards on top are components of straight R with rightwards harpoon with barb upwards on top, 
    Therefore,
    straight R with rightwards harpoon with barb upwards on top space equals space lambda subscript 1 A with rightwards harpoon with barb upwards on top space plus space beta subscript 1 B with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis 
    And also,
    straight R with rightwards harpoon with barb upwards on top space equals space lambda subscript 2 A with rightwards harpoon with barb upwards on top space plus space beta subscript 2 stack B space with rightwards harpoon with barb upwards on top space space space space space space space space space space... left parenthesis 2 right parenthesis
    So, from equations (1) and (2), we have, 
    space space space space space space space straight lambda subscript 1 straight A with rightwards harpoon with barb upwards on top space plus space straight beta subscript 1 straight B with rightwards harpoon with barb upwards on top space equals space straight lambda subscript 2 straight A with rightwards harpoon with barb upwards on top space plus space straight beta subscript 2 straight B with rightwards harpoon with barb upwards on top rightwards double arrow space space space left parenthesis space straight lambda subscript 1 minus space straight lambda subscript 2 right parenthesis straight A with rightwards harpoon with barb upwards space on top space equals space left parenthesis space straight beta subscript 2 space minus space straight beta subscript 1 right parenthesis straight B with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
    Since, vector A and B are different non-parallel vectors, so equation (3) is possible only and only if, 
    space space space space space space space space space space space straight lambda subscript 1 space minus space straight lambda subscript 2 space equals space straight beta subscript 2 space minus space straight beta subscript 1 space equals space 0 rightwards double arrow space space space space space space space space straight lambda subscript 1 space equals space space straight lambda subscript 2 space and space space straight beta subscript 2 space equals space straight beta subscript 1
    Hence, both the sets of components are identical. There is one and only one set of resolved components in a particular direction.

    Question 196
    CBSEENPH11018238
    Wired Faculty App

    What are rectangular components of a vector? How will you resolve the rectangular components of a vector in a plane?

    Solution
    When the resolved components of a vector are mutually perpendicular to each other, they are called as rectangular components. 
    Let us resolve the rectangular components of a vector. 
    The resultant of the vector is represented by OP with rightwards harpoon with barb upwards on top in XY plane. 
     
    Now, from point P, draw a perpendicular PM on X-axis and PN on Y-axis. 
    Now, using the triangle law of vectors to increment OMP comma spacewe have 
    straight R with rightwards harpoon with barb upwards on top space equals space stack O P with rightwards harpoon with barb upwards on top space equals space stack O M with rightwards harpoon with barb upwards on top space plus space stack M P with rightwards harpoon with barb upwards on top space space space space space equals space stack O M with rightwards harpoon with barb upwards on top space plus space stack O N with rightwards harpoon with barb upwards on top 

    Here, vectors OM and ON are mutually perpendicular to each other. Therefore, the  vectors OM and ON are called rectangular components of vector R.

    Now, using appropriate notations, X and Y components are represented by straight R with rightwards harpoon with barb upwards on top subscript x space a n d space R with rightwards harpoon with barb upwards on top subscript y, respectively.  

    The internationally accepted notations for unit vectors along X-axis and Y-axis are straight i with hat on top space a n d space j with hat on top respectively. Thus, 
    straight R with rightwards harpoon with barb upwards on top space equals space R with rightwards harpoon with barb upwards on top subscript x space plus space R with rightwards harpoon with barb upwards on top subscript y space equals space R with rightwards harpoon with barb upwards on top subscript x stack i space with hat on top space plus space R with rightwards harpoon with barb upwards on top subscript y j with hat on top space 
    If the vector straight R with rightwards harpoon with barb upwards on top has magnitude R, and makes an angle straight theta with positive direction of X-axis. The from increment OMP comma 
    OM space equals space straight R subscript straight x space equals space straight R space cos space straight theta MP space equals space straight R subscript straight y space equals space straight R space sin space straight theta Thus comma space straight R with rightwards harpoon with barb upwards on top space equals space straight R subscript straight x straight i with hat on top space plus space straight R subscript straight y straight j with hat on top space equals space straight R space cos space straight theta space straight i with hat on top space plus space straight R space sinθ stack space straight j with hat on top

    This is how we resolve the rectangular components of a vector in a plane. 


     

    Question 197
    CBSEENPH11018239
    Wired Faculty App

    Resolve the rectangular components of a vector in three dimensions?

    Solution

    Let us resolve the rectangular components of a vector straight R with rightwards arrow on top represented by <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> in three dimensions.
                    
    From point P, draw a perpendicular PM on XY plane and from M draw a perpendicular PM on XY plane and from M draw perpendicular ML on Y axis and MK on X axis. '
    Using triangle law to increment OKM comma
    OM with rightwards harpoon with barb upwards on top space equals space OK with rightwards harpoon with barb upwards on top space plus KM with rightwards harpoon with barb upwards on top space space space space space space space space space... thin space left parenthesis 1 right parenthesis Using space triangle space law space to space increment OMP comma OP with rightwards harpoon with barb upwards on top space equals space OM with rightwards harpoon with barb upwards on top space plus space MP with rightwards harpoon with barb upwards on top space space space space space space space space space space space... space left parenthesis 2 right parenthesis space From space equation space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space stack OP space with rightwards harpoon with barb upwards on top space equals space OK with rightwards harpoon with barb upwards on top space plus KM with rightwards harpoon with barb upwards on top space space plus MP with rightwards harpoon with barb upwards on top Here comma space vector space OK with rightwards harpoon with barb upwards on top space comma space KM with rightwards harpoon with barb upwards on top space and space MP with rightwards harpoon with barb upwards on top space are space mutually space perpendicular space to space each space other. space Therefore comma space vectors space OK with rightwards harpoon with barb upwards on top space comma space KM with rightwards harpoon with barb upwards on top space and space MP with rightwards harpoon with barb upwards on top space are space called space rectangular space components space of space vector space straight R with rightwards harpoon with barb upwards on top. space Now comma space straight X comma space straight Y space and space straight Z space components space are space represented space by space straight R with rightwards harpoon with barb upwards on top subscript straight x comma space straight R with rightwards harpoon with barb upwards on top subscript straight y space and space straight R with rightwards harpoon with barb upwards on top subscript straight z space respectively. Unit space vectors space along space straight X comma space straight Y space and space straight Z space axis space are space straight i with hat on top comma space straight j with hat on top space and space straight k with hat on top. So comma space resultant space is space given space by comma space straight R with rightwards harpoon with barb upwards on top space equals space stack straight R subscript straight x with rightwards harpoon with barb upwards on top space plus space stack straight R subscript straight y with rightwards harpoon with barb upwards on top space plus space space stack straight R subscript straight z with rightwards harpoon with barb upwards on top space If space vector space straight R with rightwards harpoon with barb upwards on top space space has space magnitude space straight R space and space makes angle space straight theta subscript straight x comma space straight theta subscript straight y space and space straight theta subscript straight z comma space with space positive space direction space of space straight X comma space straight Y space and space straight Z space axis space respectively comma space then OK space equals space straight R subscript straight x space equals space straight R space space cos space straight theta subscript straight x KM space equals space straight R subscript straight y space equals space straight R space sin space space straight theta subscript straight y MP space equals space space straight R subscript straight z space equals space straight R space sin space space straight theta subscript straight z Thus comma space straight R with rightwards harpoon with barb upwards on top space equals space straight R subscript straight x straight i with hat on top space plus space straight R subscript straight y straight j with hat on top space plus space straight R subscript straight z straight k with hat on top space space space space space space space space equals space space space Rcos space straight theta subscript straight x space straight i with hat on top plus straight R space sin space space straight theta subscript straight y straight j with hat on top space plus straight R space sin space space straight theta subscript straight z straight k with hat on top
    The above expression gives us the rectangular components of a vector in three dimension. 
                   

    Question 198
    CBSEENPH11018240
    Wired Faculty App

    Define direction cosines of a vector and show that sum of the square of the direction cosines is equal to unity.

    Solution

    The cosine of angles made by a vector with the positive directions of X, Y and Z axis is called the direction cosines of a vector. 
    Let us define a vector straight A with rightwards harpoon with barb upwards on top space equals space A subscript x i with hat on top plus A subscript y j with hat on top plus A subscript z k with hat on top, making angle straight theta subscript straight x comma space straight theta subscript straight y space and space straight theta subscript straight z respectively with the positive directions of X, Y and Z axis. 
    The direction cosines of vector are, 
    cos space theta subscript x space equals space A subscript x over A semicolon space cos space theta subscript y space equals space A subscript y over A space semicolon space cos theta subscript z space equals space A subscript z over A space  
    Now, square of magnitude of vector are, 
    straight A squared space equals space straight A subscript straight x squared space plus space straight A subscript straight y squared space plus space straight A subscript straight z squared space space space space space space equals space left parenthesis Acosθ subscript straight x right parenthesis squared space plus space left parenthesis Acosθ subscript straight y right parenthesis squared space plus space left parenthesis Acosθ subscript straight z right parenthesis squared space space space space space space equals space straight A squared left parenthesis cos squared straight theta subscript straight x space plus space cos squared straight theta subscript straight y space plus space cos squared straight theta subscript straight z right parenthesis space rightwards double arrow space cos squared space straight theta subscript straight x space plus space cos squared straight theta subscript straight y space plus space cos squared straight theta subscript straight z space equals space 1 space space space space space space space space 

    Question 199
    CBSEENPH11018241
    Wired Faculty App

    What are the rectangular components of a vector whose magnitude is 12 units and it subtends an angle of 60° with X-axis?

    Solution

    Given that,
       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                     

    ∴            straight A subscript straight x space equals space straight A space cosθ space equals space 12 space cos space 60 degree space equals space 6
    and        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 200
    CBSEENPH11018242
    Wired Faculty App

    Explain that walk of a man is the example of resolution of vector.

    Solution
    When a man walks, he pushes the ground with his feet backward at angle θ with the horizontal and man gets equal and opposite reaction space straight R with rightwards arrow on top. 
    Vector space straight R with rightwards arrow on top. can be resolved into two components. 
                  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  and straight V equals space straight R space sin space straight theta
    So,
    Horizontal component of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> helps the man to walk forward. 
    Question 201
    CBSEENPH11018243
    Wired Faculty App

    Define position vector and write the position vector of point straight P left right arrow left parenthesis straight x comma space straight y comma space straight z right parenthesis.
    Solution

    A vector drawn from reference point to a given point is known as position vector of that point. 
    Position vector of point P is space OP with rightwards arrow on top. 
    Here,     straight P left right arrow left parenthesis straight x comma space straight y comma space straight z right parenthesis
    Therefore, 
    OP with rightwards arrow on top space equals space left parenthesis straight x minus 0 right parenthesis straight i with hat on top space plus space left parenthesis straight y minus 0 right parenthesis straight j with hat on top plus left parenthesis straight z minus 0 right parenthesis straight k with hat on top 
        space equals straight x straight i with hat on top plus straight y straight j with hat on top plus straight z straight k with hat on top               

    Question 202
    CBSEENPH11018244
    Wired Faculty App

    Can a vector have magnitude smaller than its rectangular component?

    Solution
    No, a vector cannot have magnitude smaller than its rectangular components because rectangular components of vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> are space space straight A space cosθ space and space straight A space sinθ.
    And,
              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>,
                space sin space straight theta space less or equal than space 1
    ∴   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  and 
           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 203
    CBSEENPH11018245
    Wired Faculty App

    Find Z component of resultant of vectors,

    straight A with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight j with hat on top plus space straight k with hat on top and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    Solution

    Resultant of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top
         straight R with rightwards arrow on top space equals space straight A with rightwards arrow on top plus straight B with rightwards arrow on top space equals space 4 straight i with hat on top space minus space 5 straight j with hat on top space plus space 0 straight k with hat on top
    Coefficient of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is the Z component of vector.
    Therefore Z component of resultant of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is zero.

    Question 204
    CBSEENPH11018247
    Wired Faculty App

    If one of the rectangular components of a velocity of 36m/s has magnitude 26 m/s, then what is the magnitude of other rectangular component of velocity?

    Solution

    If straight v subscript 1 space and space straight v subscript 2 are the magnitudes of rectangular components of velocity vector of magnitude v, then
                       straight v squared equals straight v subscript 1 squared plus straight v subscript 2 squared
    Given,
                  straight v equals 36 space straight m divided by straight s  and      straight v subscript 1 equals 26 space straight m divided by straight s
    ∴              straight v subscript 2 squared space equals space straight v squared minus straight v subscript 1 squared space space space space space space equals space left parenthesis 36 right parenthesis squared minus left parenthesis 26 right parenthesis squared 
    rightwards double arrow           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    That is,
                   straight v subscript 2 equals 24.9 space straight m divided by straight s, is the magnitude of other rectangular component of velocity. 

    Question 205
    CBSEENPH11018249
    Wired Faculty App

    A force inclined at angle 37° with the horizontal has vertical component 42N. Find the magnitude of force and magnitude of its horizontal component. (Take sin 37° = 0·6 and cos 37° = 0·8)

    Solution
    Let F be the magnitude of force.
    If force vector subtends an angle θ with the horizontal, then horizontal and vertical components of force respectively, 
                  space space space straight F subscript straight x equals space Fcosθ     and     straight F subscript straight y space equals space Fsinθ
    Here,
    Angle with which the force is inclined to the horizontal, straight theta equals 37 degree
    Vertical component of the force, straight F subscript straight y equals 42 space straight N 
    So, magnitude of force is found using, 
    ∴               42 space equals space straight F space sin 37 degree space equals space straight F space cross times space 0.6
    Force, straight F space equals space 70 straight N
    Horizontal component, straight F subscript straight x equals Fcosθ space equals space 70 cross times 0.8 space equals space 56 straight N            
           
    Question 206
    CBSEENPH11018266
    Wired Faculty App

    Two billiard balls are rolling on a flat table. One has velocity components straight v subscript straight x 1 end subscript equals 1 straight m divided by straight s semicolon space straight v subscript straight y 1 end subscript space equals space square root of 3 space straight m divided by straight s  and other has velocity components  straight v subscript straight x 2 end subscript equals 2 space straight m divided by straight s semicolon space space space straight v subscript straight y subscript 2 end subscript space equals space 2 space straight m divided by straight s. If both the balls start moving from the same point, then what is the angle between their paths?
    Solution

    Let straight theta subscript 1  and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the angles that the two balls make with X-axis.
    Given, the velocity components of first ball are, 
    space straight v subscript straight x 1 end subscript space equals space 1 space straight m divided by straight s 
     straight v subscript straight y 1 end subscript space equals space square root of 3 space space straight m divided by straight s
    Therefore, angle is given by, 
                  tanθ subscript 1 space equals space straight v subscript straight y 1 end subscript over straight v subscript straight x 1 end subscript equals fraction numerator square root of 3 over denominator 1 end fraction equals square root of 3 
    rightwards double arrow               straight theta subscript 1 space equals space 60 degree
    The velocity components of second ball are,
     straight v subscript straight x 2 end subscript space equals space 2 space straight m divided by straight s,
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴ angle is given by,  
                       tanθ subscript 2 equals straight v subscript straight y 2 end subscript over straight v subscript straight x 2 end subscript equals 2 over 2 equals 1
                           straight theta subscript 2 space equals space 45 degree
    The angle between the paths of two ball is equal to the angle between their velocity vectors.
    ∴  The angle between the paths of two balls is,
                      straight theta subscript 1 minus straight theta subscript 2 equals 60 degree minus 45 degree space equals space 15 degree.

    Question 207
    CBSEENPH11018270
    Wired Faculty App

    On the airport, the sunshines just over the head. When the aeroplane takes off at an angle of 23° to horizontal, the shadow moves along the track with velocity 240km/hr. Find the velocity with which the aeroplane moves away from the earth. (Take space tan space 23 degree space equals space 5 divided by 12)

    Solution

    The vertical component ofvelocity of the airplane will give us the velocity with which the airplane moves away from the Earth. 
    As the aeroplane takes off at angle space 23 degree with horizontal, 
     ∴           tan space 23 space equals space straight v subscript vert over straight v subscript hori 
    ∴              space space straight v subscript vert space equals space straight v subscript hori space tan 23 degree space space space space space space space space space space equals space 100 space km divided by hr  

    Question 208
    CBSEENPH11018273
    Wired Faculty App

    Determine the direction cosines of a vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution

    The vector is given by,
                             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    The X, Y and Z components of the vector are, 
    ∴  straight A subscript straight x equals negative 2,space straight A subscript straight y equals 3,<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                   

    So, the resultant of the vector is given by,
      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>     

    Thus, 
    Direction cosine is, 
    space space cosθ subscript straight x equals fraction numerator negative 2 over denominator 7 end fraction comma space space cosθ subscript straight x equals 3 over 7 comma space space cosθ subscript straight x equals 6 over 7

    Question 209
    CBSEENPH11018274
    Wired Faculty App

    Find the resultant of forces acting at a point:
    (i)      200 square root of 2 space straight N  along NE
    (ii)     space 500 square root of 2 space straight N  along NW
    (iii)      800 N along south.

    Solution

               
    Given that, 
    Force acting along NE = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    straight F with rightwards harpoon with barb upwards on top subscript 2 space equals space 500 space square root of 2 space comma space along space NW stack straight F subscript 3 with rightwards harpoon with barb upwards on top space equals space 800 space straight N space along space South Now comma space resolving space the space rectangular space components of space straight F with rightwards harpoon with barb upwards on top subscript 1 comma space straight F with rightwards harpoon with barb upwards on top subscript 2 comma space stack straight F subscript 3 with rightwards harpoon with barb upwards on top space along space straight x space and space straight y space axis space is space given space by comma space straight F subscript 1 straight x end subscript space equals space 200 square root of 2 space cos space 45 to the power of straight o space equals space 200 space straight F subscript 1 straight y end subscript space equals space 200 square root of 2 sin space 45 to the power of straight o space equals space 200 straight F subscript 2 straight x end subscript space equals space minus space 500 space square root of 2 space cos space 45 to the power of straight o space equals space minus space 500 space straight F subscript 2 straight y end subscript space equals space 500 space square root of 2 space sin space 45 to the power of straight o space equals space 500 space straight F subscript 3 straight x end subscript space equals space 800 space cos space 90 to the power of 0 space equals space 0 straight F subscript 3 straight y end subscript space equals space minus 800 space sin space 90 to the power of straight o space equals space minus 800 space Therefore comma space straight F subscript straight x space equals space straight F subscript 1 straight x end subscript plus straight F subscript 2 straight x end subscript space plus straight F subscript 3 straight x end subscript space space space space space equals space 200 space minus space 500 space plus space 0 space space space space space space equals space minus 300 space and space straight F subscript straight y space equals space straight F subscript 1 straight y end subscript space plus space straight F subscript 2 straight y end subscript space plus space straight F subscript 3 straight y end subscript space space space space space space space space space space space space space equals space 200 space plus space 500 minus 800 space space space space space space space space space space space space equals space minus 100 Therefore comma space Resultant space Force comma space straight F with rightwards harpoon with barb upwards on top space equals space minus space 300 space straight i with hat on top space minus space 100 space straight j with hat on top space Magnitude space of space force comma space open vertical bar straight F with rightwards harpoon with barb upwards on top close vertical bar space equals square root of 300 squared plus 100 squared end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 100 square root of 10 space straight N


    Question 211
    CBSEENPH11018277
    Wired Faculty App

    Define zero vector. What is its direction? Give two examples of zero vector.

    Solution

    When the magnitude of a vector is zero, it is known as a zero vector. Zero vector has an arbitrary direction. 
    Examples: (i) Position vector of origin is zero vector.
    (ii) If a particle is at rest then displacement of the particle is zero vector. 
    (iii) Acceleration of uniform motion is zero vector.

    Question 212
    CBSEENPH11018278
    Wired Faculty App

    Define dot product. Show that dot product of two vectors follows commutative law.

    Solution

    Dot product of two vectors is defined as the product of their magnitudes and cosine of the angle between them. 
                    straight A with rightwards arrow on top. straight B with rightwards arrow on top space equals space AB space cos space straight theta 
    Now         straight A with rightwards arrow on top. straight B with rightwards arrow on top space equals space AB space cosθ 
                          equals BAcosθ space equals space straight B with rightwards arrow on top. straight A with rightwards arrow on top 
    Hence, dot product of two vectors is commutative in nature. 

    Question 213
    CBSEENPH11018280
    Wired Faculty App

    What are the characteristics of dot product?
    Solution

    The characteristics of dot product are: 
    (i)  Dot product of two vectors is commutative.
    Mathematically,   straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top times straight A with rightwards arrow on top 
    (ii)  Dot product is distributive. 
    Mathematically,  straight A with rightwards arrow on top times left parenthesis straight B with rightwards arrow on top plus straight C with rightwards arrow on top right parenthesis space equals space straight A with rightwards arrow on top times space straight B with rightwards arrow on top space plus space straight A with rightwards arrow on top times straight C with rightwards arrow on top 
    (iii) Dot product of two perpendiculars vectors is zero.
    Mathematically  straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space 0            if   straight A with rightwards arrow on top space perpendicular space straight B with rightwards arrow on top 
    (iv) Dot product of vector with itself is equal to square of magnitude of vector. 
    Mathematically,    straight A with rightwards arrow on top times straight A with rightwards arrow on top space equals space straight A squared

    Question 214
    CBSEENPH11018281
    Wired Faculty App

    What is the geometric meaning of dot product?

    Solution

    The dot product of two vectors is given by, 
                straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space AB space cosθ space equals space straight A left parenthesis straight B space cosθ right parenthesis 
    Dot product is a scalar quantity. 

    Bcosstraight theta is the magnitude of component straight B with rightwards harpoon with barb upwards on top in the direction of A with rightwards harpoon with barb upwards on top 
    straight A with rightwards harpoon with barb upwards on top. B with rightwards harpoon with barb upwards on top space equals space left parenthesis M a g n i t u d e space o f space v e c t o r space A with rightwards harpoon with barb upwards on top right parenthesis space left parenthesis m a g n i t u d e space o f space c o m p o n e n t space o f space B with rightwards harpoon with barb upwards on top space i n space t h e space d i r e c t i o n space o f space A with rightwards harpoon with barb upwards on top space right parenthesis
    Similarly, 
    <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre> 
           = (Magnitude of vector straight B with rightwards harpoon with barb upwards on top) (magnitude of component of straight A with rightwards harpoon with barb upwards on top in the direction of straight B with rightwards harpoon with barb upwards on top space right parenthesis
    )

    Question 215
    CBSEENPH11018284
    Wired Faculty App

    State and prove that dot product is distributive.

    Solution
    Dot product is distributive. Dot product of a given vector with a sum of number of other vectors is equal to the sum of the dot product of given vector with the other vectors separately. 
    That is, straight A with rightwards harpoon with barb upwards on top. space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top plus... right parenthesis space equals space A with rightwards harpoon with barb upwards on top. B with rightwards harpoon with barb upwards on top space plus space A with rightwards harpoon with barb upwards on top. C with rightwards harpoon with barb upwards on top 
    Proof: Let us consider that three vectors stack straight A comma with rightwards harpoon with barb upwards on top space B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top are represented by OL with rightwards harpoon with barb upwards on top comma space stack O P with rightwards harpoon with barb upwards on top space a n d space stack O Q with rightwards harpoon with barb upwards on top spacerespectively. 
    Let, the angle between straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top is straight alpha and that between straight A with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top space i s space beta. 
     
    In the fig. above, the resultant vector is given by straight R with rightwards harpoon with barb upwards on top. straight R with rightwards harpoon with barb upwards on top is the resultant of B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top. 
    Angle made by straight R with rightwards harpoon with barb upwards on top space w i t h space stack A space with rightwards harpoon with barb upwards on top i s space theta.

    Construction: From P and T draw perpendiculars PM and TN on OL and draw perpendicular PS from P on TN. 
    straight A with rightwards harpoon with barb upwards on top. left parenthesis B with rightwards harpoon with barb upwards on top space plus space C with rightwards harpoon with barb upwards on top right parenthesis space equals space A with rightwards harpoon with barb upwards on top. space R with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space space space space space space equals space A R space c o s theta space space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O N right parenthesis space space space space space space space space space space space space space space space space space space space space space equals space A thin space left parenthesis O M plus M N right parenthesis space space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O M plus P S right parenthesis space space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O P space c o s alpha space plus space P T space c o s beta right parenthesis space space space space space space space space space space space space space space space space space space space space space equals space A thin space left parenthesis B space c o s alpha space plus space C space c o s beta right parenthesis space space space space space space space space space space space space space space space space space space space space space equals space A B space c o s alpha space plus space A C space c o s beta space space space space space space space space space space space space space space space space space space space space equals space A with rightwards harpoon with barb upwards on top. B with rightwards harpoon with barb upwards on top space plus space A with rightwards harpoon with barb upwards on top. C with rightwards harpoon with barb upwards on top space 
    Hence, from the above result we can see that the dot product is distributive. 

                       
    Question 216
    CBSEENPH11018294
    Wired Faculty App

    If straight A with rightwards arrow on top space equals space straight A subscript straight x straight i with hat on top plus straight A subscript straight y straight j with hat on top plus straight A subscript straight z straight k with hat on top and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> then find the dot product of straight A with rightwards arrow on top space and space straight B with rightwards arrow on top?
    Solution

    Given that,
                <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and 
               straight B with rightwards harpoon with barb upwards on top space equals space B subscript x i with hat on top space plus space B subscript y space j with hat on top space plus space B subscript z space k with hat on top
    Therefore, 
    straight A with rightwards harpoon with barb upwards on top. straight B with rightwards harpoon with barb upwards on top space equals space left parenthesis straight A subscript straight x straight i with hat on top space plus space straight A subscript straight y space straight j with hat on top space plus space straight A subscript straight z straight k with hat on top right parenthesis space. space left parenthesis straight B subscript straight x straight i with hat on top space plus space straight B subscript straight y straight j with hat on top space plus space straight B subscript straight z straight k with hat on top right parenthesis space space space space space space space space space equals space straight A subscript straight x straight B subscript straight x space left parenthesis straight i with hat on top. straight i with hat on top right parenthesis space plus space straight A subscript straight y straight B subscript straight x space left parenthesis straight j with hat on top. straight i with hat on top right parenthesis space plus space straight A subscript straight z straight B subscript straight x space left parenthesis straight k with hat on top space. straight i with hat on top right parenthesis space plus space space space space space space space space space space space space straight A subscript straight x straight B subscript straight y space left parenthesis straight i with hat on top. straight j with hat on top right parenthesis space plus space straight A subscript straight y straight B subscript straight y space left parenthesis straight j with hat on top. straight j with hat on top space right parenthesis plus space straight A subscript straight z straight B subscript straight y space left parenthesis straight k with hat on top space. straight j with hat on top right parenthesis space space plus space space space space space space space space space space space straight A subscript straight x straight B subscript straight z space left parenthesis straight i with hat on top. straight k with hat on top right parenthesis plus space straight A subscript straight y straight B subscript straight z space left parenthesis straight j with hat on top. straight k with hat on top right parenthesis space plus space straight A subscript straight z straight B subscript straight z space left parenthesis straight k with hat on top space. straight k with hat on top right parenthesis space We space have comma space space left parenthesis straight i with hat on top. straight i with hat on top right parenthesis space equals space left parenthesis straight j with hat on top. straight j with hat on top space right parenthesis space equals space left parenthesis straight k with hat on top space. straight k with hat on top right parenthesis space equals space 1 space and space left parenthesis straight j with hat on top. straight i with hat on top right parenthesis space equals left parenthesis straight i with hat on top. straight j with hat on top right parenthesis space equals space 0 left parenthesis straight k with hat on top space. straight i with hat on top right parenthesis space equals space space equals space 0 left parenthesis straight k with hat on top space. straight j with hat on top right parenthesis space equals left parenthesis straight j with hat on top. straight k with hat on top right parenthesis space equals space 0 space Therefore comma space straight A with rightwards harpoon with barb upwards on top. space straight B with rightwards harpoon with barb upwards on top space equals space straight A subscript straight x straight B subscript straight x space plus space straight A subscript straight y straight B subscript straight y space plus space straight A subscript straight z straight B subscript straight z

    Question 217
    CBSEENPH11018297
    Wired Faculty App

    Find the angle between the vectors straight A with rightwards arrow on top equals straight A subscript straight x straight i with overparenthesis on top plus straight A subscript straight y straight j with hat on top plus straight A subscript straight z straight k with hat on top and straight B with rightwards arrow on top equals straight B subscript straight x straight i with hat on top plus straight B subscript straight y straight j with hat on top plus straight B subscript straight z straight k with hat on top.
    Solution

    Let straight theta be the angle between the given two vectors.
    Thus, 
                             cos space straight theta space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator AB end fraction
    Here,                      straight A equals square root of straight A subscript straight x squared plus straight A subscript straight y squared plus straight A subscript straight z squared end root
                                   straight B equals square root of straight B subscript straight x squared plus straight B subscript straight y squared plus straight B subscript straight z squared end root
    and           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴      straight theta equals cos to the power of negative 1 end exponent open parentheses fraction numerator straight A subscript straight x straight B subscript straight x plus straight A subscript straight y straight B subscript straight y plus straight A subscript straight z straight B subscript straight z over denominator square root of straight A subscript straight x squared plus straight A subscript straight y squared plus straight A subscript straight z squared end root space square root of straight B subscript straight x squared plus straight B subscript straight y squared plus straight B subscript straight z squared end root end fraction close parentheses, is the angle between the vectors. 

    Question 218
    CBSEENPH11018299
    Wired Faculty App

    What is the angle between the vectors <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and straight B with rightwards arrow on top space equals space 2 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top?
    Solution

    Given that,
               <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and      straight B with rightwards arrow on top space equals space 2 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top
    Let straight theta be the angle between the given two vectors.
    We know,  straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space AB space cos space straight theta
    ∴               cos space straight theta space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator AB end fraction
    Now           straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space left parenthesis straight i with hat on top minus 2 straight j with hat on top minus 5 straight k with hat on top right parenthesis. left parenthesis 2 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis
                             equals space 2 minus 2 plus 20 space equals space 20
    Magnitude is given by,
     open vertical bar straight A with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus left parenthesis negative 5 right parenthesis squared end root space equals space square root of 30
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
     Thus, angle between the vectors,
     cosθ space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator AB end fraction equals fraction numerator 20 over denominator square root of 30 cross times square root of 21 end fraction equals square root of 40 over 63 end root
    rightwards double arrow               <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>    

    Question 219
    CBSEENPH11018300
    Wired Faculty App

    A particle moves from point (2, 3, 5) to (8, -4, 2) when a force of 3 straight i with hat on top minus 7 straight j with hat on top minus straight k with hat on top unit is applied on it. Find the work done by the force.
    Solution

    Given, a particle moves from point P(2,3,5) to Q(8,-4,2). 
    Force applied on th particle, F = 3 straight i with hat on top minus 7 straight j with hat on top minus straight k with hat on top 
    Therefore, the displacement undergone by the particle is, 
                 straight S with rightwards arrow on top space equals space left parenthesis 8 minus 2 right parenthesis straight i with hat on top plus left parenthesis negative 4 minus 3 right parenthesis straight j with hat on top plus left parenthesis 2 minus 5 right parenthesis straight k with hat on top 
                    equals 6 straight i with hat on top minus 7 straight j with hat on top minus 3 straight k with hat on top 
    Therefore work done by force is, 
     straight W equals straight F with rightwards arrow on top times straight S with rightwards arrow on top space equals space left parenthesis 3 straight i with hat on top minus 7 straight j with hat on top minus straight k with hat on top right parenthesis times left parenthesis 6 straight i with hat on top minus 7 straight j with hat on top minus 3 straight k with hat on top right parenthesis 
                  equals 18 plus 49 plus 3 equals 70 J
     

    Question 220
    CBSEENPH11018305
    Wired Faculty App

    Show that
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> and straight B with rightwards arrow on top space equals space minus 4 straight i with hat on top plus 2 straight j with hat on top are perpendicular to each other.
    Solution

    Given,       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and            straight B with rightwards arrow on top space equals space minus 4 straight i with hat on top plus 2 straight j with hat on top
    straight A with rightwards arrow on top space and space straight B with rightwards arrow on top  have to be perpendicular to each other.
    To prove that we will find the dot product.
      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
             equals negative 8 plus 0 plus 8 equals 0
    Hence     straight A with rightwards arrow on top perpendicular straight B with rightwards arrow on top. 
    Since, the dot product is zero the two vectors are perpendicular to each other. 

    Question 221
    CBSEENPH11018307
    Wired Faculty App

    Find the unit vector in direction of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> where <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Solution

    Here,    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore the magnitude of vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
          equals square root of 9 plus 16 plus 25 end root equals square root of 50
    Now the unit vector in direction of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is, 
         straight A with hat on top space equals space fraction numerator straight A with rightwards arrow on top over denominator straight A end fraction equals fraction numerator 3 straight i with hat on top minus 4 straight j with hat on top plus 5 straight k with hat on top over denominator square root of 50 end fraction

    Question 222
    CBSEENPH11018310
    Wired Faculty App

    Find the component of vector space 3 straight i with hat on top minus 4 straight j with hat on top in direction of straight i with hat on top plus straight j with hat on top.
    Solution

    Dot product of two vectors is given by,
                   straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space AB space cosθ
    ∴      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    New component of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> in the direction of space straight B with rightwards arrow on top
            Acosθ straight B with hat on top space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator straight B end fraction straight B with hat on top
                       equals fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator straight B squared end fraction straight B with rightwards arrow on top
    Given,
                  straight A with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight j with hat on top  and  straight B with rightwards arrow on top space equals straight i with hat on top plus straight j with hat on top
    Thus,      
              straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space left parenthesis 3 straight i with hat on top minus 4 straight j with hat on top right parenthesis times left parenthesis straight i with hat on top plus straight j with hat on top right parenthesis space equals space 3 minus 4 space equals space minus 1
    Magnitude of B,  open vertical bar straight B with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 2
    So, components of vector in the direction of  straight i with hat on top plus straight j with hat on top
    ∴            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 

    Question 223
    CBSEENPH11018313
    Wired Faculty App

    For what value of c the vector 2 straight i with hat on top plus straight c straight j with hat on top will be perpendicular to:

    (i) Z-axis      (ii) Y-axis?
    Solution

    (i) Vector 2 straight i with hat on top plus straight c straight j with hat on top is perpendicular to Z-axis if its dot product with <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is zero.
    i.e.                       left parenthesis 2 straight i with hat on top plus straight c straight j with hat on top right parenthesis times straight k with hat on top equals 0
    rightwards double arrow                   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow                      2 left parenthesis 0 right parenthesis plus straight c left parenthesis 0 right parenthesis space equals space 0
    The equation is satisfied for all values of c.
    Therefore left parenthesis 2 straight i with hat on top plus straight c straight j with hat on top right parenthesis is always perpendicular to Z-axis whatever the value of c.
    (ii) Similarly        left parenthesis 2 straight i with hat on top plus straight c straight j with hat on top right parenthesis times straight j with hat on top equals 0
    rightwards double arrow                    2 straight i with hat on top. straight j with hat on top plus straight c space straight j with hat on top. space straight j with hat on top space equals space 0
    rightwards double arrow                            space space 0 plus straight c space equals space 0
    rightwards double arrow                                    straight c equals 0
    ∴  left parenthesis 2 straight i with hat on top plus straight c straight j with hat on top right parenthesis is perpendicular to Y-axis if c = 0.

    Question 224
    CBSEENPH11018314
    Wired Faculty App

    If <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> then do you think that A is necessarily equal to C?
    Solution

    Given that,
                             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Let angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top be straight alpha and that between straight B with rightwards arrow on top space and space straight C with rightwards arrow on top space be space straight beta.
    ∴     AB space cos space straight alpha space equals space CB space cos space straight beta
    rightwards double arrow      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow                straight A equals straight C fraction numerator cos space straight beta over denominator cos space straight alpha end fraction
    Now,    if <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  then A = C, and
                if space straight alpha not equal to straight beta comma  then straight A not equal to straight C
    Thus A is not necessarily equal to C.

