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Mathematics Chapter 8 Binomial Theorem

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    NCERT Solution For Class 11 Mathematics

    Binomial Theorem Here is the CBSE Mathematics Chapter 8 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 Mathematics Binomial Theorem Chapter 8 NCERT Solutions for Class 11 Mathematics Binomial Theorem Chapter 8 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 Mathematics.

    Question 1
    CBSEENMA11014353
    Wired Faculty App

    Solve 24x < 100, when

    (i) x is a natural number

    (ii) x is an integer

    Solution

    24x < 100 rightwards double arrow fraction numerator 24 straight x over denominator 24 end fraction less than 100 over 24 space rightwards double arrow space straight x less than 25 over 6 space or space straight x less than 4 1 over 6
    ∴ (i)  When space space straight x space element of space straight N comma space straight x space equals space 4 comma space 3 comma space 2 comma space 1
    ∴  Solution set 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>
    (ii)  When straight x space element of space straight Z comma space straight x space equals space 4 comma space 3 comma space 2 comma space 1 comma space 0 comma space minus 1 comma space minus 2 comma space minus 3......
    The solution set is open curly brackets.... comma negative 3 comma space minus 2 comma space minus 1 comma space 0 comma space 1 comma space 2 comma space 3 comma space 4 close curly brackets

    Question 2
    CBSEENMA11014354
    Wired Faculty App

    Solve -12x > 30, when

    (i) x is a natural number.            (ii) x is an integer.
    Solution

    -12x > 30
    rightwards double arrow      fraction numerator negative 12 straight x over denominator negative 12 end fraction less than fraction numerator 30 over denominator negative 12 end fraction                                            (∵ -12 < 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>
    ∴ (i) When straight x space element of space straight N, there is no solution
    ∴ The solution set is straight ϕ.
    (ii)    When <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 solution set 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 3
    CBSEENMA11014355
    Wired Faculty App

    Solve 5x – 3 < 7, when

    (i) x is an integer.              (ii) x is a real number.          
    Solution

    5x – 3 < 7
                   5x – 3 < 7 rightwards double arrow 5x < 10 rightwards double arrow x<2
    ∴    (i) When space space straight x element of space straight Z comma space straight x space equals space 1 comma space 0 comma space minus 1 comma space minus 2 comma space........
               The solution set is open curly brackets.... comma space minus 3 comma space minus 2 comma space minus 1 comma space 0 comma space 1 close curly brackets
          (ii) When straight x space element of space straight R comma space space straight x less than 2 space or space straight x space element of space left parenthesis negative infinity comma space 2 right parenthesis
    ∴      The solution set is space space left parenthesis negative infinity comma space 2 right parenthesis

    Question 4
    CBSEENMA11014356
    Wired Faculty App

    Solve 3x + 8 > 2, when

    (i) x is an integer.              (ii) x is a real number.          
    Solution

                   3x + 8 > 2
    rightwards double arrow              3x > 2 - 8 rightwards double arrow 3x > - 6 rightwards double arrow x > -2
    ∴  (i) When space space straight x element of space straight Z comma             
              x = -1, 0, -1, 2, 3, ........
              The solution set is open curly brackets negative 1 comma space 0 comma space 1 comma space 2 comma space 3 comma......... close curly brackets
        (ii)  When straight x space element of space straight R space rightwards double arrow space straight x greater than negative 2 space space or space straight x space element of space left parenthesis negative 2 comma space infinity right parenthesis
               The solution set 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 5
    CBSEENMA11014357
    Wired Faculty App

    Solve the following inequality for real x:

    3(x – 1) ≤ 2 (x – 3)

    Solution

    3(x – 1) ≤ 2 (x – 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>
    Hence, <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 6
    CBSEENMA11014358
    Wired Faculty App

    Solve the following inequality for real x:

    3 (2 – x) ≥ 2 (1 – x)