    Question 225
    CBSEENPH11018318
    Wired Faculty App

    Prove that:

    left parenthesis 3 straight A with rightwards arrow on top plus 2 straight B with rightwards arrow on top right parenthesis times left parenthesis 2 straight A with rightwards arrow on top minus 3 straight B with rightwards arrow on top right parenthesis space equals space 6 straight A squared minus 6 straight B squared minus 5 ABcosθ, where straight theta is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
    Solution

    By using distributive and commutative law, the dot product can be evaluated as, 
       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    equals 6 straight A with rightwards arrow on top times straight A with rightwards arrow on top plus 4 straight B with rightwards arrow on top times straight A with rightwards arrow on top minus 9 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B with rightwards arrow on top times straight B with rightwards arrow on top     
     

    equals 6 straight A squared plus 4 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 9 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B squared
    equals 6 straight A squared minus 5 straight A with rightwards arrow on top times straight B with rightwards arrow on top minus 6 straight B squared                  

    equals space 6 straight A squared minus 5 ABcosθ minus 6 straight B squared
    That is,
      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> equals 6 straight A squared minus 6 straight B squared minus 5 ABcosθ  


                        
                  
                   
              

    Question 226
    CBSEENPH11018319
    Wired Faculty App

    A particle moves in XY plane, such that its position at any instant is
          straight x equals straight alpha space sin space ωt           and       straight y equals straight alpha left parenthesis 1 minus cos space ωt right parenthesis
    Find the angle between velocity vector and acceleration vector at any instant.
    Solution

    The position of the particle at any instant is,
           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore the velocity of particle at any instant is,
          straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space straight d over dt open parentheses αsinωt straight i with hat on top plus straight alpha left parenthesis 1 minus cosωt right parenthesis straight j with hat on top close parentheses
    rightwards double arrow  space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space αω space cosωt straight i with hat on top plus αω space sinωt straight j with hat on top
    Acceleration of particle at any instant is,
          straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis equals straight d over dt open parentheses aω space cosωt space straight i with hat on top plus space αωsinωt space straight j with hat on top close parentheses
    rightwards double arrow  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Now,  
    straight a with rightwards arrow on top times straight v with rightwards arrow on top equals left parenthesis negative αω squared sinωt straight i with hat on top plus αω squared cosωt straight j with hat on top right parenthesis times left parenthesis αω space cosωt space straight i with hat on top plus αωsinωt space straight j with hat on top right parenthesis
           equals negative straight alpha squared straight omega cubed sinωtcosωt plus straight alpha squared straight omega cubed cosωtsinωt equals 0
    Since dot product of acceleration vector and velocity vector is zero, therefore the acceleration vector and velocity vector is 90 degree. 

    Question 227
    CBSEENPH11018320
    Wired Faculty App

    Find the unit vector along the resultant of left parenthesis straight i with hat on top plus straight j with hat on top right parenthesis space and space left parenthesis straight i with hat on top minus straight j with hat on top right parenthesis.
    Solution

    Resultant of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is
     straight R with rightwards arrow on top equals left parenthesis straight i with hat on top plus straight j with hat on top right parenthesis plus left parenthesis straight i with hat on top minus straight j with hat on top right parenthesis space equals space 2 straight i with hat on top
    ∴   Unit vector in direction of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 228
    CBSEENPH11018325
    Wired Faculty App

    Find the component of vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  in the direction of space space space 12 straight i with hat on top space plus space 5 straight j with hat on top space minus space square root of 126 space straight k with hat on top.
    Solution

    We know straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space ABcosθ
    ∴          space straight A space cosθ space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator straight B end fraction
    Now component of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> in the direction of space straight B with rightwards arrow on top is,
          space Acosθ space straight B with hat on top space equals space fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator straight B end fraction straight B with hat on top
                       equals fraction numerator straight A with rightwards arrow on top times straight B with rightwards arrow on top over denominator straight B squared end fraction straight B with rightwards arrow on top
    Here,     straight A with rightwards arrow on top space equals space 4 straight i with hat on top minus 3 straight j with hat on top plus square root of 14 space straight k with hat on top
    and       straight B with rightwards arrow on top equals 12 straight i with hat on top plus 5 straight j with hat on top minus square root of 126 space straight k with hat on top
    Thus,         <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and           open vertical bar straight B with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 12 right parenthesis squared plus left parenthesis 5 right parenthesis squared plus left parenthesis square root of 126 right parenthesis squared end root
                         equals square root of 295
    ∴   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 229
    CBSEENPH11018331
    Wired Faculty App

    Find a vector having magnitude equal to the magnitude of vector straight A with rightwards arrow on top space equals space minus 2 straight i with hat on top plus 1 straight j with hat on top plus 4 straight k with hat on top and parallel to vector straight B with rightwards arrow on top equals straight i with hat on top plus 2 straight j with hat on top minus 3 stack straight k. with hat on top
    Solution

    Magnitude of vector A is,
         space space space space straight A equals square root of left parenthesis negative 2 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 21
    Unit vector in direction of vector straight B with rightwards arrow on top space is comma
    space space straight B with hat on top space equals space fraction numerator straight B with rightwards arrow on top over denominator straight B end fraction equals fraction numerator straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top over denominator square root of 1 plus 4 plus 9 end root end fraction equals fraction numerator straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top over denominator square root of 14 end fraction
    The vector that has magnitude same as that of vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>and parallel to vector <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    straight A straight B with hat on top space equals space square root of 21 fraction numerator straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top over denominator square root of 14 end fraction space space space space space equals square root of 3 over 2 end root straight i with hat on top plus square root of 6 space straight j with hat on top minus square root of 27 over 2 end root straight k with hat on top
          

    Question 230
    CBSEENPH11018335
    Wired Faculty App

    If straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top times straight C with rightwards arrow on top and angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is twice the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> then show that straight C over straight B equals 2 cos straight theta over 2 minus sec straight theta over 2 comma where straight theta is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
    Solution

    It is given that angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is straight theta. 
    Therefore the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> will be straight theta divided by 2.
    Here,        straight A with rightwards arrow on top times straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top times straight C with rightwards arrow on top
    ∴         AB space cos space straight theta space equals space AC space cos left parenthesis straight theta divided by 2 right parenthesis
    rightwards double arrow               straight C over straight B equals fraction numerator cosθ over denominator cos left parenthesis straight theta divided by 2 right parenthesis end fraction space space space space space space equals fraction numerator 2 cos squared left parenthesis straight theta divided by 2 right parenthesis minus 1 over denominator cos left parenthesis straight theta divided by 2 right parenthesis end fraction

    rightwards double arrow               straight C over straight B equals 2 cos space straight theta over 2 minus sec straight theta over 2.
    Hence proved. 

    Question 231
    CBSEENPH11018341
    Wired Faculty App

    Find the magnitude and direction of resultant of two vectors without using the parallelogram of vector addition.

    Solution

    Let space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top be two vectors acting at angle straight theta.
    straight R with rightwards harpoon with barb upwards on top space is the resultant of straight P with rightwards harpoon with barb upwards on top space a n d space Q with rightwards harpoon with barb upwards on top. 
    Therefore, straight R with rightwards harpoon with barb upwards on top space equals space P with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top 

    <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>
    Let, the resultant straight R with rightwards harpoon with barb upwards on top make angle straight alpha with OA with rightwards harpoon with barb upwards on top.
    straight R with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space equals space stack left parenthesis P with rightwards harpoon with barb upwards on top space plus Q with rightwards harpoon with barb upwards on top right parenthesis space. space P with rightwards harpoon with barb upwards on top R P space cos space alpha space equals space P with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top R P thin space cos space alpha space equals space P squared space plus space P Q space cos theta space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis space
    Also, 
    open vertical bar straight R with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top close vertical bar space equals space open parentheses P with rightwards harpoon with barb upwards on top plus Q with rightwards harpoon with barb upwards on top close parentheses cross times P with rightwards harpoon with barb upwards on top space R P space sin space alpha space equals space open vertical bar P with rightwards harpoon with barb upwards on top cross times P with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top cross times P with rightwards harpoon with barb upwards on top close vertical bar R P space cos alpha space equals space P Q space sin space theta space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space D i v i d i n g space e q u a t i o n space left parenthesis 2 right parenthesis space b y space left parenthesis 1 right parenthesis comma space w e space h a v e fraction numerator R P space sin alpha over denominator R P space cos space alpha end fraction space equals space fraction numerator P Q space sin space theta over denominator P squared space plus space P Q space cos theta end fraction space i. e. comma space tan alpha space equals space fraction numerator Q space sin theta over denominator P space plus space Q cos theta end fraction space space space space space space space space space space
    Thus, for two vectors acting in the same direction, the magnitude of the resultant vector is equal to the sum of the magnitude of two vectors and acts along the direction of straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top.

    Question 232
    CBSEENPH11018342
    Wired Faculty App

    Define cross product of two vectors. Show that cross product of two vectors anti commute.

    Solution
    Cross product of two vectors is a vector whose magnitude is equal to product of the magnitude of each vector and sine of angle between them and is directed along the normal to the plane containing two vectors. 
                  straight A with rightwards harpoon with barb upwards on top space cross times B with rightwards harpoon with barb upwards on top space equals space space A B space sin theta straight c with hat on top
    The direction of the cross product of two vectors, that is straight c with hat on top is determined by the right hand screw rule or right hand thumb rule. 
    The right hand screw rule states that when you place a screw perpendicular to the plane containing vectors straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top. Rotate the cap of a screw from straight A with rightwards harpoon with barb upwards on top space t o space B with rightwards harpoon with barb upwards on top to find the direction of straight A with rightwards harpoon with barb upwards on top cross times B with rightwards harpoon with barb upwards on top and <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre> to find the direction of straight B with rightwards harpoon with barb upwards on top space cross times A with rightwards harpoon with barb upwards on top.
    The direction in which the screw will advance gives us the direction of straight c with hat on top.
    Therefore, by Right Hand Screw Rule, the direction of straight A with rightwards harpoon with barb upwards on top cross times B with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top cross times A with rightwards harpoon with barb upwards on top space i s space o p p o s i t e. space
    Now, 
                     straight A with rightwards harpoon with barb upwards on top cross times B with rightwards harpoon with barb upwards on top space equals space A B space sin theta space c with hat on top     and
    straight B with rightwards harpoon with barb upwards on top cross times A with rightwards harpoon with barb upwards on top space equals space B A space sin theta space stack c space with hat on top space space space space space space space space space space equals space minus A B space sin theta space stack c space with hat on top space space space space space space space space space space equals space minus A with rightwards harpoon with barb upwards on top space cross times space B with rightwards harpoon with barb upwards on top 
    So, cross product of two vectors anti-commute. 
    Question 233
    CBSEENPH11018346
    Wired Faculty App

    What are the characteristics of cross product?

    Solution
    Characteristics of cross product are:
    (i) Cross product of two vectors is anti commutative.
    That is, 
                             space straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space minus straight B with rightwards arrow on top cross times straight A with rightwards arrow on top
    (ii) Cross product is distributive,
    That is, 
     
                     space straight A with rightwards arrow on top cross times left parenthesis straight B with rightwards arrow on top plus stack straight C right parenthesis with rightwards arrow on top space equals space straight A with rightwards arrow on top cross times straight B with rightwards arrow on top plus straight A with rightwards arrow on top cross times straight C with rightwards arrow on top
    (iii)Cross product of two parallel vectors is zero.
    That is, 
                         space straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space 0     if <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    (iv) Cross product of two vectors is equal to the area of parallelogram formed by two vectors.
    (v) Area of triangle formed by two vectors and their resultant is equal to half the magnitude of cross product.
    Question 234
    CBSEENPH11018347
    Wired Faculty App

    Show that if two adjacent sides of parallelogram represent the two vectors, then area of the parallelogram is equal to magnitude of the cross product of two vectors. 

    Solution

    As shwn in the fig., et PQRS be a parallelogram formed by vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
                   
    From the figure,
    Area of parallelogram PQRS is given by, 
                    S = PQ. RN 
                       = PQ. space PR space sinθ space equals space straight A space straight B space sinθ 
                     space space equals open vertical bar straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close vertical bar 
    Hence the result.

    Question 235
    CBSEENPH11018348
    Wired Faculty App

    Show that if two adjacent sides of a triangle represent the two vectors, then area of the triangle is equal to half of the magnitude of the cross product of two vectors.

    Solution

    Let PQR be triangle formed by vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top shown in figure.
                     
                         
    From figure, the area of increment PQR is,
              straight S equals 1 half PQ. RN space space equals 1 half PQ. PR space sinθ
              space space equals 1 half ABsinθ space equals space 1 half open vertical bar straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close vertical bar 
                 

    Question 236
    CBSEENPH11018350
    Wired Faculty App

    The magnitude of the cross product of two vectors is equal to dot product of the same two vectors. What is the angle between two vectors?

    Solution

    Given,
    Magitude of cross product is equak to the dot product of two vectors. 
    i.e., 
        space space open vertical bar straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close vertical bar space equals space open vertical bar straight A with rightwards arrow on top times straight B with rightwards arrow on top close vertical bar
    rightwards double arrow <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow          straight theta equals 45 degree

    Question 237
    CBSEENPH11018355
    Wired Faculty App

    Find the area of parallelogram formed by space straight A with rightwards arrow on top space equals space straight i with hat on top minus straight j with hat on top plus 2 straight k with hat on top  and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    Solution

    Given,
    Here,  straight A with rightwards arrow on top equals straight i with hat on top minus straight j with hat on top plus 2 straight k with hat on top,    straight B with rightwards arrow on top space equals space minus 3 straight i with hat on top plus 4 straight j with hat on top minus straight k with hat on top
    ∴    space space space space space space space space space space space space space space space space space space space space space space space space space space straight i with hat on top space space space space space space space space space space space space stack straight j space with hat on top space space space space space space space space space space space space space straight k with hat on top straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space space space space space space space space space space 1 space space space space space space space minus 1 space space space space space space space space space space space space 2 space space space space space space space space space space space space space space space space space space space space space space space minus 3 space space space space space space space space 4 space space space space space space space space space space minus 1 
                space space equals left square bracket left parenthesis negative 1 right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 2 right parenthesis left parenthesis 4 right parenthesis right square bracket straight i with hat on top minus left square bracket left parenthesis 1 right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 2 right parenthesis left parenthesis negative 3 right parenthesis right square bracket straight j with hat on top                                          plus left square bracket left parenthesis 1 right parenthesis left parenthesis 4 right parenthesis space minus space left parenthesis negative 1 right parenthesis left parenthesis negative 3 right parenthesis right square bracket straight k with hat on top

                  equals negative 7 straight i with hat on top minus 5 straight j with hat on top plus straight k with hat on top
     Now the area of parallelogram formed by vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top is, 
       open vertical bar straight A with rightwards arrow on top cross times straight B with rightwards arrow on top close vertical bar space equals space square root of left parenthesis negative 7 right parenthesis squared plus left parenthesis negative 5 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 75 space equals space 5 square root of 3

    Question 238
    CBSEENPH11018356
    Wired Faculty App

    Show that:

    left parenthesis straight A with rightwards arrow on top times straight B with rightwards arrow on top right parenthesis squared plus left parenthesis straight A with rightwards arrow on top cross times straight B with rightwards arrow on top right parenthesis squared space equals space straight A squared straight B squared
    Solution

    L.H.S = space left parenthesis straight A with rightwards arrow on top times straight B with rightwards arrow on top right parenthesis squared plus left parenthesis straight A with rightwards arrow on top cross times straight B with rightwards arrow on top right parenthesis squared
           space space equals left parenthesis AB space cosθ right parenthesis squared plus left parenthesis AB space sinθ right parenthesis squared
             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

             equals straight A squared straight B squared = R.H.S
    Hence proved
    Question 239
    CBSEENPH11018360
    Wired Faculty App

    Show that straight A with rightwards arrow on top cross times straight B with rightwards arrow on top is perpendicular to <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    Solution

    The dot product of two perpendiculars vectors is zero.
    Therefore, to prove straight A with rightwards arrow on top cross times straight B with rightwards arrow on top to be perpendicular to space straight A with rightwards arrow on top comma we have to prove space space space space left parenthesis straight A with rightwards arrow on top cross times straight B with rightwards arrow on top right parenthesis space times space straight A with rightwards arrow on top 
    So, 
    left parenthesis straight A with rightwards harpoon with barb upwards on top space cross times space straight B with rightwards harpoon with barb upwards on top right parenthesis. space straight A with rightwards harpoon with barb upwards on top space equals space open parentheses straight A with rightwards harpoon with barb upwards on top space cross times stack space straight A with rightwards harpoon with barb upwards on top close parentheses. space straight B with rightwards arrow on top space equals space 0 space. space straight B with rightwards harpoon with barb upwards on top space equals space 0 straight i. straight e. comma space open parentheses straight A with rightwards harpoon with barb upwards on top cross times straight B with rightwards harpoon with barb upwards on top close parentheses space perpendicular space straight A with rightwards harpoon with barb upwards on top 
    Hence, the result. 

    Question 240
    CBSEENPH11018361
    Wired Faculty App

    What is the angle between straight A with rightwards arrow on top plus straight B with rightwards arrow on top space and space straight B with rightwards arrow on top cross times straight A with rightwards arrow on top ?
    Solution

    Vector straight A with rightwards arrow on top cross times straight B with rightwards arrow on top is perpendicular to the plane containing straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
              straight A with rightwards arrow on top plus straight B with rightwards arrow on top is the resultant of <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    Therefore the cross product lies in the plane containing straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
    ∴  Angle between straight A with rightwards arrow on top cross times straight B with rightwards arrow on top and straight A with rightwards arrow on top plus straight B with rightwards arrow on top is 90 degree.

    Question 241
    CBSEENPH11018362
    Wired Faculty App

    Show that:
    left parenthesis straight A with rightwards arrow on top minus straight B with rightwards arrow on top right parenthesis cross times left parenthesis straight A with rightwards arrow on top plus straight B with rightwards arrow on top right parenthesis space equals space 2 left parenthesis straight A with rightwards arrow on top cross times straight B with rightwards arrow on top right parenthesis

    Solution

    L.H.S. = left parenthesis straight A with rightwards arrow on top minus straight B with rightwards arrow on top right parenthesis cross times left parenthesis straight A with rightwards arrow on top plus straight B with rightwards arrow on top right parenthesis 
             equals straight A with rightwards arrow on top cross times straight A with rightwards arrow on top minus straight B with rightwards arrow on top cross times straight A with rightwards arrow on top plus straight A with rightwards arrow on top cross times straight B with rightwards arrow on top minus straight B with rightwards arrow on top cross times straight B with rightwards arrow on top 
           space space equals 0 plus straight A with rightwards arrow on top cross times straight B with rightwards arrow on top plus straight A with rightwards arrow on top cross times straight B with rightwards arrow on top minus 0 equals 2 left parenthesis straight A with rightwards arrow on top cross times straight B with rightwards arrow on top right parenthesis
     
             = R.H.S.
    Hence, proved. 

    Question 242
    CBSEENPH11018363
    Wired Faculty App

    Find the unit vector perpendicular to both the vectors given below:
    straight P with rightwards arrow on top space equals left parenthesis 3 straight i with hat on top minus 2 straight j with hat on top plus 4 straight k with hat on top right parenthesis  and  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution

    The unit vector perpendicular to vectors space straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top is given by,
                         straight n with hat on top equals fraction numerator straight P with rightwards arrow on top cross times straight Q with rightwards arrow on top over denominator open vertical bar straight P with rightwards arrow on top cross times straight Q with rightwards arrow on top close vertical bar end fraction
    We have,
                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore,
                   P with rightwards arrow on top cross times Q with rightwards arrow on top equals<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, is the required unit vector. 

    Question 243
    CBSEENPH11018365
    Wired Faculty App

    If diagonals of a parallelogram are straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top comma then find the area of the parallelogram.
    Solution

    Let the sides of the parallelogram be straight A with rightwards arrow on top space and space straight B with rightwards arrow on top, whose diagonals are straight P with rightwards arrow on top space and space stack straight Q. with rightwards arrow on top 
    <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>
    equals space open vertical bar fraction numerator straight P with rightwards harpoon with barb upwards on top plus straight Q with rightwards harpoon with barb upwards on top over denominator 2 end fraction cross times fraction numerator straight P with rightwards harpoon with barb upwards on top minus straight Q with rightwards harpoon with barb upwards on top over denominator 2 end fraction close vertical bar space equals space 1 fourth open vertical bar open parentheses straight P with rightwards harpoon with barb upwards on top plus straight Q with rightwards harpoon with barb upwards on top close parentheses cross times open parentheses straight P with rightwards harpoon with barb upwards on top space minus space straight Q with rightwards harpoon with barb upwards on top close parentheses close vertical bar equals space 1 fourth open vertical bar open parentheses straight P with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space plus space straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space minus space straight P with rightwards harpoon with barb upwards on top space cross times straight Q with rightwards harpoon with barb upwards on top space minus space straight Q with rightwards harpoon with barb upwards on top space cross times space straight Q with rightwards harpoon with barb upwards on top close parentheses close vertical bar equals space 1 fourth open vertical bar open parentheses straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space plus straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top close parentheses space space close vertical bar equals space 1 half space open vertical bar straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top close vertical bar space equals space 1 half open vertical bar straight P with rightwards harpoon with barb upwards on top space cross times straight Q with rightwards harpoon with barb upwards on top close vertical bar
     Hence, the area of the parallelogram is found. 

    Question 244
    CBSEENPH11018367
    Wired Faculty App

    If straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space straight C with rightwards arrow on top cross times straight B with rightwards arrow on top, the do you think that A is necessarily equal to C?
    Solution


    Here,   straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space straight C with rightwards arrow on top cross times straight B with rightwards arrow on top
    Let the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top be straight alpha and that between straight B with rightwards arrow on top and space straight C with rightwards arrow on top be straight beta.
    ∴            AB space sinα space stack straight n subscript 1 with hat on top space equals space CB space sin space straight beta space stack straight n subscript 2 with hat on top


    rightwards double arrow           space AB space sinα space equals space CB space sinβ
    and             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>


    Therefore,
                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>


    rightwards double arrow                     straight A equals straight C sinβ over sinα


    Now,   space space if space straight alpha space equals space straight beta comma space space then space straight A space equals space straight C
    and       if space straight alpha not equal to straight beta comma    then straight A not equal to straight C


    Thus, A is not necessarily equal to C.

    Question 245
    CBSEENPH11018369
    Wired Faculty App

    Show that if straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top cross times straight C with rightwards arrow on top, then straight A with rightwards arrow on top plus straight C with rightwards arrow on top space equals space straight lambda straight B with rightwards arrow on top where <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> are the three vectors lying on the plane.
    Solution

    Here,    
            straight A with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top cross times straight C with rightwards arrow on top
    i.e.,    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow     straight A with rightwards arrow on top cross times straight B with rightwards arrow on top plus straight C with rightwards arrow on top cross times straight B with rightwards arrow on top space equals space 0
    rightwards double arrow      left parenthesis straight A with rightwards arrow on top plus straight C with rightwards arrow on top right parenthesis cross times straight B with rightwards arrow on top space equals space 0
    rightwards double arrow      left parenthesis straight A with rightwards arrow on top plus straight C with rightwards arrow on top right parenthesis parallel to straight B with rightwards arrow on top
    Hence,  left parenthesis straight A with rightwards arrow on top plus straight C with rightwards arrow on top right parenthesis space equals space straight lambda straight B with rightwards arrow on top

    Question 246
    CBSEENPH11018370
    Wired Faculty App

    Find the area of triangle whose vertices are straight P left right arrow left parenthesis 1 comma space 1 comma space 1 right parenthesis comma space straight Q left right arrow left parenthesis 2 comma space 2 comma space 2 right parenthesis space and space straight R left right arrow left parenthesis 2 comma space minus 1 comma space 2 right parenthesis.
    Solution

    Here, P, Q and R are the vertices of a triangle.
    Therefore PQ with rightwards arrow on top space and space PR with rightwards arrow on top represent the sides of triangle as shown in the figure.
                  
    We have, 
    PQ with rightwards arrow on top space equals space left parenthesis 2 minus 1 right parenthesis straight i with hat on top plus left parenthesis 2 minus 1 right parenthesis straight j with hat on top plus left parenthesis 2 minus 1 right parenthesis straight k with hat on top space equals space straight i with hat on top plus straight j with hat on top plus straight k with hat on top 
    PR with rightwards arrow on top space equals left parenthesis 2 minus 1 right parenthesis straight i with hat on top plus left parenthesis negative 1 minus 1 right parenthesis straight j with hat on top plus left parenthesis 2 minus 1 right parenthesis straight k with hat on top equals straight i with hat on top minus 2 straight j with hat on top plus straight k with hat on top 
    Now, area of triangle PQR = 1 half open vertical bar stack P Q with rightwards harpoon with barb upwards on top cross times stack P R with rightwards harpoon with barb upwards on top close vertical bar space
    PQ with rightwards harpoon with barb upwards on top space cross times stack P R with rightwards harpoon with barb upwards on top space equals space open vertical bar table row i j k row 1 1 1 row 1 cell negative 2 end cell 1 end table close vertical bar space equals space 3 i with hat on top space minus space 3 k with hat on top space space space space a n d open vertical bar stack P Q with rightwards harpoon with barb upwards on top space cross times stack P R with rightwards harpoon with barb upwards on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared minus open parentheses 3 close parentheses squared end root space equals space 3 square root of 2 space 
    Therefore, area of increment PQR comma space=1 half cross times 3 square root of 2 space equals space fraction numerator 3 over denominator square root of 2 end fraction u n i t s 

    Question 247
    CBSEENPH11018371
    Wired Faculty App

    Point C is situated on the circumference of a circle of diameter AB. Show that triangle ACB is right angled triangle.

    Solution

    Let a be the radius of circle. 
    Here, OB with rightwards harpoon with barb upwards on top space equals space P with rightwards harpoon with barb upwards on top space a n d space stack O C with rightwards harpoon with barb upwards on top space equals space Q with rightwards harpoon with barb upwards on top
    therefore space OA with rightwards harpoon with barb upwards on top space equals space minus straight P with rightwards harpoon with barb upwards on top 
    Also, open vertical bar OB with rightwards harpoon with barb upwards on top close vertical bar space equals space open vertical bar stack O C with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar stack O A with rightwards harpoon with barb upwards on top close vertical bar space equals space a 
    Now, from increment AOC comma space we space have 
    space space space space space space space space space space space space space space space space space space space space stack O C with rightwards harpoon with barb upwards on top space plus space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top rightwards double arrow space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals space minus P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top space  
    In increment BOC comma space
    space space space space space space space space space space space space space OC with rightwards harpoon with barb upwards on top space plus space stack C B with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top rightwards double arrow space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals negative P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top N o w comma space space space space space open parentheses stack C A with rightwards harpoon with barb upwards on top close parentheses. open parentheses stack C B with rightwards harpoon with barb upwards on top close parentheses space equals space open parentheses negative P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses. open parentheses P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space plus space P with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P squared space plus space Q squared space equals space minus a squared space plus space a squared space equals space 0 space rightwards double arrow space space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top perpendicular stack C B with rightwards harpoon with barb upwards on top  
    Therefore, triangle ACB is a right angled triangle. 

    Question 248
    CBSEENPH11018373
    Wired Faculty App

    If a particle is in equilibrium under the action of three forces, then show that ratio of each force and sine of angle between the other two forces is constant?

    Solution

    Let the three forces <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> acting at point O be in equilibrium. 


    Taking cross product of C on both sides, we get
    open parentheses straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top space plus space straight C with rightwards harpoon with barb upwards on top close parentheses cross times straight C with rightwards harpoon with barb upwards on top space equals space 0 space cross times space straight C with rightwards harpoon with barb upwards on top space equals space 0 space rightwards double arrow space straight A with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top space plus straight C with rightwards harpoon with barb upwards on top cross times straight C with rightwards harpoon with barb upwards on top space equals space 0 space rightwards double arrow space minus space space straight C with rightwards harpoon with barb upwards on top space cross times straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top space plus space 0 space equals space 0 space rightwards double arrow space straight C with rightwards harpoon with barb upwards on top space cross times straight A with rightwards harpoon with barb upwards on top space equals space space straight B with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top rightwards double arrow space fraction numerator straight A over denominator sin space straight alpha end fraction space equals space fraction numerator straight B over denominator space sin space straight beta end fraction space space space space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis space Now comma space taking space the space cross space product space of space equation space left parenthesis 1 right parenthesis space with space straight B with rightwards harpoon with barb upwards on top comma space we space have space space space space open parentheses straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top space plus space straight C with rightwards harpoon with barb upwards on top close parentheses space cross times space straight B with rightwards harpoon with barb upwards on top space equals space 0 space cross times space straight B with rightwards harpoon with barb upwards on top space equals space 0 rightwards double arrow straight A with rightwards harpoon with barb upwards on top space cross times straight B with rightwards harpoon with barb upwards on top space plus space space straight B with rightwards harpoon with barb upwards on top space cross times space straight B with rightwards harpoon with barb upwards on top space plus straight C with rightwards harpoon with barb upwards on top space cross times straight B with rightwards harpoon with barb upwards on top space space equals space 0 space rightwards double arrow space straight A with rightwards harpoon with barb upwards on top space cross times straight B with rightwards harpoon with barb upwards on top space plus 0 space minus space straight B with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top space equals space 0 rightwards double arrow space straight A with rightwards harpoon with barb upwards on top space cross times straight B with rightwards harpoon with barb upwards on top space equals space space straight B with rightwards harpoon with barb upwards on top space cross times space straight C with rightwards harpoon with barb upwards on top rightwards double arrow space space space AB space sin space straight gamma space equals space BC space sin space straight alpha space rightwards double arrow space fraction numerator straight A over denominator sin space straight alpha space end fraction space equals space fraction numerator straight C over denominator sin space straight gamma end fraction space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 3 right parenthesis thin space From space equation space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get fraction numerator straight A over denominator sin space straight alpha space end fraction space equals space fraction numerator straight B over denominator sin space straight beta end fraction space equals space fraction numerator straight C over denominator sin space straight gamma end fraction space space space space space

    Question 251
    CBSEENPH11018378
    Wired Faculty App

    Read each statement below carefully and state, with reasons and examples, if it is true or false:

    A scalar quantity is one that

    (a) is conserved in a process.

    (b) can never take negative values.

    (c) must be dimensionless.

    (d) does not vary from one point to another in space.

    (e) has the same value for observers with different orientations of axes.

    Solution
    (a) False -  When we carry out the adiabatic process, the scalar quantity temperature does not conserve.
    In inelastic collisions, the K.E. (scalar quantity) does not conserve.
    (b) False - scalar quantity can have negative value e.g. <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>.
    If the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is greater than 90 degree commathen W will be negative e.g. work done by gravitational force when body is raised upward is negative.
    (c) False -  Scalar quantity may have dimensions, e.g. Dimensional formula of work is [M1L2T-2].
    (d) False - Temperature is a scalar quantity and it changes as we go higher, the gravitational potential is a scalar quantity and it varies with the height.
    (e) True.
    Question 252
    CBSEENPH11018379
    Wired Faculty App

    Can you associate vectors with: 

    (a) length of a wire bent into a loop
    (b) a plane area
    (c) a sphere? Explain.
    Solution

    (a) Yes, the direction of the length of a wire bent in the form of a loop is along the tangent to the loop.
    (b) Yes, area is a vector whose direction is perpendicular to the plane of area.
    (c) No, we cannot associate vector with the volume of the sphere.

    Question 253
    CBSEENPH11018381
    Wired Faculty App

    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

    (a) adding any two scalars,
    (b) adding a scalar to a vector of the same dimensions,
    (c) multiplying any vector by any scalar,
    (d) multiplying any two scalars,
    (e) adding any two vectors,
    (f) adding a component of a vector to the same vector.
    Solution

    (a)  If the scalars are similar type of quantities, adding of scalars will be meaningful. Scalars of the same dimension can only be added. 
    (b) Adding a scalar to a vector of the same dimension is meaningless. a scalar cannot be added to a vector. 
    (c) Any vector can be multiplied by a scalar. A vector when multiplied by a scalar quantity will give us a vector quantity. When the vector quantity acceleration is multiplied by m, we get a force, straight F with rightwards harpoon with barb upwards on top space equals space m a, which is a meaningful operation.
    (d) Two scalars can be multiplied together. For example, when power is multiplied by time t, we get the quatity, work done. 
    (e) Adding any two vectors is meanigless because two vectors of the same dimnsion can be added.
    (f) Adding a component of a vector to the same vector is meaningful. Because both the vectors are of the same dimensions.

    Question 254
    CBSEENPH11018382
    Wired Faculty App

    The vectors 2 straight i with hat on top minus 3 straight j with hat on top space and space minus 4 straight i with hat on top plus 6 straight j with hat on top space are ..........
    Solution
    parallel vectors
    Question 256
    CBSEENPH11018384
    Wired Faculty App

    The cross product of two parallel vectors is __________ vector

    Solution

    null

    Question 257
    CBSEENPH11018385
    Wired Faculty App

    The component of vector 6 straight i with hat on top minus 3 straight k with hat on top in Y direction is ________
    Solution

    zero.

    Question 258
    CBSEENPH11018387
    Wired Faculty App

    Read each statement below carefully and state with reasons, if it is true or false:

    (a) The magnitude of vector is always a scalar.

    (b) Each component of a vector is always a scalar.

    (c) The total path length is always equal to the magnitude of displacement vector of a particle.

    (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.

    (e) Three vectors not lying in a plane can never add up to give a null vector.

    Solution

    (a)True, because magnitude is a pure number. 
    (b)  False, each component of a vector is always a vector, not a scalar.
    (c) False, total path length can also be more than the magnitude of displacement vector of a particle. e.g. when a particle follows the arc of circle, the length of path is greater than magnitude of the displacement.
    This statement is true only if the particle is moving in a straight line. 
    (d) True, because the total path length is either greater than or equal to the magnitude of the displacement vector. 
    (e) True, as the three vectors cannot be represented by the three sides of a triangle taken in the same order. 

    Question 259
    CBSEENPH11018418
    Wired Faculty App

    Find the angle between the vectors: 

    straight A with rightwards arrow on top equals straight i with hat on top plus space 2 straight j with hat on top minus straight k with hat on top space and straight B with rightwards arrow on top space equals space minus straight i with hat on top plus straight j with hat on top minus 2 straight k with hat on top
    Solution

    Angle between the vectors is given by, 
    cos space theta space equals space fraction numerator A with rightwards harpoon with barb upwards on top space. space B with rightwards harpoon with barb upwards on top over denominator A B end fraction space space space space space space space space space space space space equals fraction numerator space left parenthesis i with hat on top space plus space 2 j with hat on top space minus k with hat on top right parenthesis space. space left parenthesis negative i with hat on top space plus space j with hat on top space minus 2 k with hat on top right parenthesis over denominator 6 end fraction space space space space space space space space space space space equals space fraction numerator negative 1 space plus space 2 space plus space 2 over denominator 6 end fraction space equals space 3 over 6 space equals space 1 half 
    That is straight theta = cos-1 1 half
    This implies, 
    straight theta = 600

    Question 260
    CBSEENPH11018422
    Wired Faculty App

    A particle has the following displacements in succession i) 12 m towards east ii) 5 m towards north and iii) 6 m vertically. Find the magnitude of the resultant displacement? 

    Solution

    Here, we have 
    x = 12 m ; y = 5 m ; z = 6m
    Then, resultant vector is given by, 
    r = (x2 + y2 + z2 ) 1/2
       = ( 122 + 52 + 62)1/2
       = (205)1/2
       = 14.31 m 

    Question 261
    CBSEENPH11018423
    Wired Faculty App

    Give two examples of motion in two dimensions.

    Solution

    Examples of mtion in two dimensions are: 
    (i) Motion of a football player in a ground.
    (ii) Projectile motion. 

    Question 263
    CBSEENPH11018425
    Wired Faculty App

    A particle is following the curved path. Can the normal acceleration speed up or slow down the particle?

    Solution
    No, the normal acceleration will not change the speed of the particle. 
    Normal acceleration will only change the direction of motion. 
    Question 264
    CBSEENPH11018426
    Wired Faculty App

    A particle is following the circular path. Which component of acceleration can speed up or slow down the particle?

    Solution
    When a particle is moving in a circular path, the  tangential component of acceleration of particle can change the speed of the particle. 
    Question 266
    CBSEENPH11018428
    Wired Faculty App

    Comment on the statement:

    Displacement vector of a particle is position vector of the particle.

    Solution
    Displacement vector of a particle is the position vector with respect to the initial position.
    Question 267
    CBSEENPH11018429
    Wired Faculty App

    The resultant of two equal forces acting at right angles to each other is 1414 N. Find the magnitude of each force? 

    Solution

    Consider the component, A=B = x. 
    The forces are acting at right angles to each other. 
    That is, 
    straight theta space equals space 90 to the power of 0
    Resultant, R = 1414 N 
    Now,

    space space space space space space space space space space space space space space space space space space space space space space space straight R space equals space square root of straight A squared plus straight B squared plus 2 AB space cos space straight theta end root space space rightwards double arrow space space space space space left parenthesis 1414 right parenthesis squared space equals space straight x squared space plus space straight x squared space plus space 2 straight x. straight x space straight x space cos space 90 to the power of straight o space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 straight x squared space rightwards double arrow space space space space space space space space space space space space space space space space space straight x space equals space fraction numerator 1414 over denominator square root of 2 end fraction space equals space 1000 space straight N
    x is the magnitude of each force. 

    Question 268
    CBSEENPH11018431
    Wired Faculty App

    A particle is moving with a velocity of 10 m/s due cast. In one second its velocity changes to 10 m/s due north. Find the change in the magnitude of velocity.

    Solution

    Here the particle is moving with a constant velocity. Hence, the magnitude of change in velocity is 0. 