    Solution

    3 (2 – x) ≥ 2 (1 – x)
    rightwards double arrow    6 minus 3 straight x greater or equal than 2 minus 2 straight x rightwards double arrow space minus 3 straight x plus 2 straight x greater or equal than 2 minus 6 space rightwards double arrow negative straight x greater or equal than negative 4 rightwards double arrow fraction numerator negative straight x over denominator negative 1 end fraction less or equal than fraction numerator negative 4 over denominator negative 1 end fraction rightwards double arrow straight x less or equal than 4
                                                                                                       (∵ -1 < 0)
    rightwards double arrow                 straight x space element of space left parenthesis negative infinity comma space 4 right square bracket
    Hence, the solution 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 7
    CBSEENMA11014359
    Wired Faculty App

    Solve the following inequality for real x:

    fraction numerator 3 space left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than fraction numerator 5 space left parenthesis 2 straight x minus 3 right parenthesis over denominator 3 end fraction

    Solution
    fraction numerator 3 space left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than fraction numerator 5 space left parenthesis 2 minus straight x right parenthesis over denominator 3 end fraction
    Mulitply by 15, we have
                           9 space left parenthesis straight x minus 2 right parenthesis space less or equal than space 25 space left parenthesis 2 minus straight x right parenthesis
    rightwards double arrow                      9 straight x minus 18 space less or equal than space 50 minus 25 straight x space rightwards double arrow space 9 straight x plus 25 straight x less or equal than 50 plus 18 space rightwards double arrow space 34 straight x less or equal than 68 space rightwards double arrow space straight x less or equal than 2
    rightwards double arrow                       straight x space element of space left parenthesis negative infinity comma space 2 right square bracket
    Hence, the solution set 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 8
    CBSEENMA11014360
    Wired Faculty App

    Solve the following inequality for real x:

    1 half open parentheses fraction numerator 3 straight x over denominator 5 end fraction plus 4 close parentheses greater or equal than 1 third left parenthesis straight x minus 6 right parenthesis

    Solution
    space space 1 half open parentheses fraction numerator 3 straight x over denominator 5 end fraction plus 4 close parentheses space greater or equal than space 1 third left parenthesis straight x minus 6 right parenthesis space rightwards double arrow space fraction numerator 3 straight x over denominator 10 end fraction plus 2 greater or equal than straight x over 3 minus 2
    Multiply by 30, we have
                         9 straight x plus 60 greater or equal than 10 straight x minus 60
    rightwards double arrow             9 straight x minus 10 straight x greater or equal than negative 60 minus 60 space rightwards double arrow space minus straight x greater or equal than negative 120 space rightwards double arrow space straight x less or equal than 120  (Divide by -1)
    rightwards double arrow                   straight x element of space left parenthesis negative infinity comma space 120 right square bracket
    Hence, the solution set 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 9
    CBSEENMA11014361
    Wired Faculty App

    Solve the following inequality for real x:

    <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
    <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>
    Multiply by L.C.M. 60 of 4, 3 and 5, we have
    rightwards double arrow           space space 15 straight x less than space 20 space left parenthesis 5 straight x minus 2 right parenthesis minus 12 space left parenthesis 7 straight x minus 3 right parenthesis space rightwards double arrow space 15 straight x space less than space 100 straight x minus 40 minus 84 straight x plus 36

    rightwards double arrow    15x - 100x + 84x < - 40 + 36 rightwards double arrow -x < -4 rightwards double arrow x > 4                      (Divide by -1)
    rightwards double arrow                         straight x space element of space left parenthesis 4 comma space infinity right parenthesis

    Hence, the solution set 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 10
    CBSEENMA11014362
    Wired Faculty App

    Solve the following inequality for real x:

    fraction numerator left parenthesis 2 straight x minus 1 right parenthesis over denominator 3 end fraction greater or equal than fraction numerator left parenthesis 3 straight x minus 2 right parenthesis over denominator 4 end fraction minus fraction numerator left parenthesis 2 minus straight x right parenthesis over denominator 5 end fraction