    Question 270
    CBSEENPH11018434
    Wired Faculty App

    Name one quantity that may remain constant when the body follows the curved path.

    Solution
    When the body follows a curved path, speed and kinetic energy may remain constant. 
    Question 271
    CBSEENPH11018435
    Wired Faculty App

    A man dive in the river perpendicular to the direction of flow. Does he reach the point directly opposite to the point of start?

    Solution
    No, since the man is diving perpendicular to the direction of flow, he will reach the point, a certain distance below the point directly opposite to the point of start.
    Question 272
    CBSEENPH11018436
    Wired Faculty App

    How a swimmer should dive to cross the river straight across?

    Solution
    To cross the river straight across, the swimmer should dive at an angle less than 90° with upstream.
    Question 273
    CBSEENPH11018437
    Wired Faculty App

    A river d km wide is flowing at rate straight v subscript 1 straight m divided by straight s. A swimmer who can swim with velocity straight v subscript 2 straight m divided by straight s relative to water wants to cross the river straight across. How long will he take to cross the river?
    Solution

    Width of the river = d km 
    Rate of flow of river = v1 m/s
    Rate at which the swimmer is swimming = v2 m/s
    So, time taken by the man to cross the river is, 
                            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 274
    CBSEENPH11018438
    Wired Faculty App

    A river d km wide is flowing at rate straight v subscript 1 straight m divided by straight s. A swimmer who can swim with velocity straight v subscript 2 straight m divided by straight s relative to water, dives into the river perpendicular to the direction of flow to cross the river. How far he will go down the river along the river.
    Solution

    Given, 
    Width f the river = d
    Rate of flow of river = v1 m/s
    Rate of flow of the swimmer = v2 m/s
    Distance moved by the swimmer along the river is given by, 
                            straight x equals straight d straight v subscript 1 over straight v subscript 2

    Question 275
    CBSEENPH11018439
    Wired Faculty App

    A river d km wide is flowing at rate <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> A swimmer which can swim with velocity straight v subscript 2 relative to water, dives into the river perpendicular to the direction of flow to cross the river. What is the displacement of swimmer when he reaches opposite side of the river?
    Solution

    Width if the river = d km 
    Rate of flow of river = v1 m/s
    Rate of flow of swimmer = v2 m/s
    Given that, the swimmer is diving in a direction perpendicular to the flow of river, to cross the river. 
    Therefore, the displacement of the swimmer when he reaches the opposite side of the river is given by, 
                             straight x equals straight d square root of 1 plus straight v subscript 1 squared over straight v subscript 2 squared end root

    Question 276
    CBSEENPH11018440
    Wired Faculty App

    A boy sitting in a train moving with constant velocity throws a ball straight up into air. Will the ball fall behind him?

    Solution
    The ball is thrown in vertically upward direction and given that the train is moving up with a constant velocity. therefore, the horizontal velocity of the train and ball are both same. And, the ball travels the same horizontal distance as that of the train. 
    Therefore, the ball will come back into the hands of thrower. 
    Question 277
    CBSEENPH11018441
    Wired Faculty App

    A particle moves in a plane with constant acceleration in direction different from initial velocity. Can ever the particle have velocity parallel to acceleration? Explain your answer. 
    Solution
    A component of initial velocity is in a perpendicular direction to the direction of acceleration. It is given that acceleration remains constant and cannot be reduced to zero. 
    Velocity will be parallel to acceleration if and only if, component of velocity perpendicular to acceleration is zero. 
    Therefore the particle moving in a plane with constant acceleration in direction different from initial velocity can never have velocity parallel to acceleration.
    Question 278
    CBSEENPH11018442
    Wired Faculty App

    A particle has arbitrary motion in the space. Which of the following statements are true or false?

    (i)  Total length of path is always equal to magnitude of displacement of particle.

    (ii)  Average speed is always less than instantaneous speed.

    (iii) Position of particle at any instant is given by straight r with rightwards arrow on top equals stack straight r subscript 0 with rightwards arrow on top plus straight u with rightwards arrow on top straight t plus 1 half straight a with rightwards arrow on top straight t squared.

    (iv) Average acceleration of motion is zero if initial and Final velocity of a particle is same.
    Solution

    (i) False, the total length of path is either greater or equal to magnitude of displacement. This statement is true only if the particle is moving in a straight path.
    (ii) False, average speed may be less, equal or greater than the speed at any instant.
    (iii) False, the given relation is true only for uniform accelerated motion.
    (iv)True. For a particle moving with a constant velocity, acceleration will remain the same. 

    Question 279
    CBSEENPH11018443
    Wired Faculty App

    Three girls skating on a circular ice ground of radius 200 m start from point P on the edge of the ground and reach a point Q diametrically opposite to P following different path as shown. What is the magnitude of displacement vector for each? For which girl is this equal to the actual length of path skate? 

                          
    Solution
    Given, 
    Radius of the skating ground = 200 m 
    The initial and final position of all grls are same. Hence, displacement of each girl is same. 
    ∴    Displacement equals PQ with rightwards arrow on top 
    Magnitude of displacemnt is given by,
    open vertical bar Displacement close vertical bar space equals space 2 space left parenthesis radius space of space track right parenthesis space equals space 400 straight m 
    From the figure, girl B skates along the straight path PQ. Therefore, for girl B, the magnitude of displacement and distance will be equal. 
     
    Question 280
    CBSEENPH11018444
    Wired Faculty App

    A truck travelling due north at 20 m/s turns left and travels at same speed. Find the change in velocity.

    Solution

    Consider, unit vector along north is straight i with hat on top and that along west is straight j with hat on top. 
    Thus,
    Velocity of truck travelling due north,  stack straight v subscript 1 with rightwards arrow on top space equals space 20 space straight i with hat on top
    Velocity of truck west,  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
     
    Now the change in velocity of truck is,
             increment straight v with rightwards arrow on top equals stack straight v subscript 2 with rightwards arrow on top minus stack straight v subscript 1 with rightwards arrow on top equals 20 straight j with hat on top minus 20 straight i with hat on top
                   equals 20 square root of 2 space due space SW 

    Question 281
    CBSEENPH11018447
    Wired Faculty App

    The position of particle at any instant is given by straight r with rightwards arrow on top equals 2 straight t squared straight i with hat on top plus 4 straight t straight j with hat on top. What is the displacement of particle in first 3 seconds?
    Solution

    Given,
    Position of particle at any instant, straight r with rightwards arrow on top left parenthesis straight t right parenthesis space equals space 2 straight t squared straight i with hat on top plus 4 straight t straight j with hat on top
    ∴
    Initial displacement of the particle is,    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and
    Displacement of particle when t =3 sec is,
         space straight r with rightwards arrow on top left parenthesis 3 right parenthesis space equals space 2 left parenthesis 3 right parenthesis squared straight i with hat on top plus 4 left parenthesis 3 right parenthesis straight j with hat on top equals 18 straight i with hat on top plus 12 straight j with hat on top
    Displacement of particle in first three sec. is,
                space space straight r with rightwards arrow on top left parenthesis 3 right parenthesis minus straight r with rightwards arrow on top left parenthesis 0 right parenthesis space equals space 18 straight i with hat on top plus 12 straight j with hat on top       

    Question 282
    CBSEENPH11018448
    Wired Faculty App

    The position of a particle is given by
                    space straight r with rightwards arrow on top equals 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top
    where t is in second and r is in meter.
    (a) Find the straight v with rightwards arrow on top space and space straight a with rightwards arrow on top of the particle.
    (b) What is the magnitude and direction of velocity of the particle at t = 2s?
    Solution

    (a)
    Given,
    Position of the particle,<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴
    Velocity,
    space space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space space space space equals straight d over dt left parenthesis 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top right parenthesis
    and,
    Acceleration,space space space straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis space equals straight d over dt left parenthesis 3 straight i with hat on top minus 4 straight t straight j with hat on top right parenthesis
                            equals negative 4 straight j with hat on top
    (b) Now,
    Velocity,space space straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight t straight j with hat on top
    At time = 2 sec, 
               straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 8 straight j with hat on top
    ∴  open vertical bar straight v with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 8 right parenthesis squared end root space equals space square root of 74 space equals space 8.54 space straight m divided by straight s
    and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    At t = 2 sec,
    Acceleration, straight a with rightwards arrow on top space equals space minus 4 straight j with hat on top
    ∴        open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 4 and directed towards negative direction of X-axis. 

    Question 283
    CBSEENPH11018449
    Wired Faculty App

    The position of particle at any instant is given by straight r with rightwards arrow on top equals 50 straight t straight j with hat on top plus 5 straight t squared straight i with hat on top. At what instant the velocity vector will make an angle 45° with X-axis?

    Solution

    Given,
    Position of the particle, straight r with rightwards arrow on top space equals space 50 straight t straight j with hat on top plus 5 straight t squared straight i with hat on top
    ∴ Velocity, straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space straight d over dt left parenthesis 50 straight t space straight j with hat on top plus 5 straight t squared straight i with hat on top right parenthesis 
                     equals 50 straight j with hat on top plus 10 straight t space straight i with hat on top
    The angle which the velocity vector makes with X-axis at any instant is,
    tanθ space equals space straight v subscript straight y over straight v subscript straight x equals fraction numerator 50 over denominator 10 straight t end fraction 
    Here, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore,
            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow          straight t equals 5 straight s
    So, when time is 5 seconds, the velocity vector will make an angle 450 with x-axis. 

    Question 284
    CBSEENPH11018451
    Wired Faculty App

    A cyclist starts from the center O of circle of park of radius 1 km, reaches the edge P of the park, then cycles along the circumference to point Q and returns to the centre along QO as shown. If the round trip takes 10 minutes, then what is:


    (a) net displacement
    (b) average velocity
    (c) average speed of the cyclist?

    Solution

    (a) The cyclist starts from O and again returns to point O. Therefore, the initial and the final point are the same. Hence, displacement of cyclist is equal to zero.
    (b) Average velocity of cyclist is zero because total displacement of cyclist is zero.
    (c) Total distance travelled by cyclist is,
                 straight S equals OP plus PQ with overparenthesis on top plus QO
                   equals straight r plus πr over 2 plus straight r equals straight r open parentheses fraction numerator 4 plus straight pi over denominator 2 end fraction close parentheses 
    ∴      Average speed  = straight S over straight t equals open parentheses fraction numerator 4 plus straight pi over denominator 2 end fraction close parentheses straight r over straight t
    Here, we have
    Radius of the circle park, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and
    Time taken for round trip,space space space straight t equals 10 min equals 600 straight s
    ∴  Average space speed space equals space open parentheses fraction numerator 4 plus straight pi over denominator 2 end fraction close parentheses 1000 over 600 
                            space space equals 5.95 space straight m divided by straight s 

    Question 285
    CBSEENPH11018456
    Wired Faculty App

    A girl walks 100 m at angle of 30° North of West. What is the ratio of the distance travelled by the girl in North to West direction?

    Solution
    Total distance covered by the girl is equal to total displacement = 100m
    Angle, straight theta = 30o
    Magnitude of the component of displacement along North is, straight S subscript 1 equals 100 space sin 30 degree
    Magnitude of the component of displacement along West is,space straight S subscript 2 equals 100 space cos space 30 degree
    Therefore, the ratio of the distance travelled by the girl in North to West direction is given by, 
      <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>
    Question 286
    CBSEENPH11018463
    Wired Faculty App

    A stone of mass 10 kg is placed on the horizontal floor and it is pulled by a force F acting at angle 45°. What is tension in the string when it breaks off from the ground? (g = 10m/s2).

    Solution
    Given, 
    Mass of the stone, m = 10kg
    Angle at which the force is acting = 45o

    When vertical component of the applied force is equal to the weight of stone, the stone will lose contact. 
    i.e., F sin45o = 100 
    rightwards double arrow space space straight F space equals space fraction numerator 100 over denominator sin space 45 to the power of straight o end fraction space equals space fraction numerator 100 over denominator begin display style bevelled fraction numerator 1 over denominator square root of 2 end fraction end style end fraction equals 100 square root of 2 space straight N
    Therefore tension in string is 100 square root of 2 straight N.
    Question 287
    CBSEENPH11018467
    Wired Faculty App

    An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10s apart is 30°, what is the speed of the aircraft?

    Solution

    The aircraft takes 10s to go from A to B and AB subtends an angle of 30° at O.
    Height of the aircraft from the groun = 3400 m
    In right angled triangle OAC,

    AC = OC tan 15°
         = 3400 x 0·2679
         = 911m
    ∴   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Now, the speed of aircraft is,
    space space straight v equals AB over straight t equals 1822 over 10 equals 182.2 space straight m divided by straight s 

    Question 288
    CBSEENPH11018473
    Wired Faculty App

    A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest taxi driver takes him along a circuitous path 23 km long and reaches the hotel in 28 minutes.

    What is:

    (a) the average speed of the taxi?

    (b) the magnitude of average velocity? Are the two equal? 
    Solution

    Given,
    Magnitude of displacement = 10 km
    Total path length = 23 km
    Time taken = 28 min = 7/15 hr.
    Therefore,
    space space Average space speed space equals space fraction numerator Distance space travelled over denominator Time end fraction
                            equals fraction numerator 23 over denominator 7 divided by 15 end fraction equals 49.3 space km divided by hr
    Magnitude of average velocity is,
          <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore, we can see that the average speed and magnitude of average velocity are not equal.

    Question 289
    CBSEENPH11018474
    Wired Faculty App

    Two particles A and B start from P and move along a circular path in anticlockwise and clockwise direction respectively. They complete one fourth of the journey in same time. Do they have same average velocities?

                   
    Solution
    The particles undergo different displacement. So, the two particles do not have the same average velocity. 
    Here, the magnitude of displacement, that is, the distance travelled by the two partcles is same. Therefore, they may have the same speed. 
    Question 290
    CBSEENPH11018475
    Wired Faculty App

    Two particles A and B start from P and move along a circular path; in anticlockwise and clockwise direction respectively as shown in above figure. Both the particles are moving with same speed. When will both the particles possess the same average velocities? 
    Solution
    The particles are moving along a circular path, but in opposite directions. So, they do not have the same average velocity. 
    The particles will have the same average velocity, when they are at a point diametrically opposite to each other. Because at this point, both particles will undergo the same displacement in the same time. Hence, at this point the particle will have same velocity. 
    Question 291
    CBSEENPH11018478
    Wired Faculty App

    What is the displacement of a point on the wheel of radius R initially in contact with ground when the wheel rolls forward half a revolution?

    Solution

    Given, 
    Radius of the wheel = R 
    When half revolution is made by the wheel, the point of contact moves from point A to C and hence undegoes the displacement stack AC. with rightwards arrow on top 

    Distance displaced by the point of contact, AB = pi R 
    Perpendicular distance traversed by the body = 2R
    Therefore, total distance, 

    A C equals space square root of AB squared plus BC squared end root space space space space space space equals space square root of left parenthesis pi R right parenthesis squared plus left parenthesis 2 R right parenthesis squared end root space space space space space space equals R square root of pi squared plus 4 end root space

    and tan space straight theta space equals space BC over AB equals 2 over straight pi 
    Thus the point of contact undergoes the displacement of magnitude straight R square root of straight pi squared plus 4 end root making an angle straight theta equals tan to the power of negative 1 end exponent 2 over straight pi with the ground.

    Question 292
    CBSEENPH11018484
    Wired Faculty App

    A man rows a boat with a speed of 18 km/hr in north-west direction. The shore line makes an angle of 15° south of west. Obtain the components of velocity of boat along shore line and perpendicular to shore line.

    Solution
    The velocity of boat is 18 km/hr in N-W direction and shore line makes angle 15° with S of W.
    Therefore the velocity vector of boat subtends an angle of 60° with shore line.

    Component of velocity of boat
    along shore line = 18 space cos space 60 degree space equals space 9 space km divided by hr.
    And component of velocity of boat perpendicular to shore line <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 293
    CBSEENPH11018486
    Wired Faculty App

    A man can swim with a speed of 4.0 km/hr in still water. How long does he take to cross the river 1.0 km wide if the river flows steadily at 3.0 km/hr and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?

    Solution
    Rate of flow of swimmer in the still water = 4.0 km/hr 
    Width of the river = 1.0 km 
    Rate of flow of river = 3.0 km/hr 
                       
    Since the swimmer dives in the river normal to the flow of river, therefore time taken by swimmer to cross the river is, 
    space space space space space space space straight t equals straight d over straight v subscript straight m equals fraction numerator 1 km over denominator 4 km divided by hr end fraction equals 1 fourth hr equals 15 space min
    In this time the swimmer will also go down the river by distance BC due to river current. 
    ∴ Distance of the river travelled by the swimmer when he reaches the other bank is given by, 
     BC space equals space straight v subscript straight r straight t space space space space space space space equals space left parenthesis 3 km divided by hr right parenthesis open parentheses 1 fourth hr close parentheses 
          equals 3 over 4 km equals 0.75 km
    Question 294
    CBSEENPH11018489
    Wired Faculty App

    A river d km wide is flowing at the rate straight v subscript 1 straight m divided by straight s. A swimmer which can swim with velocity straight v subscript 2 relative to water wants to cross the river straight across. Find how long will he take to cross the river?
    Solution
    The path AB is the shortest path to cross the river.
    To cross the river straight across along path AB, the resultant velocity of swimmer and the water flow should be along AB i.e. perpendicular to the direction of water flow.
               
    To meet this condition, let the swimmer start swimming upstream at angle 0 with AB.
    Therefore,
    tan 90 degree space equals space fraction numerator straight v subscript 2 sin left parenthesis 90 degree plus straight theta right parenthesis over denominator straight v subscript 1 plus straight v subscript 2 cos left parenthesis 90 degree plus straight theta right parenthesis end fraction rightwards double arrow space space space space space space space straight v subscript 1 plus straight v subscript 2 space cos left parenthesis 90 degree plus straight theta right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space straight v subscript 1 minus straight v subscript 2 space sinθ space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space sinθ space equals space straight v subscript 1 over straight v subscript 2 
    Now,
     
             straight v equals straight v subscript 2 space cosθ space equals space straight v subscript 2 square root of 1 minus sin squared straight theta end root space space equals space straight v subscript 2 square root of 1 minus straight v subscript 1 squared over straight v subscript 2 squared end root space equals space square root of straight v subscript 2 squared minus straight v subscript 1 squared end root 
    Time taken by the swimmer to cross the river is,
            space space space space space space space space space straight t equals straight d over straight v equals fraction numerator straight d over denominator square root of straight v subscript 2 squared minus straight v subscript 1 squared end root end fraction
    Question 295
    CBSEENPH11018490
    Wired Faculty App

    A river d km wide is flowing at a rate straight v subscript 1 straight m divided by straight s. A swimmer which can swim with velocity straight v subscript 2 relative to water, dives into the river perpendicular to the direction of flow to cross the river. How far he will go down the river along the river?
    Solution
    When the swimmer dives perpendicular to river, then time taken by swimmer to cross the river is,
                               straight t equals straight d over straight v subscript 2
                
    The distance which the swimmer will go down the river along the river is,
                          BC equals straight v subscript 1 straight t space space space space space equals straight d straight v subscript 1 over straight v subscript 2
    Question 296
    CBSEENPH11018492
    Wired Faculty App

    A river 100 m wide is running downwards at the rate of 1 km per hour and a swimmer who can swim at the rate of 3·6 km/hr wishes to reach a point just opposite. Along which direction should he strike and how long will he take to cross the river?

    Solution
    Given, 
    Width of the river = 100 m 
    Rate of flow of river = 1 km/hr
    Rate of flow of swimmer = 3.6 km/hr

    Let the swimmer dive at an angle straight theta as shown in figure to reach the point B directly opposite to A. 
    From the fig., we have
    ∴     straight v subscript straight m sinθ space equals space straight v subscript straight r 
    rightwards double arrow      sinθ space equals space straight v subscript straight r over straight v subscript straight m equals fraction numerator 1 over denominator 3.6 end fraction equals 0.278 
    rightwards double arrow           straight theta equals 16.14 degree 
    Therefore, the swimmer should dive at angle (90 + 16·14) =106·14° with the direction of flow.
    Now,
    Time taken to cross the river, t = fraction numerator AB over denominator straight v subscript straight m cosθ end fraction
                                                  equals fraction numerator 0.1 over denominator 36 cross times 16.14 end fraction
                                                  equals 0.0289 space hr equals space 104.1 space straight s
    Question 297
    CBSEENPH11018496
    Wired Faculty App

    Two men A and B can swim in still water, at the rate of 5km/hr. The man A swims upstream a distance half km equal to width of river and B swims straight across the river and back again. Assuming that the velocity of river is 3 km/hr, find the difference in the time taken by them.

    Solution

    Here, velocity of man is, v = 5 km/hr


    Velocity of river is, u = 3 km/hr, along QP.


    Swimmer A:
    Velocity of swimmer A w.r.t. bank while going from P to Q = v - u
    Velocity of swimmer A w.r.t. bank while coming back from Q to P = v + u
    Time taken by swimmer A to make the round trip is,
                 straight t subscript 1 equals fraction numerator straight d over denominator straight v minus straight u end fraction plus fraction numerator straight d over denominator straight v plus straight u end fraction equals fraction numerator 2 vd over denominator straight v squared minus straight u squared end fraction

    rightwards double arrow                               space straight t subscript 1 equals fraction numerator 2 cross times 5 cross times begin display style 1 half end style over denominator 5 squared minus 3 squared end fraction space space space equals 5 over 16 hr space space space equals space 18 space min space 45 sec

    Swimmer B:
    To cross the river straight across man will dive at angle straight theta given by,
                          sinθ space equals straight u over straight v equals 3 over 5

    Therefore time taken by man B to make the round trip is,


      straight t subscript 2 equals 2 cross times fraction numerator straight d over denominator straight v space cosθ end fraction space space equals 2 cross times fraction numerator 1 divided by 2 over denominator 5 cross times 0.8 end fraction space space equals 1 fourth hr space space equals 15 min
    The difference of the time taken by two swimmers is,
                    increment straight t space equals space straight t subscript 1 minus straight t subscript 2 space space space space space space space equals space 3 space min space 45 space sec. 

    Question 298
    CBSEENPH11018497
    Wired Faculty App

    A stream of width d is flowing with velocity v. A person who can swim with speed V relative to water wishes to row straight across it. In what direction should he strike out and how long would it take him to cross the stream?
    Solution
    To row straight across the stream, the person must strike out at angle θ upstream, such that the resultant R of his own velocity and velocity of stream is perpendicular to velocity of stream.
    ∴     tanθ equals fraction numerator straight V space sin left parenthesis 90 degree plus straight theta right parenthesis over denominator straight v plus Vcos left parenthesis 90 degree plus straight theta right parenthesis end fraction
            straight v plus straight V space cos left parenthesis 90 degree plus straight theta right parenthesis space equals 0
                        straight v minus Vsinθ space equals space 0
                                sinθ equals straight v over straight V
    ∴                         cosθ equals square root of fraction numerator straight V squared minus straight v squared over denominator straight V end fraction end root 
    The swimmer should dive at an angle straight theta equals sin to the power of negative 1 end exponent left parenthesis straight v divided by straight V right parenthesis with normal to river and upstream. 
    Now the resultant velocity of swimmer and the stream is, 
                    Error converting from MathML to accessible text.
                    Error converting from MathML to accessible text.
    rightwards double arrow            Error converting from MathML to accessible text.
    Therefore the time taken by swimmer to cross the river is,
                     Error converting from MathML to accessible text.
    Question 299
    CBSEENPH11018498
    Wired Faculty App

    Point mass slides from rest from the topmost point of vertical circle of radius r along the smooth chord. Show that the time of descend of mass along the chord is independent of chord chosen.

    Solution

    Given, 
    Radius of the circle = r 
                 
    Consider, an arbitrary chord AB which subtends an angle straight theta with vertical.
    Now, the length of chord AB is given by, 

    AB equals square root of straight r squared plus straight r squared minus 2 rrcos left parenthesis 180 degree minus 2 straight theta right parenthesis end root

       equals square root of 2 straight r squared left parenthesis 1 plus cos space 2 straight theta right parenthesis end root equals space 2 straight r space cosθ                   
    Acceleration of mass along inclined plane is, straight a equals gcosθ

    Now, using the equation of motions,
                         straight S equals ut plus 1 half at squared 
    ∴         2 straight r space cosθ space equals space 0 cross times straight t plus 1 half straight g space cosθt squared 
    rightwards double arrow      straight t equals square root of fraction numerator 4 straight r over denominator straight g end fraction end root equals 2 square root of straight r over straight g end root
    Since t is independent of straight theta, hence time of descend of mass is independent of the choice of chord. 

    Question 300
    CBSEENPH11018499
    Wired Faculty App

    What is a projectile?

    Solution
    Projectile is the name given to a body thrown with some initial velocity with the horizontal direction, and then allowed to move in two dimensions under the action of gravity alone, without being propelled by any engine or fuel. 
    Question 301
    CBSEENPH11018500
    Wired Faculty App

    What is the name given to the path followed by a projectile?

    Solution
    Trajectory.
    Question 302
    CBSEENPH11018501
    Wired Faculty App

    What is the shape of trajectory of a projectile?

    Solution
    Parabola.
    Question 303
    CBSEENPH11018502
    Wired Faculty App

    Can the trajectory of projectile be a straight line?

    Solution
    Yes, when the projectile is thrown in vertically upward direction, the trajectory is a straight line.
    Question 304
    CBSEENPH11018503
    Wired Faculty App

    Can the moon be taken as an example of a projectile? 

    Solution
    Yes, moon is orbiting around earth and has it's x and y motions inter-connected. 
    Question 305
    CBSEENPH11018504
    Wired Faculty App

    A projectile is projected with velocity u in horizontal direction. Write the expression for the trajectory of projectile.

    Solution

    When the body is projected with velocity 'u' in the horizontal direction, expression for trajectory of projectile is given by, 
                             Error converting from MathML to accessible text. 
    where, 
    'g' is acceleration de to gravity,
    'x' is the horizontal distance covered by the particle, 
    'u' is the initial velocity of the particle, and 
    'y' is the object in the vertical direction. 

    Question 306
    CBSEENPH11018505
    Wired Faculty App

    A projectile is projected in horizontal direction with velocity u from height h. How long will it take to reach the ground?

    Solution

    Velocity of the particle = u 
    Height at which the particle is projected = h
    Let, time of flight be T
    Using the equation, 
    y = yo +uyt + <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>
    We have, y=h, yo = 0 and uy = 0 
    Therefore, 
    h = 0+0 <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>
    Time taken for the particle to reach the ground, T =square root of fraction numerator 2 straight h over denominator straight g end fraction end root 

    Question 308
    CBSEENPH11018507
    Wired Faculty App

    A body is dropped from some height and the second body is projected horizontally from the same point.
    Which of the two will reach the ground first? 

    Solution
    The acceleration due to gravity acting on both media is the same. Therefore, both the bodies will reach the ground simultaneously. 
    Question 309
    CBSEENPH11018508
    Wired Faculty App

    Does the time taken by horizontal projectile to reach the ground depend on the speed of projection?

    Solution
    No, time taken by horizontal projectile, to reach the ground does not depend on the speed of projection. 
    Question 310
    CBSEENPH11018509
    Wired Faculty App

    A stone is thrown vertically upwards and then it returns to the thrower. Is it a projectile motion? Explain

    Solution
    The motion is not projectile because a projectile should have two component of velocities in two mutually perpendicular directions. But, in this case the stone has velocity only in one direction, while going up or coming downwards. 
    Question 311
    CBSEENPH11018510
    Wired Faculty App

    An airplane moving horizontally drops a packet when it is directly above the point P on the ground. Will it hit the point P? 

    Solution
    No, the packet will not directly hit the point P on the ground. The packet will travel few distance before it hits the ground. 
    Question 312
    CBSEENPH11018511
    Wired Faculty App

    Which quantities remain conserved during the projectile motion? 

    Solution
    Horizontal component of velocity and horizontal component of momentum remain conserved in projectile motion.
    Question 313
    CBSEENPH11018512
    Wired Faculty App

    A man in a moving train drops a stone. What is the path of the stone observed by the man from the train?

    Solution
    When a man drops a stone from a moving train, the path of the stone observed by the man from the train is a straight line.
    Question 314
    CBSEENPH11018513
    Wired Faculty App

    A body is projected with velocity u at angle θ with horizontal. Write the expression for the path followed by body.

    Solution

    Velocity with which the body is projected = u 
    Angle made by the body with horizontal = straight theta 
    The equation of trajectory is given by, 
                   straight y equals straight x space tanθ minus 1 half straight g fraction numerator straight x squared over denominator straight u squared cos squared straight theta end fraction
    This is an equation of a parabole, hence the path of projectile projected at some angle with the horizontal direction from the ground is a parabolic path. 

    Question 316
    CBSEENPH11018515
    Wired Faculty App

    A body is projected with velocity u at angle θ with horizontal. What is the maximum height attained by body?

    Solution

    Maximum height is the maximum vertical height attained by the object above the point of projection during it's flight.
    It is denoted by H and is given by, 
                           straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction

    Question 317
    CBSEENPH11018516
    Wired Faculty App

    A particle is projected with velocity u at angle θ with the horizontal. How long will it take to reach the ground? 

    Solution

    Time taken by the particle to reach the ground, t =fraction numerator 2 straight u space sinθ over denominator straight g end fraction 

    Question 318
    CBSEENPH11018517
    Wired Faculty App

    In how many ways the projectile can be projected to fall at the same distance?

    Solution

    A particle can be projected in two different ways to make them fall at the same distance.

    Question 319
    CBSEENPH11018518
    Wired Faculty App

    At which point of projectile motion, i)P.E is maximum ii) K.E is maximum iii) Total mechanical energy is maximum.

    Solution

    In case of projectile motion, 
    i) P.E is maximum at the highest point.
    ii) K.E of a projectile will be minimum (but not zero) at the highest point, because only vertical component of velocity is zero. K.E is maximum at the place of projection. 
    iii) Total mechanical energy is conserved at all location of it's motion.  

    Question 320
    CBSEENPH11018519
    Wired Faculty App

    At what angle a projectile must be projected so that it falls at the farthest distance?

    Solution
    For a body to fall at the farthest distance, the horizontal velocity and vertical velocity has to be equal. So, the body must be projected at an angle of 45°.
    Question 322
    CBSEENPH11018521
    Wired Faculty App

    What is the maximum distance at which we can throw the body with speed u?

    Solution
    Horizontal range will be maximum when sin 2straight theta is maximum. 
    That is, 
       sin 2straight theta = maximum 
    rightwards double arrow space space space space space space 2 straight theta space equals space sin space 90 to the power of straight o space equals space 1 or space space space space space space space space 2 straight theta space equals space 90 to the power of straight o space straight i. straight e. comma space space space space space space straight theta space equals space 45 to the power of straight o 
    Therefore, maximum horizontal range is given by, 
    straight R subscript straight m space equals space straight u squared over straight g sin space 2 cross times 45 to the power of straight o space space space space space space equals space straight u squared over straight g 
    Question 323
    CBSEENPH11018522
    Wired Faculty App

    A projectile is aimed at angle θ, to hit the target. Can you hit the same target by aiming at some other angle?

    Solution
    Yes, we can hit the same target by aiming at angle (90°–θ).
    Question 324
    CBSEENPH11018523
    Wired Faculty App

    What is the angle between the velocity vector and acceleration vector in uniform circular motion?

    Solution

    The angle between the velocity vector and acceleration vector is 90o. 

    Question 325
    CBSEENPH11018524
    Wired Faculty App

    At what point the momentum of projectile is minimum?

    Solution
    Momentum of projectile is minimum at highest point because at the highest point the speed of the projectile is minimum. 
    Question 326
    CBSEENPH11018525
    Wired Faculty App

    A body is projected from ground at certain angle with horizontal. Is there any point at which acceleration and velocity are parallel to each other?

    Solution
    No, acceleration and velocity are never parallel to each other. 
    At the highest point, acceleration and velocity are perpendicular to each other. 
    Question 327
    CBSEENPH11018526
    Wired Faculty App

    A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?

    Solution

    Let, u be the velocity of projection of the ball. The motion of the ball is along vertical direction. 
    The ball will cover maximmum horizontal distance when angle of projection with horizontal is 45o. 
     straight R subscript max space equals space straight u squared over straight g Here comma space straight u squared over straight g space equals space 100 space straight m comma straight u subscript straight y space equals space straight u comma space straight a subscript straight y space equals space minus straight g comma space straight v subscript straight y space equals space 0 straight y subscript straight o space equals space 0 comma space straight y space equals space ? comma space straight t equals ? As comma space straight v subscript straight y space equals space straight u subscript straight y plus space straight a subscript straight y straight t therefore space space space 0 space equals space straight u space plus space left parenthesis negative straight g right parenthesis straight t space rightwards double arrow space space space straight t space equals space straight u over straight g space Also comma space straight y space equals space straight y subscript straight o space plus space straight u subscript straight y straight t space plus space 1 half straight a subscript straight y straight t squared space therefore space space space space space straight y space equals space 0 space plus space straight u open parentheses bevelled straight u over straight g close parentheses space plus space 1 half open parentheses negative straight g close parentheses straight u squared over straight g squared space space space space space space space space space space space space space equals straight u squared over straight g space minus space 1 half straight u squared over straight g space space space space space space space space space space space space space space equals 1 half straight u squared over straight g space equals space 100 over 2 space equals space 50 space straight m

    Question 328
    CBSEENPH11018527
    Wired Faculty App

    Two projectiles are projected from the ground simultaneously with speeds 30 m/s and 20 m/s at angle 60o and 30o with the horizontal in the same direction. Find the maximum distance between them on ground where they strike. 

    Solution

    Horizontal range is given by, 
     straight R subscript 1 space equals space straight u subscript 1 squared over straight g space sin space 2 straight theta subscript 1 space space space space space space equals space open parentheses 30 close parentheses squared over 10 sin space 2 space straight x space 60 to the power of straight o space space space space space space space equals space 90 space straight x space square root of 3 divided by 2 space equals space 45 square root of 3 space straight m space straight R subscript 2 space equals space straight u subscript 2 squared over straight g space sin space 2 straight theta subscript 2 space space space space space space equals space fraction numerator left parenthesis 20 right parenthesis squared over denominator 10 end fraction straight x space sin space 2 space straight x space 30 to the power of straight o space space space space space equals space 20 space square root of 3 space straight m space therefore space straight R subscript 1 space minus space straight R subscript 2 space equals space space 45 square root of 3 space straight m space minus 20 space square root of 3 space straight m space space space space space space space space space space space space space space space space space space space space space space space equals space 25 square root of 3 space straight m space
    This is the required maximum distance between the two projectiles when they strike the ground. 

    Question 329
    CBSEENPH11018528
    Wired Faculty App

    What is the maximum range of projectile projected with velocity numerically equal to g?

    Solution

    The maximum range of projectile projected with velocity numerically equal to 9.8 is  'g' (i.e., 9.8) meter.

    Question 330
    CBSEENPH11018529
    Wired Faculty App

    What is the angular velocity in radian per second of a flywheel making 300 r.p.m? 

    Solution

    Here, 
    Frequency = 300 r.p.m = 300 over 60 space equals space 5 space r. p. s
    Angular velocity of the flywheel,  
    straight omega space equals space 2 πν space equals space 2 space straight x space 22 over 7 straight x space 5 space equals space 31.4 space rad divided by sec

    Question 331
    CBSEENPH11018530
    Wired Faculty App

    The horizontal range of projectile is 4 times the maximum height attained by the projectile. What is the angle of projection?

    Solution
    Given, 
    Horizontal range is equal to 4 times the maximum height attained by the projectile. 
    Therefore, 
    space space space space space space fraction numerator straight u squared over denominator 2 straight g end fraction sin 2 theta space equals 4 space fraction numerator u squared sin squared theta over denominator 2 g end fraction i. e. comma space space space space sin space 2 theta space equals space 4 sin squared theta space space left square bracket because space sin space 2 theta space equals space 2 space sin theta space cos theta right square bracket rightwards double arrow space space space space space space space space space fraction numerator sin space theta over denominator cos space theta end fraction space equals space 1 rightwards double arrow space space space space space space space space space space space tan space theta space equals space 1 rightwards double arrow space space space space space space space space space space space space theta space equals space 45 to the power of o
    So, angle of projection is 45°.
    Question 332
    CBSEENPH11018531
    Wired Faculty App

    The range of a particle, when launched at an angle of 15o with the horizontal, is 1.5 km. What is the range of the projectile when launched at an angle of 45o to the horizontal? 
    Solution

    Given that, 

    space space space space space space fraction numerator straight u squared space sin space 2 space straight x space 15 to the power of straight o over denominator straight g end fraction space equals space 1.5 space rightwards double arrow space space space space space space space space space space straight u squared over straight g space straight x space 1 half space equals space 1.5 rightwards double arrow space space space space space space space space space space space space space space straight u squared over straight g space equals space 3 space km space 
    Horizontal range for angle of projection 45o will be, 

    straight R space equals space straight u squared over straight g space sin space 2 space straight x space 45 to the power of straight o space space space space space equals straight u squared over straight g space sin space 90 to the power of straight o space equals space 3 space km space
    That is the range of projectile is 3 km. 

    Question 335
    CBSEENPH11018534
    Wired Faculty App

    How does the horizontal component and vertical component of velocity of horizontal projectile change with the time?

    Solution
    The horizontal component of velocity remains constant while vertical component of velocity increases with the time. 
    Question 336
    CBSEENPH11018535
    Wired Faculty App

    Two projectiles are fired from the same point in the same plane at angles 23° and 67°. Which of the two projectiles falls at far distance?

    Solution
    Both the projectiles will reach the ground at the same point.
    Question 337
    CBSEENPH11018536
    Wired Faculty App

    Two projectiles are fired from the same point in the same plane at angles 23° and 67°. Which of the two projectiles reaches the ground earlier?

    Solution
    The projectile which is inclined at a lesser angle will reach the ground earlier. 
    That is, the body which is projected at angle 23° will reach the ground first. 
    Question 338
    CBSEENPH11018537
    Wired Faculty App

    Two projectiles are projected at angles θ and (90°–θ) with same speed. What is the ratio of their time of flight?

    Solution
    Time of flight is given by fraction numerator 2 space straight u space sinθ over denominator straight g end fraction
    So ratio of time of flight = fraction numerator begin display style bevelled fraction numerator 2 space straight u space sinθ over denominator straight g end fraction end style over denominator begin display style bevelled fraction numerator 2 space straight u space sin space left parenthesis 90 to the power of straight o minus straight theta right parenthesis over denominator straight g end fraction end style end fraction space 
                                        equals space fraction numerator sin space straight theta over denominator cos space straight theta end fraction space equals space tan space theta space thin space colon space 1

    Question 339
    CBSEENPH11018538
    Wired Faculty App

    At what angle the projectile must be projected so that time of flight is maximum?

    Solution
    Time of flight = fraction numerator 2 usinθ over denominator straight g end fraction 
    For maximum time of flight sin straight theta = 1
    So angle of the projectile is 90o. 
    Question 340
    CBSEENPH11018539
    Wired Faculty App

    An object is projected with kinetic energy K at an angle θ. What is the kinetic energy of projectile at the highest point?

    Solution
    Kinetic energy of projectile at the highest point is given by,
    K.E = K cos2θ
    Question 341
    CBSEENPH11018540
    Wired Faculty App

    An object is projected with kinetic energy K at an angle θ. What is the ratio of kinetic energy to the potential energy of the projectile at the highest point?

    Solution
    The ratio of the kinetic energy to the potential energy of projectile at the highest point is cot2θ.
    Question 342
    CBSEENPH11018541
    Wired Faculty App

    A ball thrown up is caught by the thrower 4 s after start. What is the height to which the ball has risen?

    Solution

    Time of ascent = Time of descent. 
    Therefore, 
    t1 + t2 = 4
    t2 + t2 = 4
    This implies, 
    t2 = 2 sec
    Now, taking vertical downward motion of the ball from the highest point to surface of earth we have, 
    u = 0, a = 10 m/s, t = 2 s, s = ? 
    Now, using the formula,
                          s = ut + 1/2 at2
                             = space 0 space plus space 1 half straight x 10 space straight x space left parenthesis 2 right parenthesis squared 
                             = 20 m
    This is the height to which the ball has risen. 

    Question 343
    CBSEENPH11018542
    Wired Faculty App

    What is the angle between instantaneous velocity and acceleration at the highest point of trajectory?

    Solution
    At the highest point of trajectory, velocity becomes zero. 
    So, the angle between instantaneous velocity and acceleration at the highest point of trajectory is 90°.
    Question 344
    CBSEENPH11018543
    Wired Faculty App

    A body is projected with velocity u at angle θ with horizontal. At what instant will it make an angle α with the horizontal?

    Solution

    The instant at which the body will make an angle straight alpha with the horizontal is given by, 
                       ucosθ over straight g left parenthesis tanθ minus tanα right parenthesis

    Question 345
    CBSEENPH11018544
    Wired Faculty App

    What is a projectile and what are the assumptions made in the study of projectile motion? 

    Solution

    A body projected with some initial velocity and then allowed to fall only under gravity is called a projectile. 
    Assumptions made in the study of projectile motion: 
    (i) Acceleration due to gravity ‘g’ is constant both in magnitude and direction.
    (ii) There is no resistance due to air.
    (iii) The earth is flat.
    (iv) Rotational motion of the earth is absent.

    Question 346
    CBSEENPH11018545
    Wired Faculty App

    A projectile is projected horizontally with velocity u and allowed to fall only under gravity. Discuss the trajectory of projectile. 

    Solution
    Consider a projectile, which is projected with velocity 'u' in horizontal direction from a point at height h from the ground. 
    Let origin be the point of projection. 
    The right direction is positive direction of X-axis and vertically downward direction is positive direction of Y-axis.
    The component of velocity and acceleration is resolved into horizontal and vertical components.
    So, 
                        straight x subscript straight o equals 0 space space space space space space space space space space space space space space space space space space space straight y subscript straight o equals 0 straight u subscript straight x space equals straight u space space space space space space space space space space space space space space space space space space straight u subscript straight y space equals space 0 straight a subscript straight x space equals space 0 space space space space space space space space space space space space space space space space space straight a subscript straight y space equals space straight g 
    Motion along horizontal direction is, 
         space space space space space space space space space straight x equals straight x subscript straight o plus straight u subscript straight x straight t plus 1 half straight a subscript straight x straight t squared equals ut              ...(1) 
    Motion along vertical direction is,
                straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared equals 1 half gt squared         ...(2)
    Eliminating 't' from (1) and (2), we get
                            straight y equals open parentheses fraction numerator straight g over denominator 2 straight u squared end fraction close parentheses straight x squared 
    This is the required equation of trajectory.
    Since the equation of trajectory is an equation of parabola, therefore path of projectile is parabola.
    Question 347
    CBSEENPH11018546
    Wired Faculty App

    A projectile is projected horizontally with velocity u. What will be the velocity of the projectile after time t ?

    Solution
    Let a projectile be projected with velocity u in horizontal direction.

    If straight v subscript straight x space and space straight v subscript straight y are X and Y components of velocity at any instant, then
             straight v subscript straight x equals straight u subscript straight x plus straight a subscript straight x straight t space space equals space u space a n d straight v subscript straight y equals straight u subscript straight y plus straight a subscript straight y straight t space equals space gt therefore space space space space straight v with rightwards arrow on top space equals space straight v subscript straight x straight i with hat on top space plus space straight v subscript straight y straight j with hat on top 
    Now manitude of resultant velocity is,
           space space straight v equals square root of straight v subscript straight x superscript 2 plus straight v subscript straight y superscript 2 end root space space space space equals space square root of straight u squared plus straight g squared straight t squared end root
    The angle which the velocity vector makes with horizontal is,
                                 tanθ equals gt over straight u

    Question 348
    CBSEENPH11018547
    Wired Faculty App

    An aeroplane is flying in horizontal direction with velocity u at a height h. How far from the target, should the bomb be released from it so as to hit the target?

    Solution
    Let the aeroplane be flying at height h in horizontal direction with velocity u.
    Let the bomb be dropped from O to hit the target.
    Let t be time taken by the bomb to reach the ground at target.
                     
    We know,      straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 over straight a straight a subscript straight y straight t squared
    ∴                   straight h equals 0 plus 0 plus 1 half gt squared
    i.e.,                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Distance travelled by bomb in horizontal direction in time t is,
                       straight x equals straight R equals ut equals straight u square root of fraction numerator 2 straight h over denominator straight g end fraction end root
     Bomb must be dropped at a distance= space space straight u square root of fraction numerator 2 straight h over denominator straight g end fraction end root from the target. 
    Question 349
    CBSEENPH11018548
    Wired Faculty App

    A projectile is fired with velocity u horizontally from height h. With what velocity will it strike the ground?

    Solution

    Consider a projectile being projected with velocity 'u' horizontally from height 'h'.
    Let the projectile strike the ground with velocity 'v' after time t. 
                  
    Now, on resolving the components of velocity aong vertical and horizontal direction, we have 
    straight u subscript straight x space equals space straight u space and space straight u subscript straight y equals 0 straight a subscript straight x space equals space 0 space and space straight a subscript straight y space equals space straight g space
    Time taken by the body to reach the ground: 
    We know that, 
    space space space space space space space space space straight y space equals space straight y subscript straight o space plus space straight u subscript straight y straight t space plus space 1 half straight a subscript straight y straight t squared space rightwards double arrow space space space space space straight h space equals space 0 plus 0 plus 1 half gt squared rightwards double arrow space space space space space space straight t space equals space square root of fraction numerator 2 straight h over denominator straight g end fraction end root
    Velocity of the body at ground: 
    Let, vx and vy be the components of velocity along X-axis and Y-axis respectively. 
    vx = ux + ax t = u 
    vy = uy + ayt = gsquare root of fraction numerator 2 straight h over denominator straight g end fraction end root space equals square root of 2 g h end root
    So, velocity, v = square root of straight v subscript straight x squared plus straight v subscript straight y squared end root space equals space square root of u squared plus 2 g h end root
    tan space theta space equals space v subscript y over v subscript x space end subscript = fraction numerator square root of 2 gh end root over denominator straight u end fraction

    Question 350
    CBSEENPH11018549
    Wired Faculty App

    A ball rolls off the top of a staircase with horizontal velocity u. If the height of the step is h and the width is b then at what step the ball hits?

    Solution

    Horizontal velocity = u 
    Height of the step = h
    Let the ball hit the nth step.
    Therefore the ball transverses a horizontal distance of nb and falls vertically by nh.
    Equation of trajectory of horizontal projectile,
                   straight y equals 1 half gx squared over straight u squared


    Substituting y = nh and x = nb, we get,
                        nh equals 1 half fraction numerator straight g left parenthesis nb right parenthesis squared over denominator straight u squared end fraction


    rightwards double arrow               2 n h u squared equals g n squared straight b squared
    rightwards double arrow                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 

    Question 351
    CBSEENPH11018550
    Wired Faculty App

    A ball is dropped from a certain point. At the same time another ball is fired horizontally from the same height. Which one will hit the ground earlier? Explain.

    Solution
    Initial velocity of both the balls in vertical direction is zero. So, height by which both the balls have to fall is the same. Therefore, both the balls will hit the ground simultaneously. 
    Question 352
    CBSEENPH11018551
    Wired Faculty App

    A body is projected with velocity 19·6 m/s in horizontal direction. After what time the horizontal and vertical component of velocity will be the same?

    Solution
    Given, that the body is projected in the horizontal direction. 
    So, we have 
    straight u subscript straight x equals 0 space space space space space space space space and space space space space space straight u subscript straight y space equals space 19.6 space straight m divided by straight s
    straight a subscript straight x equals 9.8 space straight m divided by straight s squared space space space space and space space space straight a subscript straight y equals 0
    Now, the horizontal component of velocity at any instant is given by, 
    straight v subscript straight x equals space straight u subscript straight x space plus space straight a subscript straight x space straight t space equals space 0 plus 9.8 straight t space equals space 9.8 straight t space 
    Vertical component of velocity is, 
    straight v subscript straight y space equals space straight u subscript straight y plus straight a subscript straight y space straight t space equals space 19.6 space plus space 0 straight t space equals space 19.6 space 
    Since, horizontal and vertical component of velocity is the same, we have
            9.8 t = 19.6
    i.e.,        t = 2s 
    Question 353
    CBSEENPH11018552
    Wired Faculty App

    A ball is projected horizontally with a velocity 4g m/s from height 18 meter. What is the velocity of body at t = 3 sec?

    Solution
    Given,
    A ball is projected in horizontal direction with velocity 4g m/s. 
    straight u subscript straight x space equals space 4 straight g space straight m divided by straight s space space space space space space space straight u subscript straight y space equals space 0 straight a subscript straight x space equals space 0 space space space space space space space space space space space space space space space space space space space straight a subscript straight y space equals space 9.8 space straight m divided by straight s squared space space space space space space space space space space space space space space space
    Height from which the ball is projected = 18 m 
                          
    At time t=3 s, horizontal and vertical component of velocity is given by, 
    straight v subscript straight x space equals space straight u subscript straight x plus straight a subscript straight x straight t space equals space 4 straight g space plus space 0 straight t space equals space 4 cross times 9.8 space equals space 39.2 space straight m divided by straight s space straight v subscript straight y space equals space straight u subscript straight y plus straight a subscript straight y straight t space equals space 0 space plus space 9.8 straight t space equals space 9.8 cross times 3 space equals space 29.4 space straight m divided by straight s
    Therefore, magnitude of velocity is given by, 
    v = square root of straight v subscript straight x squared plus straight v subscript straight y squared end root space equals space square root of left parenthesis 4.29 right parenthesis squared plus left parenthesis 2.49 right parenthesis squared end root space equals space 49 space m divided by s 
    The angle that the velocity vector makes with horizontal is,
    space space space space space space space tan space theta space equals space v subscript y over v subscript x equals 294 over 392 equals 3 over 4 i. e.. space space space space space theta space equals space tan to the power of negative 1 end exponent open parentheses 3 over 4 close parentheses space equals space 37 to the power of o 
    Question 354
    CBSEENPH11018553
    Wired Faculty App

    A shot fired horizontally from the top of a tower with a velocity of 60 ft/s hits the ground in 2 seconds. What is the height of the tower and the distance from the foot of the tower where it strikes the ground?

    Solution

    Here,
    Time taken for the shot to reach the ground, t = 2s.
    Velocity in horizontal direction, u = 60 ft/s
    The height of the tower = distance fallen by shot in 2s 
    That is, 
    space 1 half gt squared equals 1 half cross times 32 cross times left parenthesis 2 right parenthesis squared equals 64 space ft 
    The distance from the foot of the tower where it strikes the ground, s= ut = 60 x 2 = 120 feet. 

    Question 355
    CBSEENPH11018554
    Wired Faculty App

    A projectile is projected horizontally with velocity from the top of an inclined plane of inclination θ. How far from the point of projection will the projectile hit tht plane?

    Solution
    Consider a projectile, which is projected with velocity 'u' in horizontal direction from a point O on an inclined plane and hit the inclined plane at P↔(x, y).
               
    Let O be the origin. 
    The horizontal direction is positive direction of X-axis and vertically downward direction is positive direction of Y-axis.
    On resolving the components of velocity and acceleration in horizontal and vertical direction, we have
    space space space space straight x subscript straight o equals 0 space space space space space space space space straight y subscript straight o equals 0 space space space space straight u subscript straight x equals straight u space space space space space space space space straight u subscript straight y space equals 0 space space space space space straight a subscript straight x equals 0 space space space space space space space space space straight a subscript straight y space equals space straight g 
    Motion along horizontal direction is, 
            space straight x equals straight x subscript straight o plus straight u subscript straight x straight t plus 1 half straight a subscript straight x straight t squared equals ut             ...(1) 
    Motion along vertical direction is, 
             straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared equals 1 half gt squared      ...(2) 
    Eliminating 't' from (1) and (2), 
                    straight y equals fraction numerator straight g over denominator 2 straight u squared end fraction straight x squared                        ...(3) 
    The equation of inclined plane is, 
                     space straight y equals xtanθ                         ...(4) 
    The point P is the intersection of trajectory and inclined plane, thus the coordinates of P are the solutions of equation (3) and (4).
    By solving (3) and (4), we obtain 
           space space space straight x equals fraction numerator 2 straight u squared tanθ over denominator straight g end fraction space space space space and space space space space space space straight y space equals space fraction numerator 2 straight u squared tan squared straight theta over denominator straight g end fraction 
    Therefore distance OP is, 
              OP equals square root of straight x squared plus straight y squared end root space equals space fraction numerator 2 straight u squared tanθ over denominator straight g end fraction square root of 1 plus tan squared straight theta end root 
               space space equals fraction numerator 2 straight u squared tanθsecθ over denominator straight g end fraction
    So, the distance travelled by the projectile is OP. 

    Question 356
    CBSEENPH11018555
    Wired Faculty App

    A car starts from rest and accelerates uniformly with 2 m/s2. At t = 5s a stone is dropped out of the window 15 m high of the car. What is velocity and acceleration of stone at t = 5·5 s? ( g = 10m/s2).

    Solution

    Given, 
    Car starts from rest. That is, 
    Initial velocity = 0 m/s
    Uniform acceleration of the car = 2 m/s2
    Velocity of car at t = 5s =  10 m/s
                
    When stone is dropped, the velocity of car and hence velocity of stone is 10 m/s in horizontal direction.
    Since at 5s the stone loses the contact from car, therefore, the stone will move uniformly in horizontal direction and accelerate in vertically downward direction due to gravity.
    At t = 5·5s or 0·5 s after the stone is dropped, the X and Y component of velocity are 
    straight V subscript straight x space equals space straight v subscript straight x space plus space straight a subscript straight x space straight t space equals space 10 space plus space 0 cross times 0.5 space equals space 10 space straight m divided by straight s straight V subscript straight y space equals space straight v subscript straight y space plus space straight a subscript straight y straight t space equals space 0 space plus space 10 cross times 0.5 space equals space 5 space straight m divided by straight s space  
    So, magnitude of velocity is given by, 
    straight V space equals space square root of straight V subscript straight x squared plus straight V subscript straight v squared end root space equals space square root of 10 squared plus 5 squared end root space equals space 5 square root of 5 space straight m divided by straight s
    Acceleration of the stone at t=5.5 sec is the same as acceleration due to gravity. 
    That is, acceleration = 10 m/s2 
    And, 
    Angle of inclination with the horizontal, straight theta = straight v subscript straight y over straight v subscript straight x space equals space 1 half
    i.e., straight theta space equals space 26 to the power of straight o space 33 to the power of apostrophe

    Question 357
    CBSEENPH11018556
    Wired Faculty App

    A projectile is projected with velocity u at an angle θ with horizontal. Discuss the trajectory of projectile. Also find the velocity of projectile at any instant.

    Solution
    Let us consider a projectile is projected with velocity 'u' at angle 'θ' with horizontal direction from a point on the ground.
    Let the point of projection be origin, right direction is positive direction of X-axis and vertically upward direction is positive direction of Y-axis.
    Equation of trajectory:
    Break the components of velocity and acceleration due to gravity in horizontal and vertical direction.
    x0 = 0    ; yo = 0
    ux = u cos straight theta   ; uy = u sin straight theta   
    ax = 0 ; ay = -g

    Motion along horizontal direction:
    straight x equals straight x subscript straight o plus straight u subscript straight x straight t plus 1 half straight a subscript straight x straight t squared equals straight u space cosθt            ...(1)
    Motion along vertical direction:
    space space space space straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared
        equals usinθt minus 1 half gt squared                             ...(2)
    Eliminating t from (1) and (2),
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    This the required equation of trajectory.
    Since the equation of trajectory is the equation of parabola, therefore projectile follows parabolic path.

    Velocity of projectile:
    Let at any instant,straight v with rightwards harpoon with barb upwards on top be the velocity of the projectile making an angle straight alpha with the horizontal. 
    i.e., 
    ux = u cos straight theta ,  uy = u sin straight theta 
    ax = 0 , ay = -g 
    Now, 
    vx = ux + axt    ,     vy =  uy + ayt  
    Therefore, 
    vx = u cosstraight theta     , vy = u sinstraight theta - gt
    The velocity f the projectile is given by, 
    v = square root of straight v subscript straight x squared plus straight v subscript straight y squared end root
      equals space square root of straight u squared cos squared straight theta plus left parenthesis usinθ minus gt right parenthesis squared end root equals space square root of u squared minus space 2 u g t space s i n theta plus g squared t squared end root
    Angle pf projection is given by, tan straight alpha = fraction numerator straight u space sinθ space minus space gt over denominator straight u space cos space straight theta end fraction

    Question 358
    CBSEENPH11018557
    Wired Faculty App

    A projectile is projected with velocity u at an angle θ with horizontal. Find the time of flight, maximum height attained and horizontal range of projectile.

    Solution
    Let us consider a projectile is projected with velocity u at angle θ with the horizontal direction from a point on the ground.
    Let the point of projection be origin, right direction is positive direction of X-axis and vertically upward direction is positive direction of Y-axis. 

    Break the components of velocity and acceleration due to gravity in horizontal and vertical direction.
      
    Time of flight:
    It is the time for which the projectile remains in air above the point of projection. It is represented by T. 
    The position of projectile at any instant in Y direction is given by,
    space space space space straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared
    The time of flight is the difference of instants when the Y position is zero.
    i.e.          0 equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared 
     rightwards double arrow    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    rightwards double arrow        space space straight t subscript 1 equals 0    and     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    ∴             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Maximum height:
    Maximum height is the greatest height to which the projectile rises above point of projection. It is represented by H.
    The displacement of projectile in vertical direction is given by,
                         straight S subscript straight x equals fraction numerator straight v subscript straight y squared minus straight u subscript straight y squared over denominator 2 straight a subscript straight x end fraction
    At the highest point, vertical component of velocity is zero.
    ∴       <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Horizontal range:
    It is the horizontal distance travelled by the projectile during it's flight. i.e., the horizontal distance travelled covered by object. It is represented by R.
    Obviously, it is the distance travelled by projectile in horizontal direction with constant velocity u cosθ in interval equal to time of flight.
    i.e.      straight R equals ucosθT equals ucosθ fraction numerator 2 usinθ over denominator straight g end fraction
    rightwards double arrow       straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
     
    Question 359
    CBSEENPH11018558
    Wired Faculty App

    Find the angle of projection for which horizontal range is maximum.

    Solution

    Horizontal range is given by,  straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
    R is maximum if sin space 2 straight theta is maximum.
    i.e.      space space sin space 2 straight theta space equals space 1
    rightwards double arrow              <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow               straight theta equals 45 degree
    Horizontal range is maximum when angle of projection is 45°. 

    Question 360
    CBSEENPH11018559
    Wired Faculty App

    Show that there are two angles of projection for which horizontal range is the same.

    Solution

    Horizontal range is given by,
                       straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
    where, straight theta is the angle of projection and is less than 90 degree.
    We know, sin left parenthesis 180 degree minus 2 straight theta right parenthesis space equals space sin space 2 straight theta
    rightwards double arrow             sin space 2 left parenthesis 90 degree minus straight theta right parenthesis space equals space sin space 2 straight theta
    rightwards double arrow            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>         [Where straight alpha equals 90 degree minus straight theta]
    Now,  
             straight R subscript 0 equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction  and  straight R subscript straight alpha space equals space fraction numerator straight u squared space sin 2 straight alpha over denominator straight g end fraction
    Since     sin space 2 straight theta space equals space sin space 2 straight alpha
    ∴            straight R subscript 0 equals straight R subscript straight alpha
    Thus horizontal range is the same for angle of projection θ and α = 90°–θ. 

    Question 361
    CBSEENPH11018560
    Wired Faculty App

    There are two angles of projection for which horizontal range is the same. Show that product of their time of flights is directly proportional to the horizontal range?

    Solution
    The angles of projection, for which horizontal range is same, are θ and 90° – θ.
    Time of flight for angle of projection θ is,
                           space straight T subscript 1 equals fraction numerator 2 usinθ over denominator straight g end fraction
    Time of flight corresponding to angle of projection (90°–θ) is, 
                      straight T subscript 2 equals fraction numerator 2 usin left parenthesis 90 degree minus straight theta right parenthesis over denominator straight g end fraction equals fraction numerator 2 straight u space cosθ over denominator straight g end fraction
    Now, product of their time flight is, 
        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
             space space equals fraction numerator 2 straight u squared 2 sinθ space cosθ over denominator gg end fraction        
               equals fraction numerator 2 straight u squared space sin space 2 straight theta over denominator gg end fraction equals fraction numerator 2 straight R over denominator straight g end fraction            
    That is,
                            straight T subscript 1 straight T subscript 2 space proportional to space straight R
    Hence proved the relation. 
                          
    Question 362
    CBSEENPH11018561
    Wired Faculty App

    There are two angles of projection for which horizontal range is same. Prove that the sum of maximum heights for these two angles is independent of angle of projection.

    Solution

    Given, horizontal range is same for two angles of projection. 
    The angles are θ and 90°–θ.
    Maximum height H1 for angle projection θ is,
                         straight H subscript 1 equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction 
    Maximum height H2 for angle projection 90°–θ is, 
              space space space space straight H subscript 2 equals fraction numerator straight u squared sin squared left parenthesis 90 degree minus straight theta right parenthesis over denominator 2 straight g end fraction equals fraction numerator straight u squared cos squared straight theta over denominator 2 straight g end fraction 

    Now, sum of maximum height is,  
      space space space space space straight H subscript 1 plus straight H subscript 2 equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction plus fraction numerator straight u squared cos squared straight theta over denominator 2 straight g end fraction equals fraction numerator straight u squared over denominator 2 straight g end fraction   

    Hence the result.

    Question 363
    CBSEENPH11018562
    Wired Faculty App

    Prove that the maximum horizontal range is 4 times the maximum height attained by a projectile.

    Solution
    The horizontal range is maximum when the angle of projection is 45°.
    So,
    space space space space straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction equals fraction numerator straight u squared sin left parenthesis 2 cross times 45 degree right parenthesis over denominator straight g end fraction equals straight u squared over straight g        ...(1)
    Maximum height for angle of projection 45 degree is,
    space space space straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction equals fraction numerator straight u squared sin squared 45 degree over denominator 2 straight g end fraction equals fraction numerator straight u squared over denominator 4 straight g end fraction              ...(2) 
    Therefore, from equation (1) and (2),
                          R = 4H
    That is, range is 4 times the maximum height attained by a projectile. 
    Question 364
    CBSEENPH11018563
    Wired Faculty App

    A body is projected with velocity u at an angle θ and reaches the ground at time T. What will be the time of flight if the body is projected with same speed at an angle (90°–θ)?

    Solution
    Time of flight for angle of projection straight theta is given by,
     
             straight T equals fraction numerator 2 straight u space sin space straight theta over denominator straight g end fraction                                     ...(1)
    If the body is projected at an angle <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, then the time of flight T' is
          straight T apostrophe equals fraction numerator 2 straight u space sin left parenthesis 90 degree minus straight theta right parenthesis over denominator straight g end fraction equals fraction numerator 2 straight u space cosθ over denominator straight g end fraction          ...(2)
    Dividing (2) by (1), we get
              straight T apostrophe space equals space straight T space cot space straight theta 
    Question 365
    CBSEENPH11018564
    Wired Faculty App

    A projectile is fired with ‘kinetic energy K at an angle θ with the horizontal. What will be the kinetic energy of projectile at the highest point?

    Solution
    Let a projectile of mass m be projected with velocity u at an angle θ.
    Kinetic energy of projection, space space straight K subscript straight o space equals space 1 half mu squared
    At the highest point, vertical component of velocity is zero and there is only horizontal component of velocity.
    Horizontal component = u cos θ.
    ∴ Kinetic energy at the highest point is,
     K.E = 1 half mu squared
        straight K equals 1 half space mu squared cos squared straight theta space space space equals space straight K subscript straight o cos squared straight theta
    Question 366
    CBSEENPH11018565
    Wired Faculty App

    A man sitting near the window of a moving train drops a stone. Describe the path of stone on seeing by a man standing near the track.

    Solution
    The stone which is dropped from moving train has same horizontal velocity as that of train with respect to the track. Therefore, for a person standing near the track, stone will act like a horizontal projectile and path followed is parabola. 
    Question 367
    CBSEENPH11018566
    Wired Faculty App

    Why one has to aim a little above the target?

    Solution
    If a person aims at the target, the particles will follow a parabolic trajectory and will noy hit the target. Parabolic path is followed as result of gravity. 
    Therefore to hit the target one has to aim a little above as shown in figure (ii), so that after following the parabolic path, the bullet may hit the target.

     

    Question 368
    CBSEENPH11018567
    Wired Faculty App

    A projectile is projected with velocity u at an angle θ with the vertical. At what point velocity and acceleration will be perpendicular to each other?

    Solution
    At the highest point the velocity is in horizontal direction and acceleration is in vertically downward direction. Hence, velocity and acceleration of projectile are perpendicular to each other at the highest point. 
    Question 369
    CBSEENPH11018568
    Wired Faculty App

    A particle is projected with velocity ‘u’ in two different ways: first such that horizontal range is maximum and second such that the height is maximum. What is the ratio of maximum height to the maximum horizontal range?

    Solution
    Horizontal range is maximum for angle of projection 45°.
    Maximum horizontal range is,
                 straight R subscript max equals fraction numerator straight u squared sin left parenthesis 2 cross times 45 right parenthesis over denominator straight g end fraction equals straight u squared over straight g 
    The height is maximum, if the projectile is projected in vertically upward direction, i.e. θ = 90°. 
    Maximum height is, 
             straight H subscript max equals fraction numerator straight u squared sin squared left parenthesis 90 right parenthesis over denominator 2 straight g end fraction equals fraction numerator straight u squared over denominator 2 straight g end fraction 
    So, the ratio of maximum height to the maximum horizontal range is given by, 
           straight H subscript max over straight R subscript max equals 1 half 
    Question 374
    CBSEENPH11018573
    Wired Faculty App

    A particle is projected with speed ‘u’ at an angle θ. Show that it will strike the ground with the speed u.

                

    Solution

    Let a projectile be projected with speed 'u' at angle 'θ' from O and strike the ground at P with velocity 'v'.
    The projectile will strike the ground at P after time equal to the time of the flight.
    Now, resolving the components of 'u' in the horizontal and vertical direction, we have
                       straight u subscript straight x space equals space straight u space cosθ space space space space space space space space space space space space space space space space space space space space space straight u subscript straight y space equals space straight u space sinθ straight a subscript straight x space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight a subscript straight y space equals space minus straight g
    Horizontal and vertical component is given by, 
             straight v subscript straight x equals straight u subscript straight x plus straight a subscript straight x straight t equals straight u space cosθ straight v subscript straight y space equals space straight u subscript straight y plus straight a subscript straight y straight t space equals space straight u space sinθ space minus space gT space equals space minus straight u space sinθ 
    Therefore magnitude of velocity at ground is, 
            straight v equals square root of straight v subscript straight x superscript 2 plus straight v subscript straight y superscript 2 end root space equals space square root of straight u squared space cos squared straight theta plus left parenthesis negative straight u space sin space straight theta right parenthesis squared end root 
               equals square root of straight u squared left parenthesis cos squared straight theta plus sin squared straight theta right parenthesis end root space equals space straight u         

    Question 375
    CBSEENPH11018574
    Wired Faculty App

    In long jumping does it matter how high you jump? What factor determines the span of the jump?

    Solution
    It does not matter how high an athlete jumps in a long jump. The span of long jump depends upon the initial velocity and the angle at which an athlete jumps. The span of jump is maximum for an angle of 45°.
    Question 376
    CBSEENPH11018575
    Wired Faculty App

    Show that the ratio of the horizontal range to the maximum height attained by projectile is equal to 4 cot straight theta and is independent of speed of projection.

    Solution
    The horizontal range attained by projectile projected with velocity u at angle θ is, 
                       straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction 
    Maximum height attained by projectile projected with velocity u at angle θ is, 
                         straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction
    The ratio of horizontal range to maximum height is given by, 
        straight R over straight H equals fraction numerator straight u squared sin space 2 straight theta over denominator straight g end fraction cross times fraction numerator 2 straight g over denominator straight u squared sin squared straight theta end fraction equals 4 space cotθ

    Hence, the result proved. 
    Question 377
    CBSEENPH11018576
    Wired Faculty App

    At what angle of projection the horizontal range is four times the maximum height attained by the projectile?

    Solution

    Given, horizontal range is four times the maximum height attained by the projectile.
    i.e.,              R = 4H 
    We know,   straight R over straight H equals 4 space cot space straight theta 
    Substituting R = 4H in the above equation,
    we get,
                    space space space 4 space equals space 4 space cotθ 
     rightwards double arrow         cotθ space equals space 1
    i.e.,             straight theta equals 45 degree

    Question 378
    CBSEENPH11018577
    Wired Faculty App

    Why the electrons in beam from an electron gun do not fall as much as a water molecule in stream from a hose due to gravity?

    Solution
    Vertical distance by which projectile falls while travelling a given distance in horizontal direction is, 
                          space space straight y equals 1 half straight g straight x squared over straight u squared 
    From the above relation, straight y proportional to space 1 divided by straight u squared. 
    Velocity of electron from electron gun is greater than the water molecule from hose. Therefore, for a given horizontal displacement x, the vertical distance by which the electron beam falls is less than that for water molecule. 
    Question 379
    CBSEENPH11018578
    Wired Faculty App

    Two projectiles are projected with same velocity at same angle, one on the earth and second on the moon. If the projectile acquires the maximum height of H on the earth, then what will be the maximum height attained by the projectile on the moon?

    Solution
    Maximum height attained by the projectile on the earth is,
    straight H subscript earth equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction 
    Acceleration due to gravity on the moon is g/6. Therefore maximum height attained by the projectile on the moon is, 
    straight H subscript moon equals fraction numerator straight u squared sin squared straight theta over denominator 2 left parenthesis straight g divided by 6 right parenthesis end fraction equals 6 open parentheses fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction close parentheses space equals space 6 straight H subscript earth 
    Thus maximum height attained on the moon is six times the maximum height attained on the earth.
    Question 380
    CBSEENPH11018579
    Wired Faculty App

    A body of mass 2 kg is projected with a velocity 60 m/sec at an angle of 30°. What is the height of the body from ground at t = 2·5s? (Take g = 10 m/s2)

    Solution
    The height of the body from ground at any instant is given by,
                     <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>         ... (1)
    Given that,
    Mass of the body, m = 2 kg
    Initial space velocity space of space the space body comma space straight u equals 60 space straight m divided by straight s Angle space of space projection comma space straight theta space equals space 30 degree
    Time, t = 2.5s
    Acceleration due to gravity,straight g equals 10 space straight m divided by straight s squared
    Putting this in equation (1), we will get the height of the body from the ground,
     space space space space straight h equals 60 space sin 30 degree cross times 2.5 minus 1 half cross times 10 cross times left parenthesis 2.5 right parenthesis squared
          equals 75 minus 31.25 equals space 43.75 straight m
    Question 381
    CBSEENPH11018580
    Wired Faculty App

    Water stream leaves from hose pipe with velocity u m/s at angle θ with vertical and hose pipe rotates about vertical axis with angular velocity ω rad/sec. Find the area of circle and where will the water stream strike the ground?
    Solution
    The radius of the circle is equal to the horizontal distance at which water stream strikes the ground.
    That is, 
     
                                straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
    Area of the circle is,
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 382
    CBSEENPH11018581
    Wired Faculty App

    A body is projected with velocity u at angle θ with vertical. At what height will it strike the wall, r distance away from the point of projection?

    Solution
    Consider a body which is projected from O in XY plane with velocity 'u' at angle θ with vertical, and
    The body is projected at an angle α = 90°–θ with the horizontal. 
    Let the body strike the wall at height h.
    So, co-ordinates of point P where the projectile hits the wall are ↔ (r,h).
    Equation of trajectory is given by, 
    space space space space straight y equals xtanα minus 1 half straight g fraction numerator straight x squared over denominator straight u squared cos squared straight alpha end fraction 
         equals xcotθ minus 1 half straight g. fraction numerator straight x squared over denominator straight u squared sin squared straight theta end fraction        [∵ straight alpha equals 90 degree minus straight theta] 
    Since straight P left right arrow left parenthesis straight r comma space straight h right parenthesis lies on trajectory, 
    ∴        straight h equals rcotθ minus 1 half straight g fraction numerator straight r squared over denominator straight u squared sin squared straight theta end fraction, 
    is the height at which the body will strike the wall.

    Question 383
    CBSEENPH11018582
    Wired Faculty App

    Two projectiles are projected with same speed but at different angles. Under what conditions can they land at the same point at the same time on the earth?