    Solution
    fraction numerator left parenthesis 2 straight x minus 1 right parenthesis over denominator 3 end fraction greater or equal than fraction numerator left parenthesis 3 straight x minus 2 right parenthesis over denominator 4 end fraction minus fraction numerator left parenthesis 2 minus straight x right parenthesis over denominator 5 end fraction
    Multiply by L.C.M. 60 of 3, 4, and 5, we have
    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    40 straight x minus 45 straight x minus 12 straight x greater or equal than negative 30 minus 24 plus 20 space rightwards double arrow space minus 17 straight x greater or equal than negative 34 space rightwards double arrow space fraction numerator negative 17 straight x over denominator negative 17 end fraction less or equal than fraction numerator negative 34 over denominator negative 17 end fraction space rightwards double arrow space straight x less or equal than 2

                                                    [∵ division by negative number reverses the inequality]
    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>

    Hence, the solution set 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 11
    CBSEENMA11014363
    Wired Faculty App

    Solve the following inequality:

    6 less or equal than negative 3 left parenthesis 2 straight x minus 4 right parenthesis less than 12

    Solution
    space space space space space 6 less or equal than negative 3 space left parenthesis 2 straight x minus 4 right parenthesis less than 12
    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>    ...(i)
    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> (∵ ..6 <0)
    rightwards double arrow         x > 0                                                                                                        ...(ii)
    From (i) and (ii), we have, 0 less than straight x less or equal than 1 which is the required solution of the given inequality.
    Hence, the solution set is (0, 1].
    Question 12
    CBSEENMA11014364
    Wired Faculty App

    Solve the following inequality:

    negative 3 space less or equal than space 4 minus fraction numerator 7 straight x over denominator 2 end fraction less or equal than 18

    Solution
    negative 3 space less or equal than space 4 space minus space fraction numerator 7 straight x over denominator 2 end fraction less or equal than 18
    Now,     negative 3 space less or equal than space 4 minus fraction numerator 7 straight x over denominator 2 end fraction rightwards double arrow negative 6 space less or equal than space 8 space minus 7 straight x space rightwards double arrow space 7 straight x space less or equal than space 8 space plus space 6 space rightwards double arrow space 7 straight x space less or equal than space 14 space rightwards double arrow space straight x space less or equal than space 2 ...(i)
    Also,    space space space space space space space 4 minus fraction numerator 7 straight x over denominator 2 end fraction less or equal than 18 space rightwards double arrow space 8 minus 7 straight x less or equal than 36 space rightwards double arrow space minus 7 straight x less or equal than 36 minus 8
    rightwards double arrow                space space space space space minus 7 straight x space less or equal than space 28 space rightwards double arrow space fraction numerator negative 7 straight x over denominator negative 7 end fraction greater or equal than fraction numerator 28 over denominator negative 7 end fraction rightwards double arrow space straight x greater or equal than space minus 4                                    ...(ii)
    From (i) and (ii), we have negative 4 less or equal than straight x less or equal than 2 which is the required solution of the given inequality.
    Hence, the solution is [-4, 2].
    Question 13
    CBSEENMA11014365
    Wired Faculty App

    Solve the following inequality:

    space space 7 less or equal than fraction numerator 3 straight x plus 11 over denominator 2 end fraction less or equal than 11

    Solution
    space space 7 less or equal than fraction numerator 3 straight x plus 11 over denominator 2 end fraction less or equal than 11
    Now,    7 less or equal than fraction numerator 3 straight x plus 11 over denominator 2 end fraction rightwards double arrow 14 less or equal than 3 straight x plus 11 space rightwards double arrow space minus 3 straight x less or equal than 11 minus 14

    rightwards double arrow               negative 3 straight x space less or equal than space minus 3 space rightwards double arrow space fraction numerator negative 3 straight x over denominator negative 3 end fraction greater or equal than space fraction numerator negative 3 over denominator negative 3 end fraction space rightwards double arrow space straight x space greater or equal than 1                ...(i)
    Also,            fraction numerator 3 straight x plus 11 over denominator 2 end fraction less or equal than 11 space rightwards double arrow space 3 straight x plus 11 space less or equal than space 22 space rightwards double arrow space 3 straight x space less or equal than space 22 space minus 11 space rightwards double arrow space 3 straight x less or equal than 11 space rightwards double arrow space straight x space less or equal than space 11 over 3 ...(ii)
    ∴ From (i) and (ii), 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> which is the required solution of the given inequality.
    Hence, the solution set is open square brackets 1 comma space 11 over 3 close square brackets.