    Solution
    Given, two projectiles are projected at different angles with the same speed. 
    So, the two angles of projection i.e. θ and 90°–θ, for which horizontal range is same.
    The projectiles will land at the same point if they are projected at angles θ and (90°–θ) from the same point and in the same plane. But, since the time of flight is different for the two cases, therefore the projectiles will reach the ground at different moments. 
    If the projectile that takes more time to reach the ground is projected earlier than the second, then both may reach the ground at the same moment. 
    Time of flight of first projectile projected at angle θ, straight T subscript 1 equals fraction numerator 2 straight u space sin space straight theta over denominator straight g end fraction
    Time of flight of second projectile projected at angle 90°–θ,straight T subscript 2 equals fraction numerator 2 straight u space cosθ over denominator straight g end fraction 
    If two projectiles are projected at the time interval, then they will land at the same time. 
                 straight T subscript 1 minus straight T subscript 2 equals fraction numerator 2 straight u over denominator straight g end fraction left parenthesis sin space straight theta space minus space cos space straight theta right parenthesis
    Question 384
    CBSEENPH11018583
    Wired Faculty App

    A ball is projected at angle 0 from a point O ↔ (0,0). After passing the ball through a point (x, y) it reaches the ground at a distance R from O. Show that
    straight y over straight x equals tan space straight theta open parentheses 1 minus straight x over straight R close parentheses

    Solution
    Let the ball be projected with speed u.
    The equation of trajectory is,
             straight y equals straight x space tanθ space minus space 1 half straight g fraction numerator straight x squared over denominator straight u squared cos squared straight theta end fraction
    On taking x tanstraight theta common, we get
    or      straight y equals straight x space tanθ open parentheses 1 minus 1 half straight g fraction numerator straight x over denominator straight u squared cos squared straight theta space tanθ end fraction close parentheses 
    or     space straight y equals straight x space tanθ open parentheses 1 minus straight g fraction numerator straight x over denominator 2 straight u squared space cosθ space sinθ end fraction close parentheses
    or     space space straight y equals straight x space tanθ open parentheses 1 minus fraction numerator straight x over denominator open parentheses begin display style bevelled fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction end style close parentheses end fraction close parentheses
    or      straight y over straight x equals tanθ open parentheses 1 minus straight x over straight R close parentheses
    Hence, proved the result. 
    Question 385
    CBSEENPH11018584
    Wired Faculty App

    A bullet fired from a gun with a velocity of 800 ft/s strikes the ground at the same level as the gun at a distance of 17320 ft. Find the angle of inclination with the horizontal at which the gun was fired.

    Solution

    Given that,
    Distance of the gun, R = 17320 ft
    Initial velocity, u = 800 ft./sec
    Horizontal range is given by,
                       straight R equals fraction numerator straight u squared space sin space 2 straight theta over denominator straight g end fraction
    Therefore,
    sin space 2 straight theta space equals space Rg over straight u squared space space space space space space space space space space space space equals space fraction numerator 17320 space cross times space 2 over denominator left parenthesis 800 right parenthesis squared end fraction space space space space space space space space space space space space equals space 0.866
    rightwards double arrow2 straight theta space equals space 60 degree
    rightwards double arrow  straight theta space equals space 30 degree, is the angle of inclination with the horizontal at which the gun was fired. 

    Question 386
    CBSEENPH11018585
    Wired Faculty App

    The ceiling of a long hall is 25m high. What is the maximum horizontal distance that a ball thrown with speed of 40 m/s can go without hitting the ceiling of the hall?

    Solution
    Height of the ceiling = 25 m = MAximum height attained by the ball. 
    Velocity of the ball, u = 40 m/s
    Maximum height is given by, straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction

    i.e., sin squared straight theta space equals space fraction numerator 2 gH over denominator straight u squared end fraction equals fraction numerator 2 cross times 9.8 cross times 25 over denominator left parenthesis 40 right parenthesis squared end fraction equals 0.30625
    ∴       sin space straight theta space equals space square root of 0.30625 end root space equals space 0.5534 
    <pre>uncaught exception: <b>Http Error #404</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 61<br />#0 [internal function]: com_wiris_plugin_impl_HttpImpl_0(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #404') #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('com_wiris_plugi...', Array) #2 [internal function]: _hx_lambda->execute('Http Error #404') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(532): call_user_func_array(Array, Array) #4 [internal function]: haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #404') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/php/Boot.class.php(769): call_user_func_array('haxe_Http_5', Array) #6 [internal function]: _hx_lambda->execute('Http Error #404') #7 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(27): call_user_func_array(Array, Array) #8 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #404') #9 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), NULL, NULL) #10 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(40): haxe_Http->request(true) #11 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(80): com_wiris_plugin_impl_HttpImpl->request(true) #12 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #13 {main}</pre>          straight theta equals 33.6 
    ∴ Maximum horizontal distance,
         straight R equals fraction numerator straight u squared sin space 2 straight theta over denominator straight g end fraction equals fraction numerator left parenthesis 40 right parenthesis squared space sin space 67.2 over denominator 9.8 end fraction 
           equals space 150.5 space straight m.                                    
                   
    Question 387
    CBSEENPH11018586
    Wired Faculty App

    A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?

    Solution
    Horizontal range is maximum if the angle of projection is 45°.
    Horizontal range in this case is given by,
                    straight R subscript max space equals space straight u squared over straight g
    So,      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    ∴           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    If the cricketer throws the ball vertically upward then it will attain the maximum height from the ground.
    ∴      straight H subscript max equals fraction numerator straight u squared over denominator 2 straight g end fraction equals 100 over 2 equals 50 straight m, is the height of the ball from the ground. 
    Question 388
    CBSEENPH11018587
    Wired Faculty App

    A body is projected with kinetic energy 100J at an angle 45° with horizontal. What fraction of kinetic energy will be the potential energy at the highest point?

    Solution
    If a body is projected with kinetic energy K at angle θ, then kinetic energy at highest point is K cos2θ.
    Therefore, potential energy of the body at highest point is,
    P.E. = space space straight K minus straight K space c o s squared straight theta space equals space straight K left parenthesis 1 minus c o s squared straight theta right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight K space s i n squared straight theta 
    Fractional potential energy at highest point is, 
    space space space fraction numerator Potential space Energy over denominator Total space energy space at space the space time space of space projection end fraction
      equals fraction numerator straight K space sin squared straight theta over denominator straight K end fraction equals sin squared straight theta           
    Here, we have
    K = 100 J and straight theta space equals space 45 to the power of straight o   
    Fractional potential energy at highest point is, 
                          space space space sin squared 45 degree space equals space 0.5
    At the highest point, K.E will be 0.5 times the P.E at the highest point. 
    Question 389
    CBSEENPH11018588
    Wired Faculty App

    Show that if the projectile attains the maximum height of H and falls at a distance R, then it is projected at an angle θ given by <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Solution
    Let the speed with which the projectile be projected at angle θ  be 'u'.
    The maximum height and the horizontal range is given by,
    Horizontal range, straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
    Maximum height, straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction 
    Ratio of horizontal range to maximum height is given by, 
    ∴               straight R over straight H equals fraction numerator 2 s i n 2 straight theta over denominator s i n squared straight theta end fraction equals 4 space c o t space straight theta
    rightwards double arrow          t a n space straight theta space equals space fraction numerator 4 straight H over denominator straight R end fraction
    That is, 
                      straight theta equals t a n to the power of negative 1 end exponent open parentheses fraction numerator 4 straight H over denominator straight R end fraction close parentheses, is the angle of projection. 
    Question 390
    CBSEENPH11018589
    Wired Faculty App

    A body is projected with a velocity 30 m/sec at angle 30°. After what time will it be at maximum height?

    Solution
    We know,
    A projectile is at maximum height at time equal to half of the time of flight.
    That is, 
                    space space straight t equals fraction numerator straight u space s i n space straight theta over denominator straight g end fraction
    Here, we have
    Initial space velcoity comma space straight u space equals space 30 space straight m divided by straight s Angle space of space projection comma space straight theta space equals space 30 degree
    Time when the projectile is at maximum height is, 
    ∴           straight t equals fraction numerator 30 cross times s i n space 30 degree over denominator 9.8 end fraction equals 1.53 straight s 
    Question 391
    CBSEENPH11018590
    Wired Faculty App

    Two bodies are placed at a distance 4 km and 6·5 km from the gun. When the bullet is fired at an angle of 30° with horizontal, it hits the body at 4 km. At what angle the bullet should be fired so that it may hit the second body?

    Solution

    Horizontal range is given by, 
                           straight R equals fraction numerator straight u squared sin space 2 straight theta over denominator straight g end fraction
    Therefore the ratio of horizontal ranges R1 and R2 for angles of projections θ1 and θ2 is,
                     space space straight R subscript 1 over straight R subscript 2 equals fraction numerator sin space 2 straight theta subscript 1 over denominator sin space 2 straight theta subscript 2 end fraction
    ∴           <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    We have here,
    Angle space of space projection comma space straight theta subscript 1 equals 30 degree Horizontal space range comma space straight R subscript 1 space equals space 4 space km Horizontal space range comma space straight R subscript 2 equals 6.5 space km
    Therefore, 
    sin 2 straight theta subscript 2 equals sin 60 degree cross times fraction numerator 6.5 over denominator 4 end fraction equals 1.407 greater than 1, which is not possible.
    Therefore for same speed of bullet, the bullet can not hit the body at a distance of 6·5km.

    Question 392
    CBSEENPH11018591
    Wired Faculty App

    A player P kicks the football with velocity 20 m/s at angle of 15°. Another player Q on the goal line 12 m away in the direction of kick, runs at the same instant to meet the ball. With what speed he should run so that he could catch the ball just before it hits the ground? (Take g = 10 m/s2).

    Solution

    We know that,
    Horizontal range, straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
    We have,
    Initial space velocity comma space straight u equals 20 straight m divided by straight s Angle space of space projecvtion comma space straight theta equals 15 degree a c c e l e r a t i o n space d u e space t o space g r a v i t y comma space straight g equals 10 space straight m divided by straight s squared
    ∴  
    Horizontal range, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    i.e. the ball hits the ground at a distance of 20m from the position of player P.
    Since the player Q is 12m away, therefore, he has to travel a distance 8m to catch the ball in time equal to the time of flight.
    Now the time of flight is,
            space space space space straight T equals fraction numerator 2 straight u space s i n theta over denominator straight g end fraction space space space space space space equals fraction numerator 2 cross times 20 cross times s i n 15 degree over denominator 10 end fraction space space space space space space space equals 4 s i n 15 degree
    rightwards double arrow       straight T equals 1.035 space straight s 
    The velocity of player Q is,
         straight v equals fraction numerator 8 over denominator 1.035 end fraction equals 7.727 space straight m divided by straight s

    Question 393
    CBSEENPH11018592
    Wired Faculty App

    A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m/s to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g = 10 m/s2)

    Solution

    Velocity of shell is, u  = 600 m/s
    Velocity of fighter plane is, v = 720 km/hr = 200 m/s

    Let the shell be fired at angle θ with vertical.
    To hit the fighter plane, the horizontal component of muzzle speed should be equal to the velocity of plane.
    i.e.               space space space space space straight v equals straight u space sinθ          

    rightwards double arrow              200 space equals space 600 space s i n space straight theta
    rightwards double arrow               sinθ space equals space 1 divided by 3


    rightwards double arrow                 <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                   
    The maximum height attained by the shell is,

                  straight H equals fraction numerator straight u squared sin squared left parenthesis 90 minus straight theta right parenthesis over denominator 2 straight g end fraction
                     equals fraction numerator left parenthesis 600 right parenthesis squared space s i n squared left parenthesis 70.53 right parenthesis over denominator 2 cross times 10 end fraction equals 15995 space straight m 
    That is, 
                 space space straight H asymptotically equal to 16 space km
    Therefore the minimum altitude for safe flight is 16km.

    Question 394
    CBSEENPH11018593
    Wired Faculty App

    A stone is thrown from the top of a tower 50m high with speed 30m/s at an angle 30° with horizontal. Find the distance from the foot of tower where the stone will hit the ground.

    Solution

    Given that, 

          u = 30 m/s,       straight theta equals 30 degree
    ∴ x component of velocity, 
    straight u subscript straight x equals 30 space cos space 30 degree space equals space 25.98 space straight m divided by straight s
    and  
    y component of velocity,   
    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>     

    Let the stone reaches the ground after time t.
                    space space space space space space space straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared      ... (1)

    We have,
           straight y equals 0 comma space space straight y subscript straight o equals 50 comma space space space straight a subscript straight y equals negative 10 straight m divided by straight s squared
    Putting the values in equatuion (1), we have 
               0 equals 50 plus 15 straight t minus 5 straight t squared
    i.e.,            straight t equals 5 straight s
    The distance from the foot of tower where the stone will hit the ground is,
                  s = u x t
                     = 25.98 x 5
                     = 104.9m

    Question 395
    CBSEENPH11018594
    Wired Faculty App

    A body is projected from the ground with some velocity. Three seconds later the body is moving in a direction making an angle of 30° with the horizontal and two seconds later it moves horizontally. Find the distance at which the body will fall on the ground from the point of projection.

    Solution

    Let the body be projected with velocity u making an angle θ with the horizontal.
    The direction of motion of body after t seconds of throw is given by the equation,
                 tan space straight alpha space equals space fraction numerator straight u space sin space straight theta space minus space gt over denominator straight u space cos space straight theta end fraction
    Here at time, t = 3 the value of straight alpha is 30 degree and at t = 5s the value of straight alpha is 0 degree.
    ∴               tan space 30 degree space equals space fraction numerator straight u space sinθ minus 30 over denominator straight u space cosθ end fraction           ...(1)
    and  
             tan space 0 degree space equals space fraction numerator straight u space sinθ minus 50 over denominator straight u space cosθ end fraction                       ...(2)
    From (2),
     
                    <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>                              ...(3)
    Substituting straight u space sinθ space equals 50 in equation (1), we get
                  straight u space cos space straight theta space equals space fraction numerator straight u space sinθ minus 30 over denominator tan 30 degree end fraction equals 20 square root of 3      ...(4)
    The horizontal range of the body,
                 space space space space straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction equals fraction numerator 2 straight u space sinθ space straight u space cosθ over denominator straight g end fraction
                       
                      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Therefore, the body falls at a distance of 200square root of 3 from the point of projection. 
          

    Question 396
    CBSEENPH11018595
    Wired Faculty App

    Two hunters simultaneously aim their guns at a bird sitting on a tree. The first hunter fires his shot with velocity 117·6m/s at angle 30°. The second hunter ahead of first hunter by a distance 19·6√3 m fires his shot at angle 60° and both the shots hit the bird simultaneously. Find the speed of fire of shot by second hunter, the distance of foot of tree from first hunter and height of bird.

    Solution

    The situation is as shown in figure.

    A is the point of first shot. 
    With respect to A, the position of bird is straight P left right arrow left parenthesis straight x comma space straight y right parenthesis.
    Position of first shot at any instant is, 
    straight x subscript 1 space equals space straight u subscript 1 space cosθ subscript 1 straight t space space space space space equals space 117.6 space cos space 30 to the power of straight o space straight t space space space space space equals space 58.8 square root of 3 space straight t space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space straight y subscript 1 space equals space straight u subscript 1 sinθ subscript 1 straight t space minus space 1 half gt squared space space space space space space equals space 117.6 space sin space 30 to the power of straight o space straight t space minus 1 half 9.8 space straight t squared space space space space space space equals space 58.8 space straight t space minus space 4.9 straight t squared space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space Position space of space second space shot space at space any space instant space is straight x subscript 2 space equals space straight x subscript infinity space plus space straight u subscript 2 cosθ subscript 2 space straight t space space space space space space equals space 19.6 square root of 3 space plus space straight u space cos 60 to the power of straight o straight t space space space space space equals space 19.6 space square root of 3 space plus straight u over 2 straight t space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis straight y subscript 2 space equals space straight u subscript 2 sin space straight theta subscript 2 straight t space minus space 1 half gt squared space space space space space equals space straight u space sin space 60 to the power of 0 space straight t space minus 1 half 9.8 straight t squared space space space space space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t space minus space 4.9 space straight t squared space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 4 right parenthesis
    As both the shots hit the bird simultaneously. Therefore at the instant of hitting, positions of both the shots will be same. 
    i.e. straight x subscript 1 equals straight x subscript 2 space space space and space space straight y subscript 1 equals straight y subscript 2 
    From condition straight y subscript 1 equals straight y subscript 2 comma we get 
    58.8 space straight t space minus space 4.9 space straight t squared space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t space minus space 4.9 straight t squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 58.8 space straight t space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t space space space space space straight u space equals space fraction numerator 117.6 over denominator square root of 3 end fraction space equals space 39.2 space square root of 3 space straight m divided by straight s space From space condition space straight x subscript 1 space equals space straight x subscript 2 comma space we space get space space space space space space 58.8 square root of 3 straight t space equals space 19.6 square root of 3 space plus space straight u over 2 straight t space rightwards double arrow space space space 58.8 square root of 3 straight t space equals space 19.6 square root of 3 space plus space 19.6 square root of 3 straight t space rightwards double arrow space space space space space space space straight t space equals space 1 half sec The space distance space of space foot space of space tree space from space the space first space hunter space is comma space straight S space equals space straight x subscript 1 space equals space straight u subscript 1 cosθ subscript 1 straight t space equals space 117.6 space cos 30 to the power of straight o cross times space 1 half space space space space equals space 29.4 square root of 3 space straight m space The space height space of space the space bird space is comma space straight h space equals space straight y subscript 1 space equals space 58.8 space straight t space minus space 4.9 straight t squared space space space space space space space space space space space space space space space equals 58.8 cross times 1 half space minus 4.9 open parentheses 1 half close parentheses squared space space space space space space space space space space space space space space space space equals space 28.175 space straight m
                  

    Question 397
    CBSEENPH11018596
    Wired Faculty App

    A stone is projected with velocity (√2gh) from the top of building h meter high. Show that it will fall farthest at a distance (2√2)h.
    Solution
    Let the stone be projected at angle θ to make the stone fall at farthest point and (0,R) be the coordinates of point where stone reaches the ground. 
     
    Here (0, R) lies on the trajectory. Therefore, we have, 
    0 space equals space straight h space plus space left parenthesis tanθ right parenthesis straight R space minus space open parentheses fraction numerator straight g over denominator 2 straight u squared end fraction sec squared straight theta close parentheses straight R squared space space space space space space space space... left parenthesis 2 right parenthesis space For space straight r space to space be space maximum comma space dR over dθ space equals space 0. So comma space differentiating space equation space left parenthesis 2 right parenthesis space straight w. straight r. straight t space straight theta comma space we space get 0 space equals space 0 plus left parenthesis sec squared straight theta right parenthesis straight R space plus space tanθ dR over dθ space space space space equals negative fraction numerator straight g over denominator 2 straight u squared end fraction left parenthesis sec squared straight theta right parenthesis 2 straight R dR over dθ space minus space fraction numerator straight g over denominator 2 straight u squared end fraction left parenthesis 2 space sec squared θtanθ right parenthesis straight R squared Putting space dR over dθ space equals space 0 comma space we space get space space space space space space space space left parenthesis sec squared straight theta right parenthesis straight R space minus space fraction numerator straight g over denominator 2 straight u squared end fraction left parenthesis 2 space sec squared θtanθ right parenthesis straight R squared space equals space 0 rightwards double arrow space space space space space tan space straight theta space equals space straight u squared over gR equals fraction numerator 2 gh over denominator gR end fraction equals fraction numerator 2 straight h over denominator straight R end fraction space space space space space space space... left parenthesis 3 right parenthesis Now comma space substituting space value space of space tanθ space in space equation space left parenthesis 2 right parenthesis comma space we space get space space space space space straight h space plus space straight u squared over gR straight R space minus open square brackets fraction numerator straight g over denominator 2 straight u squared end fraction open parentheses 1 plus fraction numerator straight u to the power of 4 over denominator straight g squared straight R squared end fraction close parentheses close square brackets straight R squared space equals space 0 rightwards double arrow space space straight h space plus straight u squared over straight g minus fraction numerator straight g over denominator 2 straight u squared end fraction straight R squared space minus 1 half straight u squared over straight g space equals space 0 rightwards double arrow space space space straight h space plus 1 half straight u squared over straight g space equals space fraction numerator straight g over denominator 2 straight u squared end fraction straight R squared space rightwards double arrow space space space straight h space plus 1 half fraction numerator 2 gh over denominator straight g end fraction space equals space fraction numerator straight g over denominator 2 cross times 2 gh end fraction straight R squared space rightwards double arrow space space straight R squared space equals 8 straight h rightwards double arrow space space straight R space equals 2 square root of 2 space space straight h
    Question 398
    CBSEENPH11018597
    Wired Faculty App

    A body is projected with velocity u from the top of building h meter high. Find the angle of projection so that it falls at farthest point.

    Solution
    Given, a body is projected with velocity u. 
    Height of the building = h
    To find - angle of projection theta = ? 
    Let the stone be projected at angle θ to make it fall at farthest point R from the foot of tower. 
                  
    The equation of trajectory of body is given by, 
    straight y space equals straight h space plus space xtan space straight theta space minus space fraction numerator straight g over denominator 2 space straight u squared cos squared straight theta end fraction straight x squared space space space space space equals space straight h space plus space left parenthesis tan space straight theta right parenthesis straight x space minus space space open parentheses fraction numerator straight g space sec squared straight theta over denominator 2 space straight u squared end fraction close parentheses straight x squared space space space space space space... left parenthesis 1 right parenthesis space Here comma space left parenthesis 0 comma straight R right parenthesis space lies space on space the space trajectory comma space we space have space therefore 0 space equals space straight h space plus space left parenthesis tan space straight theta right parenthesis straight R space minus space open parentheses fraction numerator straight g space sec squared straight theta over denominator 2 space straight u squared end fraction close parentheses straight R squared space space space space space space space space... left parenthesis 2 right parenthesis For space straight R space to space be space maximum comma space dR over dθ equals 0. So comma space differentiating space eq. space left parenthesis 2 right parenthesis space straight w. straight r. straight t space straight theta comma space we space get 0 space equals space 0 space plus space left parenthesis sec squared straight theta right parenthesis straight R space plus space left parenthesis tan space straight theta right parenthesis dR over dθ space space space space equals space minus fraction numerator straight g over denominator 2 straight u squared end fraction open parentheses sec squared straight theta close parentheses 2 straight R dR over dθ space minus space fraction numerator straight g over denominator 2 straight u squared end fraction open parentheses 2 sec squared straight theta space tanθ close parentheses straight R squared On space putting space dR over dθ space equals space 0 comma space we space get space space space space space space space space space space left parenthesis sec squared straight theta right parenthesis straight R space minus space fraction numerator straight g over denominator 2 straight u squared end fraction open parentheses 2 sec squared straight theta space tanθ close parentheses straight R squared space equals space 0 straight i. straight e. comma space space space space space space tanθ space equals space straight u squared over gR space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis Now comma space substituting space value space of space tan space straight theta space in space equation space left parenthesis 2 right parenthesis comma space we space get space space space space space space space space straight h space plus straight u squared over gR straight R space minus space open square brackets fraction numerator straight g over denominator 2 straight u squared end fraction open parentheses 1 plus fraction numerator straight u to the power of 4 over denominator straight g squared straight R squared end fraction close parentheses close square brackets straight R squared space equals space 0 space rightwards double arrow space space space space straight h space plus space straight u squared over straight g space minus fraction numerator straight g over denominator 2 straight u squared end fraction straight R squared space minus space 1 half straight u squared over straight g space equals space 0 rightwards double arrow space space space space space space straight u squared space plus space 2 gh space equals space fraction numerator straight g squared straight R squared over denominator straight u squared end fraction equals open parentheses gR over straight u squared close parentheses squared straight u squared space rightwards double arrow space space space space space space space space straight u squared over gR space equals space fraction numerator straight u over denominator square root of straight u squared plus 2 gh end root end fraction space space space space space space space space space space space space... left parenthesis 4 right parenthesis Now comma space from space equations space 3 right parenthesis space and space left parenthesis 4 right parenthesis comma space we space have space tanθ space equals space fraction numerator straight u over denominator square root of straight u squared plus 2 gh end root end fraction That space is comma space straight theta space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator straight u over denominator square root of straight u squared plus 2 gh end root end fraction close parentheses
    straight theta is the required angle og projection so that the body falls at the farthest point. 



    Question 399
    CBSEENPH11018598
    Wired Faculty App

    A particle is projected with velocity 2 square root of gh at such an angle that it just clears two walls of equal height h which are at a distance 2h from each other. Show that time of passing between the two walls is 2 square root of straight h over straight g end root.
    Solution
    Given, 
    Velocity with which the particle is projected = 2square root of gh 
    Height of the wall = h
    Distance between the walls = 2h
    Let the particle be projected at angle θ.
    Time taken by particle to cross the two walls each of height 'h' is equal to difference of instants when it would be at a height h above ground.
    The kinematics of motion of particle in vertical direction is given by, 
    space space space space space space straight y space equals space straight y subscript straight o space plus space straight u subscript straight y straight t space plus space 1 half straight a subscript straight y straight t squared space rightwards double arrow space straight h space equals space 0 space plus space straight u space sinθt space minus space 1 half gt squared space rightwards double arrow space gt squared space minus space 2 usinθ space straight t space plus space 2 straight h space equals space 0 The space difference space of space the space instant space when space the space particle space is space at space height space straight h space is comma space increment straight t space equals space fraction numerator square root of 4 straight u squared sin squared straight theta minus 8 gh end root over denominator straight g end fraction space space space space space space space Here comma space straight u space equals space 2 square root of gh space therefore space increment straight t space equals space fraction numerator square root of 16 ghsin squared straight theta space minus space 8 gh end root over denominator straight g end fraction space space space space... left parenthesis 1 right parenthesis The space distance space travelled space by space particle space in space horizontal space direction space in space the space interval space increment straight t space is comma space straight u space cosθ increment straight t space equals space 2 straight h straight i. straight e. comma space 2 square root of gh cosθ fraction numerator square root of 16 ghsin squared straight theta space minus space 8 gh end root over denominator straight g end fraction space equals space 2 straight h rightwards double arrow space space space cosθ square root of 16 space sin squared straight theta minus 8 end root space equals space 1 space rightwards double arrow space space left parenthesis 1 minus sin squared straight theta right parenthesis space left parenthesis 16 sin squared straight theta space minus space 8 right parenthesis space equals space 1 rightwards double arrow space space space 16 space sin to the power of 4 straight theta space minus space 24 space sin squared straight theta space plus space 9 space equals space 0 space rightwards double arrow space space space space space space sin squared straight theta space equals space fraction numerator 24 plus-or-minus square root of 576 minus 576 end root over denominator 32 end fraction space equals space 3 over 4 Substituting space the space value space of space sin squared straight theta space in space equation space left parenthesis 1 right parenthesis comma space we space get increment straight t space equals space fraction numerator square root of 12 space gh space minus space 8 gh end root over denominator straight g end fraction equals square root of fraction numerator 4 straight h over denominator straight g end fraction end root
    That is, 2 space square root of straight h over straight g end root is the time of passing between the two walls. 

    Question 400
    CBSEENPH11018599
    Wired Faculty App

    Define relative velocity.
    Solution

    Relative velocity of a body A w.r.t. B is the velocity with which A appears to move to B, if B were at rest.
    If space stack straight v subscript straight A with rightwards arrow on top is the velocity of A, and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is the velocity of B, then relative velocity of A w.r.t. B is, 
                           space space stack straight v subscript straight r with rightwards arrow on top space equals space stack straight v subscript straight A with rightwards arrow on top minus stack straight v subscript straight B with rightwards arrow on top

    Question 402
    CBSEENPH11018601
    Wired Faculty App

    A car moves a distance of 200 m. It covers the first half of the distance at speed 40 km/hr and second half of distance at speed v. The average speed is 48 km/hr. Find the value of v? 

    Solution

    We have, 
    Distance moved, s = 2 km 
    Velocity, u1 = 40 km/hr
    u2 = v 
    Average velocity = fraction numerator 0.2 over denominator 48 end fraction space equals space fraction numerator 0.1 over denominator 40 end fraction space plus space fraction numerator 0.1 over denominator v end fraction
    On solving the equation, we get
    v = 60 km/hr

    Question 403
    CBSEENPH11018602
    Wired Faculty App

    Two particles A and B are moving with speed u and v in mutually perpendicular directions. What is the magnitude of the relative velocity of B w.r.t. A? 

    Solution

    The magnitude of the relative velocity of B w.r.t A is, 
                         vr = square root of straight u squared plus straight v squared end root

    Question 404
    CBSEENPH11018603
    Wired Faculty App

    Two particles A and B are in motion. If the relative velocity of A w.r.t. B is <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, then what is the relative velocity of B w.r.t.A?
    Solution

    The relative velocity of B w.r.to A is same in magnitude but opposite in direction. 
    That is the relative velocity is -straight v with rightwards harpoon with barb upwards on top. 

    Question 405
    CBSEENPH11018604
    Wired Faculty App

    A balloon is ascending at the rate of 20 m/sec. When it is at a height 100 m from ground, a packet is dropped from the balloon.

    What will be the velocity of the packet w.r.t. 

    (i) ground

    (ii) balloon, when it was just dropped?  

    Solution

    Speed at which the balloon is ascending, v = 20 m/s
    Height of balloon from the ground = 100 m
    With respect to ground, the velocity of the packet will be 20 m/sec upward.
    With respect to balloon, its velocity will be zero.

    Question 406
    CBSEENPH11018605
    Wired Faculty App

    When the magnitude of relative velocity is equal to the sum of the magnitude of the velocities of two bodies?

    Solution
    When the two bodies are moving in opposite direction along a straight line, then the magnitude of relative velocity is equal to the sum of the magnitude of the velocities of two bodies.
    Question 407
    CBSEENPH11018606
    Wired Faculty App

    When the magnitude of relative velocity is equal to the difference of the magnitude of the velocities of two bodies?

    Solution
    When the two bodies are moving in same direction along a straight line, then the magnitude of relative velocity is equal to the difference of the magnitude of the velocities of two bodies.
    Question 408
    CBSEENPH11018607
    Wired Faculty App

    Two trains travelling on the same track are approaching each other with the equal speed of 40 m/s. The drivers of the train begin to deaccelerate simultaneously when just 2.0 km apart. Assuming deacceleration to be uniform and equal, the value of the deacceleration to barely avoid collision should be? 
    Solution
    We have, 
    Initial speed, u = 40 m/s 
    Distance to be covered by each train before rest, s = 1000 m 
    Acceleration, a = ? 
    Using the relation, 
                      v2 = u2 + 2as
    We have, 
    straight a space equals space fraction numerator left parenthesis straight v squared space minus space straight u squared right parenthesis over denominator 2 straight s end fraction space space space space equals space fraction numerator 0 space minus space left parenthesis 40 right parenthesis squared over denominator 2 space straight x space 1000 end fraction space space space space equals space 0.8 space straight m divided by straight s squared
    Question 410
    CBSEENPH11018609
    Wired Faculty App

    Rain is falling vertically downward. A man is going in the rain. Which side of the man will get wet?

    Solution
    The rain is falling vertically downwards but the man is in motion. As the man is moving forward, the probability of him getting wet is more. The man will escape the rainfalls falling in sides but, will be hit in the front. 
    Question 412
    CBSEENPH11018611
    Wired Faculty App

    A jet airplane travelling at the speed of 500 km/hr ejects its products of combustion at the speed of 1500 km/hr relative to the jet plane. What is the speed of the later with respect to an observer on the ground?

    Solution

    Let the velocity of ejecting gases with respect to ground be straight V subscript straight g . 
    Velocity of jet plane,    straight V subscript straight j equals plus 500 space km divided by hr
    Velocity of ejected products w.r.t. jet plane, 
                   straight V subscript gj equals negative 1500 space km divided by hr.
    Now, relative velocity of ejecting gases w.r.t the jet plane is given by, 
                               straight V subscript gj equals straight V subscript straight g minus straight V subscript straight j
    rightwards double arrow                        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                    equals negative 1500 plus 500 space equals space minus 1000 space km divided by hr 

    Question 413
    CBSEENPH11018612
    Wired Faculty App

    Two ships A and B start together from the same point. A sails 16 km/h towards east and B sails 20 km/h towards north. Find the relative velocity of B w.r.t. A and distance between them after 5 hours.

    Solution

    Velocity of ship A,space space space stack straight v subscript straight A space with rightwards arrow on top space equals space 16 straight i with hat on top
    Velocity of ship B,      stack straight v subscript straight B with rightwards arrow on top space equals space 20 straight j with hat on top
    The relative velocity of B w.r.t. A is,
              straight V with rightwards arrow on top space equals space stack straight v subscript straight B with rightwards arrow on top space minus space stack straight v subscript straight A with rightwards arrow on top space equals space 20 straight j with hat on top space minus 16 straight i with hat on top
    Magnitude of velocity is given by, 
    ∴      open vertical bar straight V with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 20 right parenthesis squared plus left parenthesis 16 right parenthesis squared end root space equals space 25.6 space km divided by hr
    and   tanθ space equals space 20 over 16 equals 1.25 space rightwards double arrow space straight theta space equals space 51.34 degree
    So, relative velocity of B w.r.t. A is 25.6 km/h due <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> 
    The distance between two ships after 5 hours, 
              S = Vt
                 = 25.6 x 5
                 = 128km.

    Question 414
    CBSEENPH11018613
    Wired Faculty App

    Two trains are moving in opposite directions on parallel tracks, one with a velocity twice that of the other. If the total length of two trains is 480 ft and they take 3 seconds to cross each other, find their individual velocities. 
    Solution
    Let the two trains A and B be moving in the opposite directions with velocities u and 2u respectively.

    Relative velocity, V = u + 2u = 3u 
    Total distance moved = total length of two trains.
    i.e. S = 480 ft.
    Time taken by trains to cross each other, t = 3 sec
    Now, S     = Vt
    rightwards double arrow    480 = 3u x 3
    rightwards double arrow      2u = 106.66 ft/s 
    rightwards double arrow        u = 53.33ft/s 

    Question 415
    CBSEENPH11018614
    Wired Faculty App

    In a harbour, wind is blowing at the speed of 72 km/h and a flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?

    Solution
    Let east be taken along X-axis and north along Y-axis.
     
    Velocity of wind is,
     stack straight v subscript straight w with rightwards arrow on top space equals space 72 space km divided by hr space along space NE space direction
         equals 72 space cos space 45 degree space straight i with hat on top space plus space 72 space sin space 45 degree space straight j with hat on top
         equals 50.9 straight i with hat on top space plus space 50.9 straight j with hat on top
    Velocity of boat,  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
                                  equals 51 straight j with hat on top
    The flag flutters in the direction of relative velocity of wind relative to boat.
    The relative velocity of wind w.r.t. the boat is,
    straight v with rightwards arrow on top subscript wb equals straight v with rightwards arrow on top subscript straight w minus straight v with rightwards arrow on top subscript straight b equals 50.9 straight i with hat on top minus 0.1 straight j with hat on top
    Since Y component of velocity is approximately zero, hence the flag will flutter almost due east.
    Question 416
    CBSEENPH11018615
    Wired Faculty App

    Two cars A and B are moving with same speed 30km/hr in same direction with car B 5km behind the car A. Another car C moving in opposite direction crosses the two cars at an interval of 4 minutes. Find the speed of car C.

    Solution
    Let the car C be moving with velocity u km/hr.
    Car B is 5 km behind the car A, therefore when the car C crosses car A, it is 5 km from car B.
    Let us take left direction as positive and right direction as negative. 
     
    Therefore,
    Velocity of car B = -5 km/hr
    Velocity of car C = + u km/hr
    The relative velocity of car C w.r.t car B is,
         v = Velocity of car C - Velocity of car
            = u - (-5) = (u + 5) km/hr
    Now the car C travels 5 km distance with relative velocity (u + 5) km/hr in 4 minutes.
    Therefore, using the relation s = ut, we have
                 5 equals left parenthesis straight u plus 30 right parenthesis cross times 4 over 60 
    rightwards double arrow      <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    rightwards double arrow        straight u space equals space 45 km divided by hr, is the speed of car C. 
    Question 417
    CBSEENPH11018616
    Wired Faculty App

    Two trains are heading towards each other on the same straight track with speed 15 m/sec and 10 m/sec. When they are 250 m apart the drivers of both the trains apply their brakes. If one train retards at the rate 0·5 m/s2 then what should be the retardation of the second train so that collision of two trains just averts?

    Solution
    Given, two trains are heading towards each other on the same straight track. 

    Initial relative speed of two trains,
    straight u subscript straight r equals 15 plus 10 space equals space 25 space straight m divided by sec
    Let 'a' be the retardation of second train.
    Therefore the relative retardation is,
    space space straight A equals left parenthesis straight a plus 0.5 right parenthesis space straight m divided by straight s squared

    The collision will just avert if the relative speed of the two trains just becomes zero after travelling the distance of 250m.
    Now,       space space space space straight v subscript straight r superscript 2 minus straight u subscript straight r superscript 2 space equals space 2 AS
    ∴            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    rightwards double arrow                625 space equals space 500 left parenthesis straight a plus 0.5 right parenthesis

    rightwards double arrow                   straight a equals 0.75 space straight m divided by straight s squared 
    Question 418
    CBSEENPH11018617
    Wired Faculty App

    Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km/hr in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 m/s2. If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them?

    Solution

    Here,
    Velocity of train A, straight u subscript straight A equals 72 km divided by hr equals 20 space straight m divided by straight s 
    Velocity of train B, straight u subscript straight B equals 72 km divided by hr equals 20 space straight m divided by straight s
     
    Acceleration of train A, straight a subscript straight A equals 0 space straight m divided by straight s comma     
    Acceleration of train B, straight a subscript straight B space equals space 1 space straight m divided by straight s squared

    Let initially A be ahead of B by a distance d. Let SB and SA be the distances travelled by trains B and A respectively in 50s.
    As at t = 50 s the position of both the trains is same.
    Therefore, 
                               <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Now,
      straight S subscript straight A equals straight u subscript straight A straight t plus 1 half straight a subscript straight A straight t squared 
      space space space equals 20 cross times 50 plus 1 half cross times 0 cross times left parenthesis 50 right parenthesis squared equals 1000 space straight m
     straight S subscript straight B equals straight u subscript straight B straight t plus 1 half straight a subscript straight B straight t squared
       equals 20 cross times 50 plus 1 half cross times 1 cross times left parenthesis 50 right parenthesis squared equals 2250 space straight m 
    ∴ original distance between them is, 
    space space space space straight d equals straight S subscript straight B minus straight S subscript straight A
         equals space 2250 space minus space 1000 space equals space 1250 space straight m 

    Question 419
    CBSEENPH11018618
    Wired Faculty App

    Two cars A and B are travelling in same direction with speed straight v subscript 1 space and space straight v subscript 2 respectively left parenthesis straight v subscript 1 greater than straight v subscript 2 right parenthesis. When car A is d distance behind the car B, the brakes on A are applied, causing retardation <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>  Show that to prevent a collision between A and B it is necessary that
                            left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared less than left parenthesis 2 αd right parenthesis
    Solution
    Here, inorder to prevent the collision of car A with B, the relative velocity of car A w.r.t. B should reduce to zero before car A undergoes the relative displacement of d w.r.t. car B.
    Initial velocity of car A = straight v subscript 1
    Initial velocity of car B = straight v subscript 2
    Therefore,
    Initial relative velocity of car A w.r.t. B is,
                       straight v equals straight v subscript 1 minus straight v subscript 2
    Acceleration of car A = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Acceleration of car B = 0
    Acceleration of car A w.r.t. car B = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    The relative displacement of car A w.r.t car B, when the relative velocity of car A w.r.t. B decreases to zero is given by,
                straight S equals fraction numerator 0 squared minus straight v squared over denominator negative 2 straight alpha end fraction equals fraction numerator straight v squared over denominator 2 straight alpha end fraction equals fraction numerator left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared over denominator 2 straight alpha end fraction
    For collision to avert S should be less than d.
    i.e.                        S < d
    rightwards double arrow                 fraction numerator left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared over denominator 2 straight alpha end fraction less than straight d
    rightwards double arrow                   <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Question 420
    CBSEENPH11018619
    Wired Faculty App

    A ship P is sailing due east at 30 km/h and another ship Q is going due south with a velocity of 40 km/h. Find the velocity of ship Q that will appear to an observer in P.

    Solution

    Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the unit vectors along east and south.

    The velocity of ship P, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>, along east.
    The velocity of ship Q, stack straight v subscript straight Q with rightwards arrow on top space equals space 40 straight j with hat on top, along south. 
    Velocity of the ship is given by, 
                     straight V with rightwards arrow on top space equals space stack straight v subscript straight Q with rightwards arrow on top minus stack straight v subscript straight P with rightwards arrow on top space equals space 40 straight j with hat on top minus 30 straight i with hat on top
    Mangitude of velocity,
            <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    and
          space space space tanθ equals 30 over 40 equals 0.75
      rightwards double arrow space space straight theta asymptotically equal to 37 degree
    Thus, the velocity of ship Q w.r.t. P is 50 km/hr due 37° west of south.

    Question 421
    CBSEENPH11018620
    Wired Faculty App

    A ship moves along equator to the east with velocity 30km/hr. The south-eastern wind blows at an angle of 60° to equator with velocity 15 km/hr. Find the wind velocity relative to ship and angle between the equator and the wind direction in reference frame fixed to the ship.

    Solution
    Velocity of the ship = 30 km/hr
    Velocity of south-eastern wind = 15 km/hr
    The velocity of wind relative to ship is given by,
    stack straight v subscript straight t with rightwards harpoon with barb upwards on top space equals space v with rightwards harpoon with barb upwards on top subscript w space minus space v with rightwards harpoon with barb upwards on top subscript s


    Here, we have
    Angle space between space stack straight v subscript straight w with rightwards harpoon with barb upwards on top space and space straight v with rightwards harpoon with barb upwards on top subscript straight s comma space straight theta space equals space 120 to the power of 0 Angle space between space straight v with rightwards harpoon with barb upwards on top subscript straight w space and space minus straight v with rightwards harpoon with barb upwards on top subscript straight s space equals space 60 to the power of straight o

    Now the magnitude of relative velocity of wind relative to ship is, 
    open vertical bar v with rightwards harpoon with barb upwards on top subscript r close vertical bar space equals space open vertical bar v with rightwards harpoon with barb upwards on top subscript w minus v with rightwards harpoon with barb upwards on top subscript s close vertical bar space equals space open vertical bar v with rightwards harpoon with barb upwards on top subscript w plus left parenthesis negative v with rightwards harpoon with barb upwards on top subscript s right parenthesis close vertical bar space space space space space space space space space space equals square root of open vertical bar v with rightwards harpoon with barb upwards on top subscript w close vertical bar squared plus open vertical bar negative v with rightwards harpoon with barb upwards on top subscript s close vertical bar squared plus 2 open vertical bar v with rightwards harpoon with barb upwards on top subscript w close vertical bar open vertical bar negative v with rightwards harpoon with barb upwards on top subscript s close vertical bar cos space left parenthesis 60 right parenthesis end root space space space space space space space space space space equals space square root of 15 squared plus 30 squared plus 2 cross times 15 cross times 30 cross times left parenthesis 5.0 right parenthesis end root space equals space square root of 1575 space rightwards double arrow space space open vertical bar v with rightwards harpoon with barb upwards on top subscript w minus v with rightwards harpoon with barb upwards on top subscript s close vertical bar space equals space 36.7 space m divided by s The space angle space that space the space reltive space velocity space makes space with space minus straight v with rightwards harpoon with barb upwards on top subscript straight s space is comma space space tan space alpha space equals space fraction numerator open vertical bar v with rightwards harpoon with barb upwards on top subscript w close vertical bar sin space 60 to the power of o over denominator open vertical bar negative v with rightwards harpoon with barb upwards on top subscript s close vertical bar plus open vertical bar v with rightwards harpoon with barb upwards on top subscript w close vertical bar c o s space left parenthesis 60 end fraction space equals space fraction numerator 15 cross times open parentheses begin display style bevelled fraction numerator square root of 3 over denominator 2 end fraction end style close parentheses over denominator 30 plus 15 left parenthesis 0.5 right parenthesis end fraction space equals space fraction numerator square root of 3 over denominator 5 end fraction rightwards double arrow space space space alpha space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 5 end fraction close parentheses space equals space 19.1
    Question 422
    CBSEENPH11018621
    Wired Faculty App

    A particle A is moving along a straight line with velocity 5 m/s and another particle B has a velocity 8 m/s at angle 37° to the path of A. Find the relative velocity of B w.r.t. A.

    Solution

    Consider, <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the velocities of particles A and B respectively.
    The relative velocity of B relative to A is given by,
                             space space stack straight v subscript straight B with rightwards arrow space on top minus stack straight v subscript straight A with rightwards arrow on top. 
    To find stack straight v subscript straight B with rightwards arrow on top minus stack straight v subscript straight A with rightwards arrow on top comma we will first change stack straight v subscript straight A with rightwards arrow on top into negative stack straight v subscript straight A with rightwards arrow on top and then add to <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> as shown in figure.


    We can see that,  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> makes an angle of 37 degree with stack straight v subscript straight A with rightwards arrow on top comma.
    Therefore <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> will subtend an angle <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> with <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>
    Now,   open vertical bar stack straight v subscript straight B with rightwards arrow on top minus stack straight v subscript straight A with rightwards arrow on top close vertical bar space equals space open vertical bar stack straight v subscript straight B with rightwards arrow on top plus left parenthesis negative stack straight v subscript straight A with rightwards arrow on top right parenthesis close vertical bar
    So,
    open vertical bar stack straight v subscript straight B with rightwards arrow on top plus left parenthesis negative stack straight v subscript straight A with rightwards arrow on top right parenthesis close vertical bar space equals space square root of straight v subscript straight B squared plus straight v subscript straight A squared plus 2 straight v subscript straight B straight v subscript straight A space cos left parenthesis 180 minus 37 degree right parenthesis end root          
    rightwards double arrow  open vertical bar stack straight v subscript straight B with rightwards arrow on top minus stack straight v subscript straight A with rightwards arrow on top close vertical bar space equals space square root of 8 squared plus 5 squared plus 2 cross times 8 cross times 5 cross times left parenthesis negative 0.8 right parenthesis end root
                      equals square root of 25
    rightwards double arrow   open vertical bar stack straight v subscript straight B with rightwards arrow on top minus stack straight v subscript straight A with rightwards arrow on top close vertical bar space equals space 5 space straight m divided by straight s
    The angle that the relative velocity makes with <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> is,
              tanα equals fraction numerator open vertical bar negative stack straight v subscript straight A with rightwards arrow on top close vertical bar space sin left parenthesis 180 degree minus 37 degree right parenthesis over denominator open vertical bar stack straight v subscript straight B with rightwards arrow on top close vertical bar plus open vertical bar negative stack straight v subscript straight A with rightwards arrow on top close vertical bar space cos left parenthesis 180 degree minus 37 degree right parenthesis end fraction
                    equals fraction numerator 5 cross times 0.6 over denominator 8 plus 5 left parenthesis negative 0.8 right parenthesis end fraction equals 3 over 4
    rightwards double arrow        <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>

    Question 423
    CBSEENPH11018622
    Wired Faculty App

    On a certain day, rain was falling vertically with a speed 30 m/s. The wind starts blowing after some time with a speed 10 m/s in N to S direction. In which direction should a boy, waiting at bus stop, hold his umbrella to save himself from rain?

    Solution

    Speed of rainfall = 30 m/s
    Speed of wind  = 10 m/s 

    The rain will appear to fall in the direction of the resultant of rain and wind, to the boy waiting at the bus stop.
    Let θ be the angle which the resultant velocity subtends with the vertical.
    From the figure, 
           space space space space space space tanθ equals straight v subscript straight w over straight v subscript straight r equals 10 over 30 equals 1 third
                      straight theta equals 18 degree 26 apostrophe
    Therefore, the boy should hold the umbrella at angle 18 degree 26 apostropheorth of vertical.

    Question 424
    CBSEENPH11018623
    Wired Faculty App

    Rain is falling vertically with speed of 30 m/s. A woman rides a bicycle with a speed of 10 m/s in N to S direction. In which direction she should hold the umbrella?

    Solution

    Let stack straight v subscript straight r with rightwards arrow on top be the velocity of rain and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre> be the velocity of the woman.
                     
    To the woman the rain will appear to be coming from the direction of stack straight v subscript straight r with rightwards arrow on top minus straight v with rightwards arrow on top. 
    Therefore, woman should hold the umbrella in the direction of left parenthesis stack straight v subscript straight r with rightwards arrow on top minus straight v with rightwards arrow on top right parenthesis which is given by, 
                space space space space tanθ equals straight v over straight v subscript straight r equals 10 over 30 equals 1 third
    i.e. she should hold the umbrella at an angle straight theta equals tan to the power of negative 1 end exponent left parenthesis 1 divided by 3 right parenthesis south of vertical.

    Question 425
    CBSEENPH11018624
    Wired Faculty App

    Two trains, each of length 100 m, are running on parallel tracks. One overtakes the other in 20 seconds and one crosses the other in 10 second. Calculate the velocities of each train? 

    Solution

    Let u and v be the velocity of two trains A and B. While overtaking the relative velocity of train A w.r.t B = u -v. 
    While crossing, the relative velocity of train A w.r.t B = u+ v.
    Total distance to be travelled by car A while crossing = 100 + 100 = 200 
    So, 
    space space space space space space space space space space 20 space equals space fraction numerator 200 over denominator straight u space minus space straight v end fraction space rightwards double arrow space space straight u space minus straight v space equals space 10 Also comma space space space space space space space space space space space space space space 10 space equals space fraction numerator 200 over denominator straight u space plus space straight v end fraction rightwards double arrow space space space space straight u plus straight v space equals space 20 space On space solving comma space we space get straight u space equals space 15 space straight m divided by straight s comma space is space the space velocity space of space first space train comma straight v space equals space 5 space straight m divided by straight s comma space is space the space velocity space of space the space second space train. space

    Question 426
    CBSEENPH11018625
    Wired Faculty App

    A body travelling along a straight line traversed one-third of the total distance with a velocity 4 m/s. The remaining part of the distance was covered with a velocity 6 m/s for the other half of time. What is the mean velocity averaged over the whole time of motion? 

    Solution

    Let s be the total distance travelled by body.
    If t1 is the time taken to cover first one-third distance, then 

                   straight t subscript 1 space equals space fraction numerator straight s divided by 3 over denominator 4 end fraction space equals space straight s over 12 space
    Let t2 be the time for each of the remaining two journeys. 
    Then, 

                     fraction numerator 2 straight s over denominator 3 end fraction space equals space 2 t subscript 2 space plus space 6 t subscript 2 space space space space space space space space space equals space 8 t subscript 2 

                        straight t subscript 2 space equals space fraction numerator 2 straight s over denominator 3 space straight x 8 end fraction space equals space straight s over 12 space 
    Therefore, 
    Average velocity = fraction numerator Total space displacement over denominator total space time end fraction 
                             = fraction numerator straight s over denominator straight t subscript 1 space plus space 2 straight t subscript 2 end fraction space equals space fraction numerator s over denominator left parenthesis s divided by 12 right parenthesis space plus space 2 left parenthesis s divided by 12 right parenthesis end fraction space equals space 4 space m divided by s

    Question 427
    CBSEENPH11018626
    Wired Faculty App

    Two persons A and B are walking with speed 4 km/hr and 5 km/hr respectively in the same direction. Find how far will be B from A after 3 hours? 

    Solution

    Relative velocity of B w.r.to A,  
    vBA = vB - vA
          = 5- 4 = 1 km/hr 
    Therefore, distance of B ahead of A in time is, 
    d = vAB x t
       = 1 x 3
        = 3 km

    Question 428
    CBSEENPH11018627
    Wired Faculty App

    Wind is blowing west to east along two parallel tracks. Two trains moving with the same speed in opposite directions have the same steam tracks, if one is double than the other, what is the speed of each train? 
    Solution

    Let v and w be the velocity of each train and wind respectively. 
    According to the question, we have
    (v+w) = 2 (v-w)

    rightwards double arrow space straight v space equals space 3 straight w 
    That is, velocity of each train is 3 times the velocity of wind. 

    Question 429
    CBSEENPH11018628
    Wired Faculty App

    Is it necessary to aim a little above the target hit at ? Explain.

    Solution

    Yes, for example if you are throwing a dart at the board, position the dart a little above the target in order to hit the target. This will enable you to hit the target accurately as the dart will follow a projectile motion. 

    Question 430
    CBSEENPH11018629
    Wired Faculty App

    A particle starts moving from position of rest under a constant acceleration. If it travels a distance x in t second, what distance will it travel in next t second? 
    Solution

    Given, 
    Initial velocity, u = 0
    Distance travelled, s = x
    Time taken to travel x distance, t = t sec
    Now, using the relation, 
    straight s space equals space ut plus space 1 half at squared comma we have
    x = 0 + 1/2 at2                     ... (1)
    Let the body travel a distance y in next t seconds. 
    The total distance travelled in t+t = 2t sec will be (x+y).
    Therefore, 
    (x+y) = 1/2 a (2t)2
              = 1/2a x 4t2                 ... (2)
    Dividing equation (2) by (1), we have
    space space space space space space fraction numerator left parenthesis straight x plus straight y right parenthesis over denominator straight x end fraction space equals space 4 space rightwards double arrow space space space space space space space space space space space straight y space equals space 3 straight x  
     
      


    Question 431
    CBSEENPH11018630
    Wired Faculty App

    A heavy and a light body are dropped from the top of a tower simultaneously. Which will reach the ground first?

    Solution

    During the vertical fall of a body, the time taken for the body to fall is given by, 
                         straight t space equals space square root of fraction numerator 2 straight h over denominator straight g end fraction end root ,
    which is independent of the mass of the body. 
    Hence, both the bodies will reach the ground simultaneously. 

    Question 432
    CBSEENPH11018631
    Wired Faculty App

    A train 100 m long is moving with a velocity of 60 km/hr. Find the time it takes to cross the bridge 1 km long. 

    Solution

    Total distance to be travelled, s = 100 + 1000 = 1100 m 

    Velocity, v = 60 km/hr = 50/3 m/s 
    Therefore, 
    Time taken to cross the bridge, t = straight s over straight v space equals space fraction numerator 1100 over denominator bevelled 50 over 3 end fraction space equals space 66 space sec 

    Question 433
    CBSEENPH11018632
    Wired Faculty App

    The blades of an aeroplane propeller are rotating at the rate of 600 revolutions per minute. Calculate its angular velocity? 

    Solution

    Here, 
    Angular frequency, straight nu = 600 r.pm 
                                    = 600 over 60 space r e v o l u t i o n divided by s e c 
    We know, 
    straight omega space equals space 2 πν space equals space 2 straight pi space straight x space 600 divided by 60 space space space space space equals space 20 space straight pi space rad divided by sec
    This is the required angular velocity. 

    Question 434
    CBSEENPH11018633
    Wired Faculty App

    A particle starts with a velocity of 200 cm/s and moves in a straight line with a retardation of 10 cm/s2. Find the time it takes to describe 1500 cm and explain the double answers. 

    Solution

    Here, we have
    Initial velocity, u = 200 cm/s 
    Acceleration, a = -10 cm/s2 
    Distance travelled, s = 1500 cm 
    Time taken, t = ? 
    Using the relation, 
            s = ut + 1/2 at2
      1500 = 200 t + 1/2(-10)t2
    On solving the equation,we get

    t = 10 s or t = 30 s
    Here, the value of 10 s corresponds to the time when the particle first arrives at the given location. It then crosses this point and at the end of 20 s, its velocity is just 0. 
    The particle then returns and at the end of 10 s more, is again at the given location from the starting point. 

    Question 435
    CBSEENPH11018634
    Wired Faculty App

    The K.E of a particle moving along a circle of radius r depends upon the distance covered s as, K = as2. What is the force acting on the particle? 
    Solution

    Given, 
    K.E, K = 1/2 mv2 
    This implies, 
    mv2 = 2K 
    When the body is moving in a circle, 

    straight F space equals space mv squared over straight r space equals space fraction numerator 2 straight K over denominator straight r end fraction space equals space fraction numerator 2 as squared over denominator straight r end fraction, is the required force. 

    Question 436
    CBSEENPH11018635
    Wired Faculty App

    Two trains 120 m and 80 m in length are running in opposite directions with velocities 42 km/hr and 30 km/hr. In what time they will completely cross each other?

    Solution

    Relative velocity of one train w.r.to second is, 
    = 42 - (-30)
    = 72 km/hr = 20 m/s
    Total distance to be travelled = 120 +80 = 200 m
    Time taken, t = 200 over 20 space equals space 10 space s 
    In 10 sec, the two trains will completely cross each other. 

    Question 438
    CBSEENPH11018637
    Wired Faculty App

    A train is moving with a velocity of 30 km/hr due east and a car is moving with a velocity of 40 km/hr due north. What is the velocity of car as  appears to a passenger in the train? 

    Solution

    Velocity of train, vT = 30 km /hr
    Velocity of car, vc = 40 km/hr
    To find: relative velocity of car w.r.t. train. To bring the train at rest, apply equal and opposite velocity of train on car, which is 30 km/hr. 
    So, relative velocity of car w.r.t train is, 
    vCT = OD = (OB2 + BD2)1/2
                   = (402 + 302)1/2 
                   = 50 km/hr 
    Let <DOB = straight theta, then 
     tan space straight theta space equals space BD over OB space equals space 30 over 40 space equals space 3 over 4 space space space space space space space space space space space equals space 0.75 space equals space tan space 36 to the power of straight o space 52 apostrophe space space space space space space space space space space equals space 36 to the power of straight o space 52 apostrophe comma space west space of space north. space 

    Question 439
    CBSEENPH11018638
    Wired Faculty App

    Two persons P and Q are standing 54 m apart on a long moving belt.  Person P rolls a round stone towards a person Q with a speed of 9 m/s with respect to belt. If the belt is moving with a speed of 4 m/s in the direction from P to Q. What is the speed of the stone w.r.to an observer on a stationary platform? 
    Solution

    Let the distance from P to Q be positive. 
    Given, speed of the belt, vb = +4 m/s
    Speed of the stone w.r.t belt, vs = + 9 m/s
    Speed of the stone w.r.t a stationary observer= vs + vb = 9+ 4 = 13 m/s

    Question 441
    CBSEENPH11018640
    Wired Faculty App

    Two trains 120 m and 80 m in length are running in opposite direction with velocities 42 km/hr and 30 km/hr. In what time they will complete cross each other? 

    Solution

    Relative velocity of one train w.r.t second, 
    = 42 - (-30)
    = 72 km /hr
    = 20 m/s 
    Total distance to be travelled = 120 + 80 = 200 m
    Time taken  = 200/20 = 10 sec

    Question 442
    CBSEENPH11018641
    Wired Faculty App

    Two buses started simultaneously towards each other from towns A and B which are 480 km apart. It took the first bus travelling from A to B eight hours to cover the distance and the second bus travelling from B to A, ten hours. Determine when the buses will meet after starting and at what distance from A. 
    Solution

    Here, distance between two towns A and B, S = 480 km 
    v1 = S/8   and
     
    v2 = S/10
    If these buses meet after time t,  then 
    S/8 x t + S/10 x t = S 
    On solving, we get 
    t = 40/9 hours 
    Therefore, distance from A = 480/80 x 40/9 = 800/3 = 266.7 km 

    Question 443
    CBSEENPH11018642
    Wired Faculty App

    A cyclist starts from the centre O of a circular track of radius 1 km, reaches the edge A of the track and then cycles along the circumference and stops at point B. If the total time taken is 10 minutes, what is the a) average velocity b) average speed of the cyclist. 

    Solution

    a) Displacement in 10 min (=1/6 hr) = OB = r = 1 km
    Therefore, 
    Average velocity = fraction numerator 1 space km over denominator 1 divided by 6 space straight h end fraction space equals space 6 space k m divided by h r
    b) Distance travelled in 1/6 hr = 0 to A (along radius)  + A to B (along the arc)
                                                  = r + rπ over 3
                                                  = 1 km + 1 km + fraction numerator 22 over denominator 7 space straight x space 3 end fraction
                                                   = 2.05 km
    Therefore, 
    Averge speed = fraction numerator 2.05 over denominator left parenthesis 1 divided by 6 right parenthesis straight h space end fraction = 12.30 km/hr

    Question 444
    CBSEENPH11018643
    Wired Faculty App

    A car travelling at 60 km/h overtakes another car travelling at 42 km/hr. Assuming each car to be 5.0 m long, find the time taken during the overtaking and the total road distance used for the overtake? 

    Solution

    Total distance to be travelled  for overtaking the car = 5.0  +5.0 = 10. 0 m 
    Relative velocity of first car w.r.t second car = 60 - 42 = 8 km/hr
         = 8 x 1000/ (60 x 60)
          = 50 m/s
    Time taken, t = 10/5 = 2 s.
    Road distance used for overtaking = distance travelled by first car in 2 s + length of first car
    = [ 60 x 1000 / (60 x60)] x 2+ 5
    = 38.33 m

    Question 445
    CBSEENPH11018644
    Wired Faculty App

    A police jeep is chasing a culprit going on a motorbike. The motorbike crosses a turning at a speed of 72 km/hr. the jeep follows it at a speed of 108 km/hr; crossing the turning 10 seconds later than the bike. Assuming that they travel at a constant speeds, how far from the turning will the jeep catch up with the bike. 

    Solution

    Speed of the motorbike  = 72 km/hr  = 20 m/s 
    Speed of police jeep = 108 km/hr = 30 m/s 
    Let the jeep catch up with the bike after t seconds. 
    Therefore, 
           30 t = 20 ( t + 10)

    rightwards double arrow        t = 20 sec 
    Distance travelled by jeep in 20 sec, 
             s =30 x 20 = 600 m 

    Question 446
    CBSEENPH11018645
    Wired Faculty App

    A person standing on a road has to hold his umbrella at 300 with the vertical to keep the rain away. He throws the umbrella and starts running at 10 m/s. He finds that raindrops are hitting his head vertically. Find the speed of raindrops with respect to, 

    a) the road,
    b) the moving person.

    Solution

    When a person is at rest with respect to ground, the rain is coming to him at an angle 30o with the vertical, i.e., along OB. 
    As the person moves along OA with velocity 10 m/s, the relative velocity of rain w.r.t person is along OC as shown in fig. below.
                            
    Here, o 
    Therefore, velocity of rain w.r.t ground = straight v with rightwards harpoon with barb upwards on top subscript r g end subscript space equals space stack O B with rightwards harpoon with barb upwards on top 
    Velocity of person w.r.t. ground = straight v with rightwards harpoon with barb upwards on top subscript p g end subscript space equals space stack O A with rightwards harpoon with barb upwards on top comma space w h e r e space O A space equals space 10 space m divided by s space equals space C B
    Velocity of rain w.r.t person = straight v with rightwards harpoon with barb upwards on top subscript r p end subscript space equals space stack O C with rightwards harpoon with barb upwards on top 
    a) increment OCB comma space OB space equals space CB divided by space sin space 30 to the power of 0 space space space space space space space space space space space space space space space space space space space space space equals space 10 divided by space left parenthesis 1 divided by 2 right parenthesis thin space equals space 20 Therefore comma space velocity space of space rain space straight w. straight r. straight t space ground space equals space 20 space straight m divided by straight s
    b)
    In increment space OCB comma space OB space equals space CB divided by tan space 30 to the power of straight o 
                              = 10/ (1/square root of 3) = 10square root of 3 
    Therefore, velocity of train w.r.t person = 10square root of 3 m/s  

    Question 447
    CBSEENPH11018646
    Wired Faculty App

    The blades of an aeroplane propeller are rotating at the rate of 600 revolutions per minute. Calculate its angular velocity. 

    Solution

    Here, 
    Frequency, straight nu space equals space 600 space revolutions space per space minute 
                      equals 600 over 60 revolutions per second 
    We know, 
    Angular frequency, straight omega = 2 πν space equals space 2 space straight pi space straight x space 600 over 60 
                                  = 20 straight pi rad sec-1 

    Question 448
    CBSEENPH11018647
    Wired Faculty App

    The radius of the earth's orbit around the sun is 1.5 x 1011 m. Calculate the angular and linear velocity of the earth? 

    Solution

    Given, 
    Radius of Earth's orbit, r = 1.5 x 1011 m 
    Time period of revolution of earth around the sun is 1 year. 
    That is, 
    T =  1 year = 365 x 24 x 60 x 60 s
    Therefore, 
    Angular velocity, straight omega space equals space fraction numerator 2 straight pi over denominator straight T end fraction
    equals space fraction numerator 2 space cross times space open parentheses begin display style bevelled 22 over 7 end style close parentheses over denominator 365 space straight x space 24 space straight x space 60 space straight x 60 end fraction space equals space 1.99 space cross times space 10 to the power of negative 7 end exponent space sec
    Linear velocity, v = straight omega space straight r 
                            equals space 1.99 space cross times space 10 to the power of negative 7 end exponent space cross times space 1.5 space cross times space 10 to the power of 11 equals space 2.99 space cross times space 10 to the power of 4 space straight m divided by straight s

    Question 449
    CBSEENPH11018648
    Wired Faculty App

    A plane flying horizontally at 100 m/s at a height of 1000 m releases a bomb from it. Find the time it takes to reach the ground. 

    Solution

    We have, 
    Initial distance, yo = 0
    Height attained after time t, y = 1000 m 
    Initial velocity, uy = 0 m/s 
    Acceleration, ay = 9.8 m/s2 
    Time = t 
    Now, using the equation of motion, we have 
    space space space space straight y space equals space straight y subscript straight o space plus space straight u subscript straight y straight t space plus space 1 half straight a subscript straight y straight t squared space therefore space 1000 space equals space 0 space plus space 0 space left parenthesis straight t right parenthesis space plus space 1 half cross times 9.8 cross times space straight t squared space space space space space space space space space space space space space equals space 4.9 space straight t squared space rightwards double arrow space straight t space equals space square root of fraction numerator 1000 over denominator 4.9 end fraction end root space equals space 100 over 7 space equals space 14.28 space sec
    Therefore, 14.28 sec ia taken for the bomb to reach the ground. 

    Question 450
    CBSEENPH11019940
    Wired Faculty App

    Pick out the two scalar quantities in the following list:

    force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.

    Solution
    Work and current are scalar quantities.

    The dot product of force and displacement gives us the amount of work done. And, we know that the dot product of two quantities is always a scalar. Therefore, work is a scalar physical quantity.
    Current is described only by its magnitude. Its direction is not taken. Hence, it is a scalar quantity.
    Question 451
    CBSEENPH11019941
    Wired Faculty App

    Pick out the only vector quantity in the following list:

    Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.

    Solution
    Impulse = change in momentum = force × time.
    Momentum and force are both vector quantities. Hence, impulse is the vector quantity.
    Question 452
    CBSEENPH11019942
    Wired Faculty App

    Establish the vector inequality:

    (a) |a + b| ≤ |a| + |b|
    Solution
    Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.



    Here, we can write: 
    OM = | a |                                 ...(i) 
        
    MN = OP = | b |                         ...(ii) 
    ON = | a + b |                            ...(iii) 
    In a triangle, each side is smaller than the sum of the other two sides. 
    Therefore, in ΔOMN, we have,
    ON < (OM + MN) 
    | a + b | < | a | + | b |                 ...(iv) 
    If the two vectors a and b act along a straight line in the same direction, then we can write: 
    | a + b | = | a | + | b |                  ... (v) 
    Combining equations (iv) and (v), we get: 
    | a + b | ≤ | a | + | b |
    Question 453
    CBSEENPH11019943
    Wired Faculty App

    Establish the vector inequalities geometrically or otherwise : 

    (b) |a+b| > ||a| −|b||

    Solution

    Let two vectors a and b be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure. 

     

    Here, we have: 
    | OM | = | a | 
    | MN | = | OP | = | b | 
    | ON | = | a + b | 
    In a triangle, each side is smaller than the sum of the other two sides. 
    Therefore, in ΔOMN, we have,
    ON + MN > OM 
    ON + OM > MN 
    | ON | > | OM - OP |   (∵ OP = MN) 
    | a + b | > | | a | - | b | |                 ...(iv) 
    If the two vectors a and b act along a straight line in the same direction, then we can write: 
    | a + b | = | | a | - | b | |                 ...(v) 
    Combining equations (iv) and (v), we get: 
    | a + b | ≥ | | a | - | b | |  

    Question 454
    CBSEENPH11019944
    Wired Faculty App

    Establish the vector inequalities geometrically or otherwise :

    (c) |a−b| < |a| + |b|

    Solution
    Let two vectors a and b be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

     
    Here we have: 
    | OR | = | PS | = | b |              ...(i) 
    | OP | = | a |                          ...(ii) 
    In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ΔOPS, we have: 
    OS < OP + PS 
    | a - b | < | a | + | -b | 
    | a - b | < | a | + | b |            ... (iii) 
    If the two vectors act in a straight line but in opposite directions, then we have,
    | a - b | = | a | + | b |             ... (iv) 
    Combining equations (iii) and (iv), we get, 
    | a - b | ≤ | a | + | b | 
    Question 455
    CBSEENPH11019945
    Wired Faculty App

    Establish the vector inequalities geometrically or otherwise :

    (d) |a−b| > ||a| − |b||

    Solution
    Let two vectors a and b be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.


    The following relations can be written for the given parallelogram. 
    OS + PS > OP                               ...(i) 
    OS > OP - PS                                ...(ii) 
    | a - b | > | a | - | b |                    ...(iii) 
    The quantity on the LHS is always positive and that on the RHS can be positive or negative.
    To make both quantities positive, we take modulus on both sides.
    That is, 
    | | a - b | | > | | a | - | b | | 
    | a - b | > | | a | - | b | |                 ...(iv) 
    If the two vectors act in a straight line but in the opposite directions, then we can write,
    | a - b | = | | a | - | b | |                   ...(v)
    Combining equations (iv) and (v), we get, 
    | a - b | ≥ | | a | - | b | |  
     
    Question 456
    CBSEENPH11019946
    Wired Faculty App

    Given a + b+ c + d = 0, which of the following statements
    are correct :

    (a) a, b, c, and d must each be a null vector,

    (b) The magnitude of (a + c) equals the magnitude of
    ( b + d),

    (c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,

    (d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear ?
    Solution

    a)

    Incorrect

    In order to make vectors a + b + c + d = 0, it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.

    (b)

    Correct

    a + b + c + d = 0

    a + c = – (b + d)

    Taking modulus on both the sides, we get,

    | a + c | = | –(b + d)| = | b + d |

    Hence, the magnitude of (a + c) is the same as the magnitude of (b + d).

    (c)

    Correct

    a + b + c + d = 0

    a = - (b + c + d)

    Taking modulus both sides, we get,

    | a | = | b + c + d |

    | a | ≤ | a | + | b | + | c | ... (i)

    Equation (i) shows that the magnitude of a is equal to or less than the sum of the magnitudes of b, c, and d.

    Hence, the magnitude of vector a can never be greater than the sum of the magnitudes of b, c, and d.

    (d)

    Correct

    For a + b + c + d = 0

    Question 457
    CBSEENPH11019947
    Wired Faculty App

    On an open ground, a motorist follows a track that turns to his left by an angle of 600 after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
    Solution

    The path followed by the motorist is a regular hexagon with side 500 m, as shown in the given figure.

    Let the motorist start from point P.

    The motorist takes the third turn at S.

    ∴ Magnitude of displacement = PS = PV + VS = 500 + 500 = 1000 m

    Total path length = PQ + QR + RS = 500 + 500 +500 = 1500 m

    The motorist takes the sixth turn at point P, which is the starting point.

    ∴ Magnitude of displacement = 0

    Total path length = PQ + QR + RS + ST + TU + UP

                          = 500 + 500 + 500 + 500 + 500 + 500

                          = 3000 m

    The motorist takes the eight turn at point R.

    ∴ Magnitude of displacement = PR
    P R space equals space square root of PQ squared plus QR squared plus 2 left parenthesis PQ right parenthesis left parenthesis QR right parenthesis space cos space 60 to the power of straight o end root space space space space space space space equals space square root of left parenthesis 500 right parenthesis squared plus left parenthesis 500 right parenthesis squared plus left parenthesis 2 cross times 500 cross times 500 cross times c o s space 60 to the power of o end root space space space space space space space equals square root of 250000 plus 250000 plus left parenthesis 500000 cross times 1 half right parenthesis end root space space space space space space space equals space 866.03 space m space beta space equals space t a n to the power of negative 1 end exponent space open parentheses fraction numerator 500 space s i n space 60 to the power of o over denominator 500 space plus space 500 space c o s space 60 to the power of o end fraction close parentheses space equals space 30 to the power of o  
    Therefore, the magnitude of displacement is 866.03 m at an angle of 30° with PR.
    This implies AC makes an angle 30° with the initial direction.
    Total path length = 8 × 500 = 4000 m. 

    Question 458
    CBSEENPH11019948
    Wired Faculty App

    Rain is falling vertically with a speed of 30 m s-1. A woman rides a bicycle with a speed of 10 m s-1 in the north to south direction. What is the direction in which she should hold her umbrella?

    Solution
    Let, 
    vc = Velocity of the cyclist, and
    vr = Velocity of falling rain 
    The woman must hold her umbrella in the direction of the relative velocity (v) of the rain, in order to protect herself from the rain. 
    straight v space equals space straight v subscript straight r space end subscript plus space left parenthesis negative straight v subscript straight c right parenthesis space space space space equals space 30 space plus space left parenthesis negative 10 right parenthesis space space space space equals space 20 space straight m divided by straight s space tan space straight theta space equals fraction numerator space straight v subscript straight c over denominator space straight v subscript straight r space end subscript end fraction space equals fraction numerator space 10 space over denominator 30 end fraction rightwards double arrow space straight theta space equals space tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 space over denominator 3 end fraction close parentheses space space space space space space space space space space equals tan to the power of negative 1 end exponent space left parenthesis 0.333 right parenthesis space almost equal to space 18 to the power of 0 space

    Therefore, the woman must hold the umbrella towards the south, at an angle of 18o.
    Question 459
    CBSEENPH11019950
    Wired Faculty App

    An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900
    km/h. Compare its centripetal acceleration with the acceleration due to gravity
    Solution
    Radius of the loop, r = 1 km = 1000 m 
    Speed of the aircraft, v = 900 km/h = 900 × 5 / 18  =  250 m/s
    Centripetal acceleration, straight a subscript straight c space equals space straight v squared over straight r
    = fraction numerator left parenthesis 250 right parenthesis squared over denominator 1000 end fraction space equals space 62.5 space m divided by s squared
    Acceleration due to gravity, g = 9.8 m/s2
    Therefore, 

    ac / g = 62.5 / 9.8  =  6.38
    ac = 6.38 g, is the required centripetal acceleration. 
    Question 460
    CBSEENPH11019951
    Wired Faculty App

    Read the statement below carefully and state, with reasons, if it is true or false: 

    A.

    (a) The net acceleration of a particle in circular motion is always along the radius of
    the circle towards the centre
    Solution
    A. FALSE
    The net acceleration of a particle is towards the centre only in case of a uniform circular motion. 

    Tips: -

    The net acceleration of a particle is towards the centre only in case of a uniform circular motion.
    Question 461
    CBSEENPH11019952
    Wired Faculty App

    Read the statement below carefully and state, with reasons, if it is true or false:


    A.

    The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point

    Solution
    A. TRUE

    Tips: -

    Because while leaving the circular path, the particle moves tangentially to the circular path.
    Question 462
    CBSEENPH11019953
    Wired Faculty App

    Read the statement below carefully and state, with reasons, if it is true or false:

    A.

    The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector

    Solution
    A. TRUE

    Tips: -

    The direction of acceleration vector in a uniform circular motion is directed towards the centre of circular path. It is constantly changing with time. The resultant of all these vectors will be zero vector.
    Question 466
    CBSEENPH11019957
    Wired Faculty App

     For any arbitrary motion in space, which of the following relations are true : 


    A.

    (a) vaverage = (1/2) (v (t1) + v (t2))
    Solution
    A. FALSE
     It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.
    Question 468
    CBSEENPH11019959
    Wired Faculty App

    For any arbitrary motion in space, which of the following relations are true :

    A.

    v (t) = v (0) + a t

    Solution
    A. FALSE

    Tips: -

    The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.
    Question 469
    CBSEENPH11019960
    Wired Faculty App

    For any arbitrary motion in space, which of the following relations are true :

    A.

     r (t) = r (0) + v (0) t + (1/2) a t2

    Solution
    A. FALSE

    Tips: -

    The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.
    Question 470
    CBSEENPH11019961
    Wired Faculty App

    For any arbitrary motion in space, which of the following relations are true :

    A.

     aaverage = [ v (t2) - v (t1 ) ] / ( t2 – t1)

    Solution
    A. TRUE

    Tips: -

     The arbitrary motion of the particle can be represented by this equation.
    Question 471
    CBSEENPH11019962
    Wired Faculty App

    A vector has magnitude and direction. Does it have a location in space ? Can it vary with time ? Will two equal vectors a and b at different locations in space necessarily have identical physical effects ? Give examples in support of your answer.

    Solution
    i) A vector in general has no definite location in space because a vector remains unaffected whenever it is displaced anywhere in space provided its magnitude and direction do not change. However, a position vector has a definite location in space.
    (ii) A vector can vary with time as example the velocity vector o an unaffected particle varies with time.
    (iii) Two equal vectors at different locations in space do not necessarily have same physical effects. For example, two equal forces acting at two different points on a body which can cause the rotation of a body about an axis will not produce equal turning effect.
    Question 472
    CBSEENPH11019963
    Wired Faculty App

    A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to the fixed, and neglect air resistance.

    Solution
    Given,
    Range, R = 3 km
    Angle of projection, θ = 30° 
    Acceleration due to gravity, g = 9.8 m/s2 

    Horizontal range for the projection velocity u0, is given by the relation: 
           R = u02 Sin 2θ / g 
    i.e.,  3 = u02 Sin 600 / g 
             u02 / g = 2√3                                    ...(i) 
    The maximum range (Rmax) is achieved by the bullet when it is fired at an angle of 45° with the horizontal, that is, 
               Rmax = u02 / g                                 ...(ii)
    On comparing equations (i) and (ii), we get:
    Rmax = 3√3 

            = 2 X 1.732 = 3.46 km 
    Hence, the bullet will not hit a target 5 km away.
    Question 473
    CBSEENPH11019964
    Wired Faculty App

    A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?
    Solution

    Speed of the cyclist, v = 27 km/h = 7.5 m/s
    Radius of the circular turn, r = 80 m
    Centripetal acceleration is,
         straight a subscript straight c space equals space straight v subscript straight c squared over straight r equals fraction numerator left parenthesis 7.5 right parenthesis to the power of 2 space end exponent over denominator space 80 space end fraction equals space 0.7 space ms to the power of negative 2 end exponent
    The question can be illustrated in the figure as shown below:

    Consider, the cyclist begins cycling from point P and moves toward point Q. 
    At point Q, he applies the breaks and decelerates the speed of the bicycle by 0.5 m/s2. 
    This acceleration is along the tangent at Q and opposite to the direction of motion of the cyclist.
    The angle between ac and aT is 900. Therefore, the resultant acceleration a is given by,
    straight a space equals space left parenthesis straight a subscript straight c to the power of 2 space end exponent plus space straight a subscript straight T squared right parenthesis to the power of 1 divided by 2 end exponent space space space space equals space open square brackets left parenthesis 0.7 right parenthesis squared space plus space left parenthesis 0.5 right parenthesis squared space close square brackets to the power of 1 divided by 2 end exponent space space space space space equals space left parenthesis 0.74 to the power of right parenthesis 1 divided by 2 end exponent space space space space space equals 0.86 space ms to the power of negative 2 end exponent space tan space straight theta space equals space fraction numerator straight a subscript straight c over denominator space straight a subscript straight T end fraction comma space space straight theta space is space the space angle space of space the space resultant space with space the space direction space of space velocity. space space space space space space space tan space straight theta space equals space fraction numerator 0.7 over denominator 0.5 end fraction space equals space 1.4 space rightwards double arrow space straight theta space space equals space space space tan to the power of negative 1 end exponent space left parenthesis 1.4 right parenthesis space equals space 54.56 to the power of straight o Therefore comma space the space magnitude space of space the space net space acceleration space of space the space cyclist space is space 0.86 space ms to the power of negative 2 end exponent space making space an space angle space of space 54.56 to the power of straight o space with space the space direction space of space velocity. space
    Question 474
    CBSEENPH11019965
    Wired Faculty App

    (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by,

                                  straight theta left parenthesis straight t right parenthesis thin space equals space tan to the power of negative 1 end exponent space open square brackets fraction numerator straight v subscript oy space minus space gt over denominator straight v subscript ox end fraction close square brackets

    Solution
    Let v0x and v0y respectively be the initial components of the velocity of the projectile motion along horizontal (x) and vertical (y) directions. 

    Let vx and vy be the horizontal and vertical components of velocity at a point P respectively. 
    Time taken by the projectile to reach point P = t
    Using the first equation of motion along the vertical and horizontal directions, we get
    vy = v0y = gt, and
    vx = v0x
    therefore space tan space straight theta space equals space straight v subscript straight y over straight v subscript straight x space equals space fraction numerator open parentheses straight v subscript 0 subscript straight y space end subscript minus space gt close parentheses over denominator straight v subscript 0 straight x end subscript end fraction rightwards double arrow space straight theta space equals space tan to the power of negative 1 end exponent space open parentheses fraction numerator straight v subscript 0 subscript straight y space end subscript minus space gt over denominator straight v subscript 0 straight x end subscript end fraction close parentheses
    Question 476
    CBSEENPH11020250
    Wired Faculty App

    A particle of mass m is moving along the side of a square of side ‘a’, with a uniform speed v in the x-y plane as shown in the figure:

    Which of the following statements is false for the angular momentum → L about the origin?

    • bold L space equals space minus fraction numerator mv over denominator square root of 2 end fraction space straight R space straight k with hat on top comma space when space the space paticle space is space moving space from space straight A space to space straight B
    • straight L space equals space mv space open parentheses fraction numerator straight R over denominator square root of 2 end fraction space plus straight a close parentheses space bold k with bold hat on top comma space when space the space particle space is space moving space from space straight B space to space straight C
    • straight L space equals space mv space open parentheses fraction numerator straight R over denominator square root of 2 end fraction space minus straight a close parentheses space straight k with hat on top comma space when space the space particle space is space moving space from space straight B space to space straight C
    • bold L space equals space minus fraction numerator mv over denominator square root of 2 end fraction space straight R space straight k with hat on top comma space when space the space paticle space is space moving space from space straight D space to space straight A

    Solution

    B.

    straight L space equals space mv space open parentheses fraction numerator straight R over denominator square root of 2 end fraction space plus straight a close parentheses space bold k with bold hat on top comma space when space the space particle space is space moving space from space straight B space to space straight C

    For a particle of mass, m is moving along the side of a square a. Such that
    Angular momentum L about the origin
    equals space straight L space equals space straight r space straight x space straight p space equals space straight r subscript straight p space sin space straight theta space straight n with hat on top space or space straight L space equals space straight r space left parenthesis straight p right parenthesis straight n with hat on top When space straight a space particle space is space moving space from space straight D space to space straight A straight L space equals space fraction numerator straight R over denominator square root of 2 end fraction space mv space left parenthesis negative straight k with hat on top right parenthesis straight A space particle space is space moving space from space straight A space to space straight B straight L space equals space fraction numerator straight R over denominator square root of 2 end fraction space mv space left parenthesis negative straight k with hat on top right parenthesis and space it space moves space from space straight C space to space straight D space straight L space equals space open parentheses fraction numerator straight R over denominator square root of 2 end fraction plus straight a close parentheses mv space left parenthesis straight k with hat on top right parenthesis For space straight B space to space straight C comma space we space have space straight L space space equals space open parentheses fraction numerator straight R over denominator square root of 2 end fraction plus straight a close parentheses mv space left parenthesis straight k with hat on top right parenthesis

    Question 479
    CBSEENPH11020273
    Wired Faculty App

    Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is

    • square root of GM over straight R end root
    • square root of 2 square root of 2 GM over straight R end root
    • square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root
    • 1 half square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root

    Solution

    D.

    1 half square root of GM over straight R left parenthesis 1 plus 2 square root of 2 right parenthesis end root

    Net force acting on any one particle M,
     =fraction numerator GM squared over denominator left parenthesis 2 straight R right parenthesis squared end fraction space plus space fraction numerator GM squared over denominator left parenthesis straight R square root of 2 right parenthesis squared end fraction cos space 45 to the power of straight o space plus space fraction numerator GM squared over denominator left parenthesis straight R square root of 2 right parenthesis squared end fraction Cos space 45 to the power of straight o space equals space GM squared over straight R squared open parentheses 1 fourth plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses

    This force will equal to centripetal force
    So, Mv squared over straight R space equals space GM squared over straight R squared open parentheses 1 fourth plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses
    straight v space equals space square root of fraction numerator GM over denominator 4 straight R end fraction left parenthesis 1 plus 2 square root of 2 right parenthesis end root space equals space 1 half square root of GM over straight R left parenthesis 2 square root of 2 plus 1 right parenthesis end root

    Question 480
    CBSEENPH11020302
    Wired Faculty App

    A boy can throw a stone up to a maximum height of 10 m. The maximum horizontal distance that the boy can throw the same stone up to will be

    • 20 square root of 2m

    • 10 m

    • 10 square root of 2 m

    • 20 m
    Solution

    D.

    20 m

    Maximum speed with which the boy can throw stone is
    straight u space equals space square root of 2 gh end root space equals space square root of 2 space straight x 10 straight x 10 end root space equals 10 square root of 2 space straight m divided by straight s
    Range is maximum when projectile is thrown at angle of 45o, Thus,
    straight R subscript max space equals space fraction numerator straight u squared over denominator straight g space end fraction space equals space fraction numerator left parenthesis 10 square root of 2 right parenthesis squared over denominator 10 end fraction space equals space 20 space straight m

    Question 481
    CBSEENPH11020303
    Wired Faculty App

    A particle of mass m is at rest at the origin at time t = 0. It is subjected to a force F (t) = F0e–bt in the x direction. Its speed v(t) is depicted by which of the following curves?
    Solution

    C.

    straight F equals space straight F subscript 0 straight e to the power of negative bt end exponent rightwards double arrow space straight a space equals space straight F over straight m space equals space straight F subscript 0 over straight m straight e to the power of negative bt end exponent rightwards double arrow space dv over dt space equals straight F subscript 0 over straight m straight e to the power of negative bt end exponent integral dv space equals space integral subscript 0 superscript straight t straight F over straight m straight e to the power of negative bt end exponent dt rightwards double arrow space straight v space equals straight F over straight m open square brackets table row cell negative 1 end cell row straight b end table close square brackets open square brackets table row cell straight e to the power of negative bt end exponent end cell end table close square brackets subscript 0 superscript straight t rightwards double arrow space straight v equals space straight F over mb left square bracket straight e to the power of negative bt end exponent right square bracket straight v equals space 0 space at space straight t equals space 0 and space straight v rightwards arrow space straight F over mb space as space straight t rightwards arrow infinity
    So, velocity increases continuously and attains a maximum value of v= F/mb as t ∞
    Question 483
    CBSEENPH11020328
    Wired Faculty App

    The figure shows the position –time (x – t) graph of one–dimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is

    • 0.4 (Ns)

    • 0.8 Ns

    • 1.6 Ns

    • 0.2 Ns
    Solution

    B.

    0.8 Ns

    From the graph it is a straight line, so uniform motion. Because of impulse direction of velocity changes as can be seen from the slope of the graph.
    (∴ impulse (l) = f x t
    ⇒ I  = mat
     space equals space fraction numerator straight m space left parenthesis straight v subscript 2 minus straight v subscript 1 right parenthesis over denominator straight t end fraction space straight t space equals space mv subscript 2 space minus space mv subscript 1
    Initial velocity, v1 = 2/2 = 1 ms-1
    Final velocity v2 = 2/2 = - 1 ms-1
    pi  = mv1 = 0.4 N-s
    pf  = mv2 = -0.4 N-s
    J = pf-pi = -0.4-0.4
     = - 0.8 N-s
    |J| = 0.8 N-s

    Question 484
    CBSEENPH11020333
    Wired Faculty App

    A point p moves in counter -clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out the length s = t3 + 5, where s is metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2 s is nearly

    • 13 ms-2

    • 12 ms-2

    • 7.2 ms-2

    • 14 ms-2
    Solution

    D.

    14 ms-2

    Given that, s  =t3 +5
    therefore speed v, = ds/st = 3t2
    and rate of change of speed, at = dv/dt = 6t
    ∴ Tangential acceleration at t =2s
    at = 6 x 2  = 12 ms-1
    and at t = 2s, v = 3 (2)2 = 12 ms-1
    ∴ Centripetal acceleration, ac = v2/R = 144/20 ms-2
    ∴ Net acceleration  = square root of straight a subscript straight t superscript 2 space plus straight a subscript straight c superscript 2 end root space almost equal to space 14 space ms to the power of negative 2 end exponent

    Question 487
    CBSEENPH11020354
    Wired Faculty App

    A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time?
    Solution

    A.

    Velocity at any time t is given by

    v = u + at

    v = v0 + (–g)t
    v = v0 – gt

    Straight line negative slope.

    Question 496
    CBSEENPH11020432
    Wired Faculty App

    The relation between time t and distance x is t=ax2 +bx where a and b are constants. The acceleration is

    • −2abv2

    • 2bv3

    • −2av3

    • 2av2
    Solution

    C.

    −2av3

     if t = ax2+bx
    then, differentiate both side with respect to time (t)
     
    dtdt = 2axdxdt +bdxdt1 = 2axv +bvv.(2ax + b) = 1(2ax + b) = 1/v
     
    Again differentiate both side with respect to time (t)
    2adxdt = -v2.dvdt2av = - v-2.accelerationacceleration = -2av3
    Hence, retardation= -2av3
    Question 500
    CBSEENPH11020464
    Wired Faculty App

    If space straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top space equals space straight B with rightwards arrow on top space straight x space straight A with rightwards arrow on top then the angle between A and B isπ

    • π

    • π/3

    • π/2

    • π/4
    Solution

    A.

    π

    open parentheses straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top close parentheses space equals space left parenthesis straight B with rightwards arrow on top space straight x straight A with rightwards arrow on top right parenthesis rightwards double arrow space open parentheses straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top close parentheses minus space left parenthesis straight B with rightwards arrow on top space straight x straight A with rightwards arrow on top right parenthesis space equals space 0 rightwards double arrow space open parentheses straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top close parentheses plus space left parenthesis straight B with rightwards arrow on top space straight x straight A with rightwards arrow on top right parenthesis space equals space 0 open square brackets therefore space left parenthesis straight B with rightwards arrow on top space straight x space straight A with rightwards arrow on top right parenthesis space equals space minus space left parenthesis straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top right parenthesis close square brackets 2 space open parentheses straight A with rightwards arrow on top space straight x space straight B with rightwards arrow on top close parentheses space equals 0 rightwards double arrow space space 2 AB space sin space straight theta space equals space 0 sin space straight theta space equals space 0 space space left square bracket space because space vertical line straight A with rightwards arrow on top vertical line space equals space straight A space not equal to 0 comma space vertical line straight B with rightwards arrow on top vertical line space equals space straight B not equal to space 0 right square bracket straight theta space equals space 0 space or space straight pi space
    Question 501
    CBSEENPH11020465
    Wired Faculty App

    A projectile can have the same range R for two angles of projection. If T1 and T2 be the time of flights in the two cases, then the product of the two time of flights is directly proportional to

    • 1/R2

    • 1/R

    • R

    • R2
    Solution

    C.

    R

    We know in advance that range of projectile is same for complementary angles i.e. for θ and (900 - θ )
    straight T subscript 1 space equals space fraction numerator 2 straight u space sin space straight theta over denominator straight g end fraction straight T subscript 2 space equals space fraction numerator 2 straight u space sin space left parenthesis 90 to the power of 0 minus straight theta right parenthesis over denominator straight g end fraction space equals space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction and space straight R space equals space fraction numerator straight u squared space sin space 2 straight theta over denominator straight g end fraction Therefore comma space straight T subscript 1 space straight T subscript 2 space equals space fraction numerator 2 space straight u space sin space straight theta over denominator straight g end fraction space straight x space fraction numerator 2 space straight u space cos space straight theta over denominator straight g end fraction space equals space fraction numerator 2 straight u squared space left parenthesis 2 space sin space straight theta space cos space straight theta right parenthesis over denominator straight g squared end fraction space equals space fraction numerator 2 straight u squared space left parenthesis sin space 2 straight theta right parenthesis over denominator straight g squared end fraction space equals space 2 straight R divided by straight g space equals space straight T subscript 1 straight T subscript 2 space proportional to space straight R

    Question 502
    CBSEENPH11020466
    Wired Faculty App

    Which of the following statements is false for a particle moving in a circle with a constant angular speed?

    • The velocity vector is tangent to the circle.

    • The acceleration vector is tangent to the circle.

    • The acceleration vector points to the centre of the circle.

    • The velocity and acceleration vectors are perpendicular to each other.
    Solution

    B.

    The acceleration vector is tangent to the circle.

    For a particle moving in a circle with constant angular speed, the velocity vector is always tangent to the circle and the acceleration vector always points towards the centre of the circle or is always along the radius of the circle. Since, the tangential vector is perpendicular to radial vector, therefore, velocity vector will be perpendicular to the acceleration vector. But in no case acceleration vector is tangent to the circle.
    Question 503
    CBSEENPH11020472
    Wired Faculty App

    A body of mass m accelerates uniformly from rest to v1 in time t1. The instantaneous power delivered to the body as a function of time t is

    • fraction numerator mv subscript 1 straight t over denominator straight t subscript 1 end fraction
    • fraction numerator mv subscript 1 superscript 2 space straight t over denominator straight t subscript 1 superscript 2 end fraction
    • fraction numerator mv subscript 1 straight t squared over denominator straight t subscript 1 end fraction
    • fraction numerator mv subscript 1 superscript 2 straight t over denominator straight t subscript 1 end fraction

    Solution

    B.

    fraction numerator mv subscript 1 superscript 2 space straight t over denominator straight t subscript 1 superscript 2 end fraction

    Let the constant acceleration of body of mass m is a.
    From equation of motion
    v1 = 0 + at1
    ⇒ a = t2/t1 = ...... (i)
    At an instant t, the velocity v of the body v = 0 + at
    straight v space equals space straight v subscript 1 over straight t subscript 1 straight t space space.... space left parenthesis ii right parenthesis therefore space instantaneous space power straight p space equals Fv space equals mav space space left parenthesis because space straight F space equals space ma right parenthesis equals space straight m open parentheses straight v subscript 1 over straight t subscript 1 close parentheses space straight x open parentheses straight v subscript 1 over straight t subscript 1 space straight x space straight t close parentheses space left square bracket space From space equ space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis space equals space fraction numerator mv subscript 1 superscript 2 straight t over denominator straight t subscript 1 superscript 2 end fraction

    Question 504
    CBSEENPH11020473
    Wired Faculty App

    A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle, the motion of the particle takes place in a plane. It follows that

    • its velocity is constant

    • its acceleration is constant

    • its kinetic energy is constant

    • it moves in a straight line.
    Solution

    C.

    its kinetic energy is constant

    When a force of constant magnitude acts on the velocity of particle perpendicularly, then there is no change in the kinetic energy of the particle. Hence, kinetic energy remains constant.

    Question 505
    CBSEENPH11020498
    Wired Faculty App

    Two masses m1 = 5 kg and m2 = 10 kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of the horizontal surface is 0.15. The minimum weight m that should be put on top of m2 to stop the motion is :

    • 10.3 kg

    • 18.3 kg

    • 27.3 kg

    • 43.3 kg
    Solution

    C.

    27.3 kg

    Given: m1 = 5kg;
    m2 = 10 kg
    μ = 0.15
    For m1, m1g -T =m1a
    = 50-T = 5 x a
    and T - 0.15 (m+10) g = (10 + m)a
    For rest a = 0
    or 50 = 0.15 (m +10) 10


    ⇒ 5 = 320 (m + 10)1003 = m + 10m = 23.3 kg

    The minimum value from the options, satisfying the above condition is, m = 27.3 kg

    Question 506
    CBSEENPH11020514
    Wired Faculty App

    A body of mass 1 kg begins to move under the action of a time dependent force F = left parenthesis 2 straight t space straight i with hat on top space plus space 3 straight t squared space straight j with hat on top right parenthesis N, where straight i with hat on top space a n d space j with hat on top are units vectors along X and Y axis. What power will be developed by the force at the time (t)?

    • (2t2 + 4t4) W

    • (2t3 + 3 t4) W

    • (2t3 + 3t5) W

    • (2t + 3t3)W
    Solution

    C.

    (2t3 + 3t5) W

    A body of mass 1 kg begins to move under the action of time dependent force,
    F = (2t straight i with hat on top+3t2 straight j with hat on top) N,
    where stack straight i space with hat on top a n d space j with hat on top are unit vectors along X and Y axis.
    F = ma
    rightwards double arrow space straight a space equals space straight F over straight m
    rightwards double arrow space straight a space equals space fraction numerator left parenthesis 2 straight t space straight i with hat on top space plus space 3 straight t squared space straight j with hat on top right parenthesis over denominator 1 end fraction space space left square bracket space straight m space equals 1 space kg right square bracket rightwards double arrow space straight a space equals space left parenthesis 2 straight t space straight i with hat on top space plus space 3 straight t squared space straight j with hat on top right parenthesis space straight m divided by straight s squared
    Acceleration, a = dv over dt
    rightwards double arrow space dv space equals space straight a. space dt space space space space space space... space left parenthesis straight i right parenthesis
    Integrating both sides, we get
    integral d v space equals space integral a. space d t space space space space space space space space space equals space integral left parenthesis 2 t space i with hat on top space plus space 3 t squared space j with hat on top right parenthesis space d t v space equals space t squared space space i with hat on top space plus straight t cubed space straight j with hat on top
    Power developed by the force at the time t will be given as,
    P = F.v = (2t straight i with hat on top + 3t2 straight j with hat on top).(straight t squared space straight i with hat on top space plus space straight t cubed space straight j with hat on top)
       = (2t. t2 + 3t2.t3)
    P = (2t3 + 3t5) W

    Question 507
    CBSEENPH11020520
    Wired Faculty App

    If the velocity of a particle is v = At + Bt2, where A and B are constants, then the distance travelled by it between 1 s and 2 s is,

    • 3A + 7B

    • fraction numerator 3 over denominator 2 space end fraction A space plus space 7 over 3 B
    • straight A over 2 plus B over 3
    • fraction numerator 3 over denominator 2 space end fraction A space plus space 4 B

    Solution

    B.

    fraction numerator 3 over denominator 2 space end fraction A space plus space 7 over 3 B

    Velocity of the particles is given as,
    v = At + Bt2, where a and B are constants.
    rightwards double arrow space dx over dt space equals space At space plus space Bt squared space space open square brackets because space straight v space equals dx over dt close square brackets rightwards double arrow space dx space equals space left parenthesis At space plus space Bt squared right parenthesis space dt
    Integrating both sides, we get
    space space space space space space integral subscript straight x subscript 1 end subscript superscript straight x subscript 2 end superscript space d x space equals space integral subscript 1 superscript 2 space left parenthesis A t space plus space B t squared right parenthesis space d t increment x space equals space x subscript 2 space minus space x subscript 1 space space space space space space equals space A integral subscript 1 superscript 2 t. space d t space plus space B integral subscript 1 superscript 2 t squared. d t space space space space space space equals space A open square brackets t squared over 2 close square brackets subscript 1 superscript 2 space plus space B space open square brackets t cubed over 3 close square brackets subscript 1 superscript 2 space space space space space space equals space A over 2 left parenthesis 2 squared space minus space 1 squared right parenthesis space plus space B over 3 left parenthesis 2 cubed minus space 1 cubed right parenthesis
    Therefore, distance travelled between 1 s and 2 s is,
    increment straight x equals straight A over 2 straight x left parenthesis 3 right parenthesis plus straight B over 3 left parenthesis 7 right parenthesis space space space space space space equals space fraction numerator 3 straight A over denominator 2 end fraction plus fraction numerator 7 straight B over denominator 3 end fraction

    Question 508
    CBSEENPH11020523
    Wired Faculty App

    A particle  moves so that its position vector is given by r = cos ωt space straight x with hat on top space plus space sin space ωt space straight y with hat on top where straight omega is a cosntant. which of the following is true?

    • velocity and acceleration both are parallel to r

    • velocity is perpendicular to r and acceleration is directed towards the origin

    • velocity is perpendicular to r and acceleration is directed away from the origin

    • velocity and acceleration both are perpendicular to r
    Solution

    B.

    velocity is perpendicular to r and acceleration is directed towards the origin

    Position vector of the article is given by,
    straight r space equals space cos space ωt space stack straight x space with hat on top space plus space sin space ωt space straight y with hat on top
    where straight omega is a constant.
    Velocity of the particle is,
    straight v space equals space dr over dt space equals space straight d over dt open parentheses cos space straight omega space straight t space straight x with hat on top space plus space sin space ωt space straight y with hat on top close parentheses
    straight v space equals space left parenthesis negative sinωt right parenthesis straight omega stack straight x space with hat on top space plus space left parenthesis cos space ωt right parenthesis straight omega straight y with hat on top space space space equals space minus space straight omega space left parenthesis space sin space ωt space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis Acceleration space of space the space particle comma straight a space equals space dv over dt space space equals straight d over dt open square brackets negative straight omega space sin space ωt space straight x with hat on top space plus straight omega space cos space ωt space straight y with hat on top close square brackets space equals space minus straight omega squared space cos space ωt space straight x with hat on top space minus space straight omega squared space sin space ωt space straight y with hat on top space equals space minus straight omega squared space open parentheses cos space ωt space straight x with hat on top space space minus sin space ωt space straight y with hat on top space close parentheses straight a space equals space minus straight omega squared space straight r space equals straight omega squared space left parenthesis negative straight r right parenthesis space
    The particle is at a point P. That is, its position vector is directed as shown below:

    Therefore, acceleration is directed towards -r, that is towards "O"
    We have
    straight v with rightwards harpoon with barb upwards on top. r with rightwards harpoon with barb upwards on top space equals space minus straight omega space left parenthesis sinω space straight t space straight x with hat on top space minus space cos space ωt space straight y with hat on top right parenthesis. space space space space space space space space space space space space space space space space space space space left parenthesis cos space ωt space straight x with hat on top space plus space sin space ωt space straight y with hat on top right parenthesis 
           equals space minus straight omega space left square bracket space sin space straight omega. space cosωt space plus space space space space space space space space space space space space 0 plus 0 space minus space sin space ωt. space cos space ωt right square bracket 
    equals space minus straight omega space left parenthesis 0 right parenthesis space equals space 0 rightwards double arrow space straight v perpendicular space straight r
    Question 510
    CBSEENPH11020545
    Wired Faculty App

    What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

    • square root of 2 gR end root
    • square root of 3 gR end root
    • square root of 5 gR end root
    • square root of gR

    Solution

    C.

    square root of 5 gR end root

    The question is illustrated in the figure below,

    Let, the tension at point A be TA.
    Using Newton's second law, we have
    straight T subscript straight A space minus space mg space equals space mv subscript straight c squared over straight R
    Energy at point A = 1 half m v subscript o squared space space space space space space space space space... space left parenthesis i right parenthesis
    Energy at point C is,
    1 half m v subscript c squared space plus space m g space x space 2 R space space space space space space space space space space... space left parenthesis i i right parenthesis
    At point C, using Newton's second law,
    straight T subscript straight c space plus space mg space equals space mv subscript straight c squared over straight R
    In order to complete a loop, Tc greater-than or slanted equal to space 0
    So,
     mg space equals space mv subscript straight c squared over straight R rightwards double arrow space straight v subscript straight c space equals space square root of gR space space space space space space space space... space left parenthesis iii right parenthesis
    From equation (i) and (ii)
    Using the principle of conservation of energy,
    1 half m v subscript o squared space equals space 1 half m v subscript c squared space plus space 2 space m g R rightwards double arrow space 1 half m v subscript o squared italic space italic equals italic space italic 1 over italic 2 m g R italic space italic plus italic space italic 2 m g R italic space x italic space italic 2 italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space open square brackets v subscript c italic space italic equals italic space square root of g R end root close square brackets italic rightwards double arrow italic space v subscript o to the power of italic 2 italic space italic equals italic space g R italic space italic plus italic space italic 4 g R italic rightwards double arrow italic space italic space v subscript o italic space italic equals italic space square root of italic 5 g R end root
    Question 511
    CBSEENPH11020558
    Wired Faculty App

    A particle of mass 10 g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration, if the kinetic energy of the particle becomes equal to 8 x 10-4 J by the end of the second revolution after the beginning of the motion?

    • 0.15 m/s2

    • 0.18 m/s2

    • 0.2 m/s2

    • 0.1 m/s2
    Solution

    D.

    0.1 m/s2

    Given, mass of particle. m = 0.01 kg
    Radius of circle along which particle is moving , r = 6.4 cm
    Kinetic energy of particle, K.E. = 8 x 10-4 J
    rightwards double arrow space 1 half mv squared space equals space 8 space straight x space 10 to the power of negative 4 end exponent space straight J rightwards double arrow space straight v squared space equals space fraction numerator 16 space straight x space 10 to the power of negative 4 end exponent over denominator 0.01 end fraction space space space space space space space space space space space equals 16 straight x 10 to the power of negative 2 end exponent space space space space space space space space space space space... space left parenthesis straight i right parenthesis
    Given that, KE of particle is equal to 8 x 10-4 J by the end of second revolution after the beginning of the motion of particle.
    It means, initial velocity (u) is 0 m/s at this moment.
    Now, using the Newton's 3rd equation of motion,
    v2 = u2 + 2as
    rightwards double arrow space space space straight v squared space equals space 2 straight a subscript straight t straight s rightwards double arrow space straight v squared space equals space 2 straight a subscript straight t space left parenthesis 4 πr right parenthesis rightwards double arrow space straight a subscript straight t space equals space fraction numerator straight v squared over denominator 8 πr end fraction space space space space space space space space space equals space fraction numerator 16 space straight x space 10 to the power of negative 2 end exponent over denominator 8 space straight x space 3.14 space straight x space 6.4 space straight x space 10 to the power of negative 2 end exponent end fraction space space space space straight a subscript straight t space equals space 0.1 space straight m divided by straight s squared

    Question 513
    CBSEENPH11020568
    Wired Faculty App

    A balloon with mass m is descending down with an acceleration a (where a <g). How much mass should be removed from it so that it starts moving up with an acceleration a?

    • fraction numerator 2 ma over denominator straight g plus straight a end fraction
    • fraction numerator 2 ma over denominator straight g minus straight a end fraction
    • fraction numerator ma over denominator straight g plus straight a end fraction
    • fraction numerator ma over denominator straight g minus straight a end fraction

    Solution

    A.

    fraction numerator 2 ma over denominator straight g plus straight a end fraction

    When the balloon is descending down with acceleration a,

    So, mg - B = mx A             ... (i)
    where B is the buoyant force.
    We assume here that while removing same mass, the volume of balloon and hence buoyant force will not change.
    Let, us assume the new mass of the balloon is m'.
    So, mass removed is (m-m')
    Therefore,
    B-m'g = m'x a         ... (ii)
    On solving equations (i) and (ii), we have
    mg - B = m x a
    B - m'g = m' x a
    mg - m'g = ma +  m'a
    (mg - ma) = m' (g+a) = m (g-a) = m' (g+a)
    That is,
    straight m apostrophe space equals space fraction numerator straight m space left parenthesis straight g minus straight a right parenthesis over denominator straight g plus straight a end fraction
    That is, mass removed is m-m'
    equals space straight m space open square brackets fraction numerator 1 minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets equals space straight m open square brackets fraction numerator left parenthesis straight g plus straight a right parenthesis minus left parenthesis straight g minus straight a right parenthesis over denominator left parenthesis straight g plus straight a right parenthesis end fraction close square brackets equals space straight m open square brackets fraction numerator straight g plus straight a minus straight g plus straight a over denominator straight g plus straight a end fraction close square brackets equals space increment straight m space equals space fraction numerator 2 ma over denominator straight g plus straight a end fraction

    Question 514
    CBSEENPH11020590
    Wired Faculty App

    A particle has initial velocity (2i+3j) and acceleration (0.31i +0.2j). The magnitude of velocity after 10 s will be

    • 9 square root of 2 units
    • 5 square root of 2 units

    • 5 units

    • 9 Units
    Solution

    B.

    5 square root of 2 units

    From the equation of motion, we knows,
    v=u+ at
    v= (2i+3j)
    a=(0.3i +0.2j)
    v=(2i+3j) +(0.3i +0.2j) x 10
    v=5i+5j
    straight v equals space 5 square root of 2 space units

    Question 515
    CBSEENPH11020602
    Wired Faculty App

    The velocity of a projectile at the initial point A is (2 i + 3 j) m/s. Its velocity (in m/s) at point B is,

    • -2 i - 3 j

    • -2 i + 3 j

    • 2 i - 3 j

    • 2 i + 3 j
    Solution

    C.

    2 i - 3 j

    From the fig,, the X component remain unchanged, while the Y-component is reverse. Then, the velocity at point B is (2 i - 3 j) m/s.

    Question 516
    CBSEENPH11020632
    Wired Faculty App

    A  particle of mass m is released from rest and follows a parabolic path is shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time?
    Solution

    D.

    Potential V (x) versus x is parabolic, SHM starts from an extreme position and x versus t should be a cosin curve.

    Question 518
    CBSEENPH11020664
    Wired Faculty App

    Two particles A and B, move with constant velocities v1 and v2. At the initial moment, their position vectors are r1 and r2 respectively. The condition for particles A and B for their collision is 

    • fraction numerator straight r subscript 1 space equals space straight r subscript 2 over denominator vertical line straight r subscript 1 minus straight r subscript 2 vertical line end fraction space equals space fraction numerator v subscript 2 space equals space v subscript 1 over denominator vertical line v subscript 2 minus v subscript 1 vertical line end fraction

    • r1.v1 = r2.v2

    • r1 x v1 = r2 x v2

    • r1 - r2 = v1-v2
    Solution

    A.

    fraction numerator straight r subscript 1 space equals space straight r subscript 2 over denominator vertical line straight r subscript 1 minus straight r subscript 2 vertical line end fraction space equals space fraction numerator v subscript 2 space equals space v subscript 1 over denominator vertical line v subscript 2 minus v subscript 1 vertical line end fraction

    For two particles A and B moves with constant velocitites v1 and v2, such that two particles to collide, the direction of the relative velocity of one with respect to other should be directed towards the relative position of the other particle.i.e., fraction numerator straight r subscript 1 minus straight r subscript 2 over denominator vertical line straight r subscript 1 minus straight r subscript 2 vertical line end fraction space rightwards arrow spacedirection of relative position of 1 w.r.t
    similarly, fraction numerator straight v subscript 1 minus straight v subscript 2 over denominator vertical line straight v subscript 1 minus straight v subscript 2 vertical line end fraction space rightwards arrow direction of velocity of 2 w.r.t 1
    so, for collision of A and B, we get

    fraction numerator straight r subscript 1 space equals space straight r subscript 2 over denominator vertical line straight r subscript 1 minus straight r subscript 2 vertical line end fraction space equals space fraction numerator v subscript 2 space equals space v subscript 1 over denominator vertical line v subscript 2 minus v subscript 1 vertical line end fraction

    Question 521
    CBSEENPH11020691
    Wired Faculty App

    A particle moves a distance x in time t according to equation x  = (t +5)-1. The acceleration of particle is proportional to

    • (velocity)3/2

    • (distance)2

    • (distance)-2

    • (velocity)2/3
    Solution

    A.

    (velocity)3/2

    Given, distance x = ( t + 5)-1   ...... (i)
    Differentiating eq. (i) w.r.t, we get
    dx over dt space equals space straight v space equals space fraction numerator negative 1 over denominator left parenthesis straight t plus 5 right parenthesis squared end fraction space space space space.... space left parenthesis ii right parenthesis Again space comma space differentiating space eq space left parenthesis straight i right parenthesis thin space straight w. straight r. straight t comma space we space get fraction numerator straight d squared straight x over denominator dt squared end fraction space equals space left parenthesis straight a right parenthesis space equals space fraction numerator 2 over denominator left parenthesis straight t space plus 5 right parenthesis cubed end fraction Comparing space eqs. space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis comma space we space get left parenthesis straight a right parenthesis space proportional to space left parenthesis straight v right parenthesis to the power of 3 divided by 2 end exponent 

    Question 522
    CBSEENPH11020692
    Wired Faculty App

    Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be

    • 2v

    • 0

    • 1.5 v

    • v
    Solution

    B.

    0

    As initially, both the particles were at rest, therefore, velocity of the centre of mass was zero and there is no external force on the system so the speed of the centre of mass remains constant i.e, it should be equal to zero.

    Question 523
    CBSEENPH11020695
    Wired Faculty App

    Six vectorsbold a with bold rightwards arrow on top and bold f with bold rightwards arrow on tophave the magnitudes and directions indicated in the figure. Which of the following statements is true?


    • bold b with bold rightwards arrow on top bold space bold plus bold c with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top
    • bold d with bold rightwards arrow on top bold space bold plus bold c with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top
    • stack bold d bold space with bold rightwards arrow on top bold space bold plus bold space bold e with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top
    • bold b with bold rightwards arrow on top bold space bold plus bold space bold e with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top

    Solution

    C.

    stack bold d bold space with bold rightwards arrow on top bold space bold plus bold space bold e with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top

    If two non -zero vectors are represented by the two adjacent sides of a parallelogram, then the resultant is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors
    therefore, stack bold d bold space with bold rightwards arrow on top bold space bold plus bold space bold e with bold rightwards arrow on top bold space bold equals bold space bold f with bold rightwards arrow on top

    Question 524
    CBSEENPH11020704
    Wired Faculty App

    The speed of the projectile at its maximum height is half of its initial speed. The angle of projection is

    • 60o

    • 15o

    • 30o

    • 45o
    Solution

    A.

    60o

    The speed of projectile at its maximum height
    v' = vo cos θ
    vo/2 = vo cos θ
    cos θ = 1/2
    θ = 60o

    Question 526
    CBSEENPH11020723
    Wired Faculty App

    A particle starts its motion from rest under the action of a constant force. If the distance covered in first 10 s is s1 and that covered in the first 20 s is s2, then

    • s2 = 2s1

    • s2 = 3s1

    • s2 = 4 s1

    • s2 = s1
    Solution

    C.

    s2 = 4 s1

    If the particle is moving in a straight line under the action of a constant force then distance covered s = ut + at2 /2
    since the body start from rest u = 0
    therefore, s = at2/2
    Now,
    straight s subscript 1 space equals 1 half straight a left parenthesis 10 right parenthesis squared space space space space.... space left parenthesis straight i right parenthesis and space straight s subscript 2 space equals space 1 half space straight a space left parenthesis 20 right parenthesis squared space.... space left parenthesis ii right parenthesis Dividing space eq space left parenthesis straight i right parenthesis space and space eq space left parenthesis ii right parenthesis comma space we space get straight s subscript 1 over straight s subscript 2 space equals space fraction numerator left parenthesis 10 right parenthesis squared over denominator left parenthesis 20 right parenthesis squared end fraction straight s subscript 2 space equals space 4 straight s subscript 1

    Question 527
    CBSEENPH11020726
    Wired Faculty App

    Two bodies of mass 1 kg have position vectors,bold i with bold hat on top bold space bold plus bold space 2 bold j with bold hat on top bold space bold plus bold k with bold hat on top bold space bold and bold space bold minus 3 bold i with bold hat on top bold space bold minus 2 bold j with bold hat on top bold space bold plus bold k with bold hat on top  respectively.The centre of mass of this system has a position vector

    • negative 2 bold i with hat on top space plus 2 space bold k with hat on top
    • negative 2 bold i with bold hat on top minus bold j with bold hat on top space plus bold k with hat on top
    • 2 bold i with bold hat on top minus bold j with bold hat on top space minus bold k with hat on top
    • negative bold i with bold hat on top plus bold j with bold hat on top space plus bold k with hat on top

    Solution

    B.

    negative 2 bold i with bold hat on top minus bold j with bold hat on top space plus bold k with hat on top

    The position vector of centre of mass

    straight r with rightwards arrow on top space equals space fraction numerator straight m subscript 1 stack bold r subscript bold 1 with bold rightwards arrow on top space plus straight m subscript 2 stack bold r subscript blank subscript bold 2 end subscript with bold rightwards arrow on top over denominator straight m subscript 1 plus straight m subscript 2 end fraction equals space fraction numerator 1 space left parenthesis straight i with hat on top plus 2 straight j with hat on top space plus space straight k with hat on top right parenthesis space plus 3 left parenthesis negative 3 bold i with hat on top space minus 2 stack bold j bold space with hat on top plus bold k with hat on top right parenthesis over denominator 1 plus 3 end fraction equals space 1 fourth space left parenthesis negative 8 bold i with bold hat on top minus 4 bold j with bold hat on top space plus 4 bold k with bold hat on top right parenthesis equals negative 2 bold i with hat on top space minus bold j with bold hat on top space plus bold k with hat on top
    Point to know:
    The centre of mass changes its position only under the translatory motion. There is no effect of rotatory motion on centre of mass of the body.

    Question 528
    CBSEENPH11020739
    Wired Faculty App

    If bold F with bold rightwards arrow on top is the force acting on a particle having position vector straight r with rightwards arrow on top space a n d space tau with rightwards arrow on top be the the torque of this force about the origin, then

    • straight r with rightwards arrow on top. straight tau with rightwards arrow on top space not equal to space 0 space and space bold F with bold rightwards arrow on top space. space straight tau with rightwards arrow on top space equals space 0
    • straight r with rightwards arrow on top. straight tau with rightwards arrow on top space greater than space space 0 space and space bold F with bold rightwards arrow on top space. space straight tau with rightwards arrow on top space less than space 0
    • straight r with rightwards arrow on top. straight tau with rightwards arrow on top space equals space space 0 space and space bold F with bold rightwards arrow on top space. space straight tau with rightwards arrow on top space equals space 0
    • straight r with rightwards arrow on top. straight tau with rightwards arrow on top space equals space space 0 space and space bold F with bold rightwards arrow on top space. space straight tau with rightwards arrow on top space not equal to space 0

    Solution

    C.

    straight r with rightwards arrow on top. straight tau with rightwards arrow on top space equals space space 0 space and space bold F with bold rightwards arrow on top space. space straight tau with rightwards arrow on top space equals space 0

    Torque is an axial vector ie, its direction is always perpendicular to the plane containing vectors  straight r with rightwards arrow on top space a n d space F with rightwards arrow on top
    stack straight tau space equals with rightwards arrow on top space bold r with bold rightwards arrow on top bold space bold x bold space bold F with bold rightwards arrow on top bold space Torque space is space perpendicular space to space both space bold r with bold rightwards arrow on top space and space bold F with bold rightwards arrow on top therefore comma straight tau with rightwards arrow on top. bold r with bold rightwards arrow on top space equals space 0 stack straight F. space with rightwards arrow on top straight tau with rightwards arrow on top space equals space 0

    Question 529
    CBSEENPH11020748
    Wired Faculty App

    The distance travelled by a particle starting from rest and moving with an acceleration 4/3 ms-2 in the third second is

    • 6 m

    • 4 m

    • 10/3

    • 19/3
    Solution

    C.

    10/3

    Distance travelled by  the particle in nth second
    straight S subscript straight n to the power of th end subscript space equals space straight u space plus 1 half space straight a space left parenthesis 2 straight n minus 1 right parenthesis
    Where u is initial speed and a is an acceleration of the particle. 
    Here, n = 3, u =0, a = 4/3 m/s2
    straight S subscript 3 rd end subscript space space equals space 0 space plus space 1 half space straight x 4 over 3 space straight x space left parenthesis 2 space straight x space 3 minus 1 right parenthesis space equals space 4 over 6 space straight x space 5 space equals space 10 over 3 space straight m

    Question 530
    CBSEENPH11020750
    Wired Faculty App

    A particle shows distance -time curve as given in this figure. The maximum instantaneous velocity of the particle is around the point

    • B

    • C

    • D

    • A
    Solution

    A.

    B

    The particle has maximum instantaneous velocity at that point at which its slope is maximum.
    Therefore, vmax = dx/dt = maximum slope
    From the figure, it is obvious that at point C, the slope is maximum, hence at this point velocity is maximum.

    Question 531
    CBSEENPH11020756
    Wired Faculty App

    A particle moves in a straight line with a constant acceleration. It changes its velocity from 10 ms-1 to 20 ms-1 while passing through a distance 135 m in t second.  The value of t is 

    • 10

    • 1.8

    • 12

    • 9
    Solution

    D.

    9

    The problem requires kinematics equations of motion
    Let u and v be the first and final velocities of particle an a and s be the constant acceleration and distance covered by it. 
    From third equation of motion
    v2 = u2 + 2as
    (20)2 = (10)2 + 2a x 135
    or
    a = 300/2 x135 = 10/9 ms-2
    Now using first equation of motion,


    v= u +at
    or 
    straight t space equals space fraction numerator straight v minus straight u over denominator straight a end fraction space equals space fraction numerator 20 minus 10 over denominator left parenthesis 10 divided by 9 right parenthesis end fraction space equals space fraction numerator 10 space straight x 9 over denominator 10 end fraction space equals space 9 straight s
    Question 532
    CBSEENPH11020758
    Wired Faculty App

    A particle of mass is projected with the velocity v making an angle of 45o with the horizontal. 
    When the particle  lands on the level ground the magnitude of the change in its momentum will be

    • 2 mv

    • mv divided by square root of 2
    • square root of 2 mv

    • zero
    Solution

    C.

    square root of 2 mv
    The horizontal momentum does not change. The change in vertical momentum is 
    mv sin θ - (-mv sin θ ) 
    =2 space mv fraction numerator 1 over denominator square root of 2 end fraction space equals space square root of 2 space mv
    Question 533
    CBSEENPH11020765
    Wired Faculty App

    The position x of a particle with respect to time t along x- axis is given by x = 9t2 -t3 where x is in meter and t in second. What will be the position of this particle when it achieves maximum speed along the +x direction? 

    • 32 m

    • 54 m

    • 81 m 

    • 24 m
    Solution

    B.

    54 m

    At the instant when speed is maximum, its acceleration is zero.
    Given, the position x of particle with respect to time t along x- axis
    x = 9t2-t3 ... (i)
    differentiating Eq. (i) with respect to time, we get speed, ie,
    straight v space equals space dx over dt space equals space straight d over dt left parenthesis 9 straight t squared minus straight t cubed space right parenthesis straight v space equals space 18 space straight t space minus space 3 straight t squared space... space left parenthesis ii right parenthesis
    Again differentiating Eq. (ii) with respect to time, we get acceleration, ie,
    straight a space equals space dv over dt space equals straight d over dt left parenthesis 9 straight t squared minus straight t cubed right parenthesis straight a space equals space 18 minus 6 straight t
    Now, when speed of particle is maximum, its acceleration is zero, ie,
    a= 0
    18-6t = 0 or t = 3s
    Putting in eq (i) We, obtain the position of a particle at that time. 
    x = 9 (3)2 - (3)3 = 9 (9) -27 
    = 81-27 = 54 m

    Question 534
    CBSEENPH11020767
    Wired Faculty App

    A particle starting from the origin (0,0) moves in a straight line in the (x,y) plane. Its coordinates at a later time are  ( square root of 3 comma space 3). The path of the particle makes with the x -axis an angle of:

    • 30o

    • 45o C

    • 60o C

    • 00
    Solution

    C.

    60o C

    The slope of the path of the particle gives the measure of angle required.

    Draw the situation as shown. OA represents the path of the particle starting from origin O (0,0). Draw a perpendicular from point A to X- axis. Let path pf the particle makes and angle θ with the x -axis, then 
    tan θ = slope of line OA

    AB over OB space equals space fraction numerator 3 over denominator square root of 3 end fraction space equals space square root of 3 theta space equals space 60 to the power of 0

    Question 535
    CBSEENPH11020770
    Wired Faculty App

    A car moves from X to Y with a uniform speed vu and returns to Y with a uniform speed vd. The average speed for this round trip is:

    • fraction numerator 2 straight v subscript straight d straight v subscript straight u over denominator straight v subscript straight d plus straight v subscript straight u end fraction
    • square root of straight v subscript straight u straight v subscript straight d end root
    • fraction numerator straight v subscript straight d straight v subscript straight d over denominator straight v subscript straight d plus straight v subscript straight u end fraction
    • fraction numerator straight v subscript straight u space plus straight v subscript straight d over denominator 2 end fraction

    Solution

    A.

    fraction numerator 2 straight v subscript straight d straight v subscript straight u over denominator straight v subscript straight d plus straight v subscript straight u end fraction

    The average speed of a body in a given time interval is defined as the ratio of distance travelled to the time taken.
    Average space speed space equals space fraction numerator Distance space travelled over denominator Time space taken end fraction
    Let t1 and t2 be times taken by the car to go from X and Y and then Y to X respectively.
    Then, 
    straight t subscript 1 space plus straight t subscript 2 space equals space XY over straight v subscript straight u space plus XY over straight v subscript straight d space equals space XY space open parentheses fraction numerator straight v subscript straight u space plus straight v subscript straight d over denominator straight v subscript straight u straight v subscript straight d end fraction close parentheses Total space distance space travelled equals space XY space plus space XY space equals space 2 XY Therefore comma space average space speed space of space the space car space for space this space round space trip space is straight v subscript av space equals space fraction numerator 2 XY over denominator XY open parentheses begin display style fraction numerator straight v subscript straight u plus straight v subscript straight d over denominator straight v subscript straight u straight v subscript straight d end fraction end style close parentheses end fraction or straight v subscript av space equals space fraction numerator 2 straight v subscript straight u straight v subscript straight d over denominator straight v subscript straight u plus straight v subscript straight d end fraction

    Question 538
    CBSEENPH11020791
    Wired Faculty App

    For angles of projection of a projectile at angles (45° - θ) and (45° + θ), the horizontal ranges described by the projectile are in the ratio of

    • 1:1

    • 2:3

    • 1:2

    • 2:1
    Solution

    A.

    1:1

    For complementary angles of projection, their horizontal ranges will be same.
    We know that, horizontal ranges for complementary angles of projection will be same.
    The projectiles are projected at angles left parenthesis 45 degree minus straight theta right parenthesis and left parenthesis 45 degree plus straight theta right parenthesis which are complementary to each other i.e., two angles add up to give 90 degree. Hence, horizontal ranges will be equal. Thus, the required ratio is 1:1.

    Question 539
    CBSEENPH11020804
    Wired Faculty App

    A car runs at aconstant speed on a circular track of radius 100 m, taking 62.8 s for every circular lap. The average velocity and average speed for each circular lap respectively is

    • 0, 0

    • 0, 10 m/s

    • 10 m/s,  10m/s

    • 10 m/s,  0
    Solution

    B.

    0, 10 m/s

    In completing a circular lap, car's displacement is zero.
    Average velocity is defined as the ratio of displacement to time taken while the average displacement to time taken while the average speed of a particle in a given interval of time is defined as the ratio of distance travelled to the time taken.
    On a circular path in completing one turn, the distance travelled is 2 πr while displacement is zero.
    Hence, average velocity = fraction numerator displacement over denominator time minus interval end fraction
                                 equals 0 over straight t space equals 0
    Average space speed space equals space fraction numerator Distance over denominator Time minus interval end fraction space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 πr over denominator straight t end fraction equals space fraction numerator 2 cross times 3.14 cross times 100 over denominator 62.8 end fraction space equals space 10 straight m divided by straight s

    Question 540
    CBSEENPH11020807
    Wired Faculty App

    The vectors bold A with bold rightwards arrow on top space and space bold B with bold rightwards arrow on top are such that a:
    open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals space open vertical bar straight A with rightwards arrow on top space minus space straight B with rightwards arrow on top close vertical bar
    The  angle between the two vectors is

    • 90 degree
    • 60 degree
    • 75 degree
    • 45 degree

    Solution

    A.

    90 degree

    As we have given
            open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals open vertical bar straight A with rightwards arrow on top minus straight B with rightwards arrow on top close vertical bar or space space space square root of straight A squared plus straight B squared plus 2 AB space Cosθ end root space equals square root of straight A squared plus straight B squared minus 2 AB space Cos space straight theta end root
    where straight theta is the angle between straight A with rightwards arrow on top space and space straight B with rightwards arrow on top
    Squaring both sides, we have
    straight A squared plus straight B squared plus 2 AB space Cosθ space equals space straight A squared plus straight B squared minus 2 AB space cosθ or space space space space space space space space space space space space space space space space space space 4 AB space cosθ space equals space 0 As space space space space space space space space space space space space space space space space space AB not equal to 0 therefore space space space space space space space space space space cosθ space equals space 0 space equals space cos space 90 degree therefore space space space space space space space space space space space space space space space space space space space space space space space straight theta space equals space 90 degree space space space space space space space space space space space space
    Hence, angle between stack straight A space with rightwards arrow on top and space straight B with rightwards arrow on top space is space 90 degree

    Question 542
    CBSEENPH11020855
    Wired Faculty App

    The moment of the force, F→ = 4i^ + 5j^ -6k^  at (2, 0, 3) about the point (2,-2,2), is given by

    • -8i^ -4j^ -7k^

    • -4i^ -j^ -8k^

    • -7i^ -4j^ -8k^

    • -7i^ -8j^ -4k^
    Solution

    C.

    -7i^ -4j^ -8k^

    Moment of force,
    τ→ = r→ x F→

    τ→ = (r→ -r0→) x F→r→ -r0→ = (2i^ + 0j^ - 3k^) - (2i^ -2j^-2k^)=0i^ + 2j^ -k^τ→ = (0i^ + 2j^ -k^) (4i^ + 5j^-6k^)τ→ = i^j^k^02-145-6 = - 7i^ - 4j^ -8k^

    Question 543
    CBSEENPH11020861
    Wired Faculty App

    A body is projected vertically upwards. The times corresponding to height h while ascending and while descending are t1 and t2, respectively.

    • gt1t22

    • g(t1 +t2)2

    • gt1t2

    • gt1t2(t1+t2)
    Solution

    B.

    g(t1 +t2)2

    Let v be the initial velocity of vertical projection and t be time taken by the body to reach a height h from the ground.

    Here u = u,

    a = -g,

    s = h,

    t = t

    Using, 

    ut + 12 (-g)t2 or gt2 - 2ut + 2h = 0∴ t = 2u± 4u2 - 4g x 2h2g = u± u2 -2ghgIt means t has two values, i.e.,t1 = u + u2 -2ghgt2 = u- u2-2ghgt1 +t2 = 2ug or u = g (t1 +t2)2

    Question 544
    CBSEENPH11020863
    Wired Faculty App

    The x and y coordinates of a particle moving in a plane are given be x(t) = acos(pt) and y(t) = bsin(pt) where a, b (<a) and p are positive constants of appropriate dimensions and t is time. Then , which of the following is not true?

    • The path of the particle is an ellipse

    • Velocity and acceleration of the particle are perpendicular to each other at t = π/2p

    • Acceleration of the particle is always directed towards a fixed point

    • Distance travelled by the particle in the time interval between t = 0 and t = π/2p is a
    Solution

    D.

    Distance travelled by the particle in the time interval between t = 0 and t = π/2p is a

    x = a cos (pt), y =b sin (pt)

    ∴ x2a2 + y2b2 = 1, i.e. equation of ellipseNow, r = xi^ + yj^  = a cos (pt) i^  + b sin (pt)j^v = drdt = -pasin (pt) i^ + pb cos (pt) j^a = dvdt = - p2a cos (pt) i^ - p2 b sin (pt)j^at t = π2pv = - pa  i^   and a = - p2b j^ v ⊥ a 

    also, a = - p2r, i.e directed towards a fixed point.

    Question 545
    CBSEENPH11020882
    Wired Faculty App

    A stick of length L and mass M lies on a frictionless horizontal surface on which it is free to move in any way. A ball of mass m moving with speed V collides elastically with the stick as shown in fig below. If after the collision, the ball comes to rest, then what should be the mass of the ball?

     

     

    • m = 2M

    • m = M

    • m= M/2

    • m = M/A
    Solution

    D.

    m = M/A

    Applying the law of conservation of momentum

    mV = MV ... (i)

    By conservation of angular momentum 

    mV (L/2) = ML212 ω .. (ii)

    As the collision in elastic, we have

    12mV2 = 12MV2 + 12Iw2   (iii)..substituting the values, we getm = M/4

    Question 546
    CBSEENPH11026113
    Wired Faculty App

    The maximum range of a gun on horizontal terrain is 16 km if g=10m/s2. What must be the muzzle velocity of the shell?

    • 200 m/s

    • 100 m/s

    • 400 m/s

    • 300 m/s
    Solution

    C.

    400 m/s

    The velocity with which a bullet or shell leaves the muzzle of a gun.
     
    We  know that in projection of the particle for maximum range, θ = 45o
     
    Now maximum rangeR =μ2sin 2θgR=μ2 sin 90og        sin 90o =1μ =Rg               .....(1)Given :- Rmax = 16 km= 16×103 mg = 10 m/s2Now from equation(1)μ =16×103×10=400 m/s
     
     
    Question 547
    CBSEENPH11026162
    Wired Faculty App

    If a vector  2 i^ + 3 j^ + 8 k^  is perpendicular to the vector 4 j^ - 4 i^ + α k^, then the value of α is

    • -1

    • 12

    • -12

    • 1
    Solution

    C.

    -12

    Two vectors must be perpendicular if their dot product is zero.

    let a→ = 2i^ + 3j^ +8 k^b→ = 4 j^ - 4 i^ + αk^     = -4i^ + 4j^ + αk^According to the above hypothesis a→ ⊥ b→⇒ a→ . b→ = 0⇒ 2i ^ + 3j^ + 8k^ . -4i^ + 4j^ + αk^ = 0⇒ -8 + 12 +8α = 0⇒ 8α = -4α =-48 = -12 

    a→ . b→ = ab cosθ. Here a and b are always positive as they are the magnitudes as a→ and b→.

    Question 548
    CBSEENPH11026171
    Wired Faculty App

    A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are 3, 3. The path of the particle makes with the x-axis an angle of

    • 30o

    • 45o

    • 60o

    • 0o
    Solution

    C.

    60o

    Draw the situation as Y shown. OA represents the path of the particle starting from origin O (0 , 0 ).  Draw a perpendicular from point A to x-axis. Let path of the particle makes an angle 0 with the x-axis, then 

    tan θ = slope of line OA 

    tan θ = ABOB          =33tan θ  = 3

    ⇒ θ  = 60o

    Question 549
    CBSEENPH11026172
    Wired Faculty App

    A wheel has angular acceleration of 3.0 rad/s2 and an initial angular speed of 2.00 rad/s. In a time of 2 s it has rotated through an angle (in radian) of

    • 6

    • 10

    • 12

    • 4
    Solution

    B.

    10

    According to Kinematic equations for uniformly accelerated motion

    θ = ωo t + 12α t2

    Where ,The angular acceleration is α = 3 rad/s2

    the initial angular speed is ωo =  2 rad/s

    The time is t = 2 s

    Given:- α = 3.0 rad/s2 , ωo = 2.0 rad/s, t = 2 s

    Hence

    θ = 2 × 2 + 12 × 3 × 22

    θ = 4 + 6

    θ = 10 rad

    Question 550
    CBSEENPH11026173
    Wired Faculty App

    Under the influence of a uniform magnetic field a charged particle is moving in a circle of radius  R with constant speed v. The time period of the motion

    • depends on v and not on R

    • depends on both R and v

    • is independent of both R and v

    • depends on R and not on v
    Solution

    C.

    is independent of both R and v

    When magnetic field is perpendicular to motion of charged particle, then

    Centripetal force = magnetic force

    i.e  m v2R = BqV⇒ R = mvBq

    Wherw B is the magnetic field,

    q-  charge on particle

    Further, time period of the motion

    T = 2 πRv     = 2π mvBqv⇒ T = 2 πmBq

    It is independent of both R and v.

    Question 551
    CBSEENPH11026208
    Wired Faculty App

    A projectile is thrown in the upward direction making an angle of 60° with the horizontal direction with a velocity of 147 ms-1 Then the time after which its inclination with the horizontal is 45°, is

    • 15 s

    • 10.98 s

    • 5.49 s

    • 2.745 s
    Solution

    C.

    5.49 s

    At the two points of the trajectory during projection, the horizontal component of the velocity is the same. 

    The horizontal component of velocity at angle 60o  =  Horizontal component of velocity at 45o

      ⇒           u cos 60o = v sin 45o

                    147 × 12 = ν × 12           v   = 1472 m/s 

    Vertical component of u = u sin 60o

                                 = 147 32 m

    Vertical component of v = v sin 45o

                             147 2 × 12 = 1472m

    but             vy = uy + α t

    ∴                  1472 = 14732 - 9.8 t ⇒                 9.8t  = 1472 3 - 1

                        t = 5.49 s

    Question 552
    CBSEENPH11026212
    Wired Faculty App

    The potential energy of particle varies with distance x fixed a origin as v = Axx + B; where A and B are constants. The dimensions of AB are

    • [ M L5/2 T-2 ]

    • [ M L2 T-2 ]

    • [ M3/2 L3/2 T-2 ]

    • [M L7/2 T-2 ]
    Solution

    D.

    [M L7/2 T-2 ]

    Given;- v = A xx + B             .... (i)

    Dimensions of v = dimensions of potential energy

                             = [ M1 L2 T-2 ]

    From equqtion (i) 

    Dimensions of B = Dimensions of x = [Mo L1 To ]

    ∴ Dimensions of A =dimension of v × dimensions of x + Bdimensions of x

                                = M1 L2 T-2 Mo L1 ToM1 L12 To

                                 = [ M1 L5/2 T-2 ]

    Hence dimensions of AB 

                                = [ M1 L5/2 T-2 ] [ Mo L1 To ]

                                = [ M L7/2 T-2 ]

    Question 553
    CBSEENPH11026235
    Wired Faculty App

    A solid sphere and a hollow sphere of the same material and of a same size can be distinguished without weighing

    • by determining their moments of inertia about their coaxial axe

    • by rolling them simultaneously on an inclined plane

    • by rotating them about a common axis of rotation

    • by applying equal torque on them
    Solution

    B.

    by rolling them simultaneously on an inclined plane

    The acceleration of a body rolling down the plane = α

               

    ∴                α = g sinθ1 + K2R2

    where K is radius of gyration and R the radius of sphere.

    For solid sphere,

                          K2R2 = 25

                           α = g sinθ1 + 25

                           α = g sin θ75                         

     ∴                       α = 57g sinθ

    For hollow sphere,

                            K2R2 = 23

                             α = g sin θ1 + 23

                              α = 35 g sin θ

                               α = 0.6 g sinθ 

    Since, acceleration of solid sphere is more than of hollow sphere, it rolls faster, and reaches the
    bottom of the inclined plane earlier. Hence, solid sphere and hollow sphere can be distinguished by rolling them simultaneously on an inclined plane.

    Question 554
    CBSEENPH11026257
    Wired Faculty App

    A certain vector in the xy-plane has an x-component of 4 m and a y-component of 10 m. It is then rotated in the xy-plane so that its x-component is doubled. Then, its new y-component is (approximately)

    • 20 m

    • 7.2 m

    • 5.0 m

    • 4.5 m
    Solution

    B.

    7.2 m

    Here, A= 4 i + 10 j

      A = 42 + 102

           = 16 + 100

      A = 116 m

    Now, according to question,

       A = 8 i + n j

    ⇒ A = 82 + n2

        116 = 82 + n2

    Squaring on both sides

       116 = 64 + n2

    ⇒ π2 = 116 - 64

         n = 7.2 m

    Question 555
    CBSEENPH11026275
    Wired Faculty App

    A coin is dropped in a lift. It takes time t1  to reach the floor when lift is stationary. It takes time t2 when lift is moving up with constant acceleration. Then,

    • t1 ≥ t2

    • t2 > t1

    • t1 = t2

    • t1 >> t2
    Solution

    A.

    t1 ≥ t2

    An object that is in flight after being thrown or projected is called projectile.

    For stationary lift t1 = 2hg

    and when the lift is moving up with constant acceleration of coin is more than g hence will take less time.

                 t2 = 2hg

    ∴            t1 > t1

    Question 556
    CBSEENPH11026284
    Wired Faculty App

    A body is thrown with a velocity of 9.8 m/s making an angle of 30° with the horizontal. It will hit the ground after a time

    • 1.5 s

    • 1 s

    • 3 s

    • 2 s
    Solution

    B.

    1 s

    The time of flight = 2 u sinθg

                             = 2 × 9.8 × sin30o9.8

                              =  1 s                 ...( sin 30o = 12 ) 

    Question 557
    CBSEENPH11026315
    Wired Faculty App

    Three points charges +q, -2q and +q are placed at points (x=0,  y=a,  z=0),  ( x=0,  y=0,  z=0 )  and  (x=a,  y=0,  z=0), respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are

    • 2 qa along + y direction

    • 2qa along the line joining point

    • qa along the line joining points (x = 0,y = 0, z = 0) and (x = a, y = a, z = 0)

    • 2qa along + x direction
    Solution

    B.

    2qa along the line joining point

    0 is the origin at which -2q charge is placed. The system is equivalent to two dipoles along x and y-directions respectively. The dipole moments of two dipoles are shown in figure            

    The resultant dipole moment will be directed along OP where P = (a, a, 0). The magnitude of resultant dipole moment is

        p' = p2 + p2

            = qa2 + qa2

         p' = 2q a

    Question 558
    CBSEENPH11026319
    Wired Faculty App

    The angle between two linear transmembrane domains is defined by following vectors a = i^ + j^ - k^ and  b = i^ - j^  +  k^

    • cos-1 13

    • cos-1 -13

    • sin-1 -13

    • sin-1 13
    Solution

    B.

    cos-1 -13

    The angle between two vectors is given by

           cosθ = a.bab

                   = 1 - 1 - 13 × 3

    ⇒      cosθ = -13

    ⇒     θ = cos-1 -13

    Question 559
    CBSEENPH11026327
    Wired Faculty App

    A projectile is fired with a velocity u at angle θ with the ground surface. During the motion at any time it is making an angle a with the ground surface. The speed of particle at this time will be

    • ucosθ secα

    • ucosθ tanα

    • u2 cos2α sin2α

    • usinθ.sinα
    Solution

    A.

    ucosθ secα

    Projectile motion is the motion of an object thrown or projected into the air, subjected to an only acceleration of gravity. The object is called projectile and its path is called trajectory.

    Horizontal component of velocity is always same

    ⇒          v cosα = u cosθ

                 v = u cosθcosα

    ⇒            v= u cosθ secα

    Question 560
    CBSEENPH11026337
    Wired Faculty App

    Water level is maintained in a cylindrical vessel upto a fixed height H. The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel, so that the water stream coming out of the hole strikes the horizontal plane of the greatest distance from the vessel?

          

    • h = H2

    • h = 3H2

    • h = 2 H3

    • h = 34H
    Solution

    A.

    h = H2

    In general

    Below equation is obtained from Kinematic equation

        v2 = u2 + 2ah

    Instead of acceleration 'a' we used 'g'

    The velocity with which water will came out

         v = 2 g ( H - h )

    The time of flight of water ejected from the hole.

        t = 2hg = time of flight

    ∴ distance = velocity × time

    ⇒      x = v × t

             = 2 g H - h × 2hg

    ⇒       x = 4  Hh - h2 

    For x to be maximum

         dxdh = 0

    ⇒   d h  Hh - h2 dH = 0

    ⇒   0 = H - 2h

    ⇒    h = H2

    Question 561
    CBSEENPH11026344
    Wired Faculty App

    What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle θ2 with the horizontal?

    • r = v2 cos2θg cos2 θ2

    • r = 2 v sinθg tanθ

    • r =v cosθg sin2θ2

    • r  = 3v cos θg cot θ
    Solution

    A.

    r = v2 cos2θg cos2 θ2

    Let v be the velocity at the point where it makes an θ2angle with the horizontal.

    The horizontal component remains unchanged. So

       

              vm cosθ2 = u cosθ

    ⇒                  v = u cosθcosθ2                      .....(i)

    From figure 

                 mg cosθ2 = m v2r

                     r = v2g cos θ2 

    Putting the value of v from Eq (i), we get

                 r = v2 cos2θ g cos2 θ2       

    Question 562
    CBSEENPH11026366
    Wired Faculty App

    A particle slides down on a smooth incline of inclination 30°, fixed in an elevator going up with an acceleration 2 m/s2. The box of incline has a length 4m. The time taken by the particle to reach the bottom will be

       

    • 89 3 s

    • 98 3 s

    • 43 32s

    • 34 32s
    Solution

    C.

    43 32s

                   

    In the frame of elevator, 

    a = acceleration of the particle with respect to the elevator

          m sin30 ( g + 2) = ma

                        a  = ( g + 2 ) sin30o

                            = ( 10 + 2 ) 12

                             = 6 m/s2

    The distance travelled by the particle from the top to the bottom

         d = 4cos 30o 

            = 432

       d = 8 33 m

    Using kinematic equation

       s = ut + 12 at2

       8 33 = 0 × t + 12 at2

    ⇒     t2 = 16 33 × 6s

    ⇒     t = 16183

    ⇒    t = 169 × 23

    ⇒  t =  43 32s

    Question 563
    CBSEENPH11026367
    Wired Faculty App

    A stone projected with a velocity u at an angle 0 with the horizontal reaches maximum height H,. When it is projected with velocity u at an angle π2 - θ with the horizontal, it reaches maximum height H2. The relation between the horizontal range R of the projectile, H1 and H2 is 

    • R = 4 H1 H2

    • R = 4 ( H1 - H2 )

    • R = 4( H1 + H2 )

    • R = H12H22
    Solution

    A.

    R = 4 H1 H2

    We know that,

    Maximum height of projectile equation is given by

          H= u2 sin2θ2 g

    Where u = velocity of stone

             θ = angle 

     H = maximum height

           H1 = u2 sin2θ2 g

    and   H2 = u2 sin2 90- θ2 g

    since   H1H2 = u2 sin2θ2 g × u2 cos2θ2 g

                        = 44 × u2 sin2θ2 g × u2 cos2θ2g

                 H1H2 = u2 sin 2θ 216 g2

         [ since sin2θ = 2 sinθ cosθ ]

                 H1H2 = R216

               R2 = 16H1 H2

               R = 4H1H2

    Question 564
    CBSEENPH11026392
    Wired Faculty App

    A projectile is given an initial velocity of i^ + 2 j^ The cartesian equation of its path is ( Take g= 10 ms2 )

    • y = x - 5x2

    • y = 2x - 5x2

    • y = 2x - 15 x2

    • y = 2x - 25x2
    Solution

    B.

    y = 2x - 5x2

    Given:-

    The initial velocity of a projectile is given by

        u→ = i^ + 2 j^

            = ux i^ + uy j^

    Then,

       ux = 1 = u cosθ

        uy = 2 = u sinθ

    ∴       tanθ = u sinθu cosθ

                     = 21

            tanθ = 2

    The equation of trajectory of a projectile motion is

       y = x tanθ -  g x22 u2 cos2θ

           = x tanθ -  g x22 u cosθ 2

    ∴  y =  (x × 2 ) -  10 × x22 12

       y = 2x - 5x2

    Question 565
    CBSEENPH11026413
    Wired Faculty App

    If  A→ and B→ are non-zero vectors which obey the relation   A→ + B→  =  A→ - B→ , then the angle between them is

    • 0o

    • 60o

    • 90o

    • 120o
    Solution

    C.

    90o

          A→ + B→ = A→ - B→

    ⇒ A→2 + B→2 + 2 A→ B→ cosθ = A→2 + B→2 - 2 A→ B→ cosθ   

    ⇒        cosθ = 0

    ⇒           θ = 90o   

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