    Question 14
    CBSEENMA11014366
    Wired Faculty App

    Solve the following inequality:

    space space minus 15 less than fraction numerator 3 left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than 0

    Solution
    space space minus 15 less than fraction numerator 3 left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than 0
    Now,    negative 15 less than fraction numerator 3 space left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction rightwards double arrow space minus 75 less than 3 space left parenthesis straight x minus 2 right parenthesis space rightwards double arrow negative 75 space less than space 3 straight x minus 6 rightwards double arrow space minus 3 straight x less than negative 6 plus 75

    rightwards double arrow                -3x < 69         rightwards double arrow space space space space space space space space fraction numerator negative 3 straight x over denominator negative 3 end fraction greater than fraction numerator 69 over denominator negative 3 end fraction                  (∵ -3 < 0)
    rightwards double arrow                  x > -23                                                                                                   ...(i)
    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>     ...(ii)
    From (i) and (ii), we have -23 < x less or equal than 2 which is the required solution of the given inequality.
    Hence, the solution set is (-23, 2].
    Question 15
    CBSEENMA11014367
    Wired Faculty App

    Solve the following inequality and show the graph of the solution on the number line.

    3 (1 - x) < 2 (x + 4)
    Solution

    3 (1 - x) < 2 (x + 4)
    rightwards double arrow          3 minus 3 x less than 2 x plus space 8 space rightwards double arrow space minus 3 x minus 2 x space less than space 8 minus 3 space rightwards double arrow negative 5 x less than 5
    rightwards double arrow                fraction numerator negative 5 straight x over denominator negative 5 end fraction greater than fraction numerator 5 over denominator negative 5 end fraction                                                (∵ -5 < 0)
    rightwards double arrow                     x > - 1
    Hence, the solution of the inequality is left parenthesis negative 1 comma space infinity right parenthesis
    Graph on the number line is:
                        

    Question 16
    CBSEENMA11014368
    Wired Faculty App

    Solve the following inequality and show the graph of the solution on the number line.

    straight x over 2 less than fraction numerator left parenthesis 5 straight x minus 2 right parenthesis over denominator 3 end fraction minus fraction numerator left parenthesis 7 straight x minus 3 right parenthesis over denominator 5 end fraction
    Solution
    straight x over 2 less than fraction numerator left parenthesis 5 straight x minus 2 right parenthesis over denominator 3 end fraction minus fraction numerator left parenthesis 7 straight x minus 3 right parenthesis over denominator 5 end fraction
    Multiply by 30 (the L.C.M. of 2, 3 and 5), we have
                                      15 straight x less than 10 space left parenthesis 5 straight x minus 2 right parenthesis space minus space 6 space left parenthesis 7 straight x minus 3 right parenthesis
    rightwards double arrow                             15 straight x less than 50 straight x minus 20 minus 42 straight x plus 18 space rightwards double arrow space 15 straight x minus 50 straight x plus 42 straight x less than negative 20 plus 18
    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>
    Hence, the solution of the inequality is open parentheses negative infinity comma space minus 2 over 7 close parentheses and the graph on the number line is:
                      
    Question 17
    CBSEENMA11014369
    Wired Faculty App

    Solve the following inequality and represent the solution graphically on the number line. 

    5 straight x plus 1 greater than negative 24 comma space space space space 5 straight x minus 1 less than 24
    Solution
    <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)
    Also,      space space 5 straight x minus 1 less than 24 space space space rightwards double arrow space 5 straight x less than 25 space rightwards double arrow space straight x space less than 5                                           ...(ii)
    From (i) and (ii), we have -5 < x < 5
    ∴ the solution set is (-5, 5). Graphically, we have:
                           

    Sponsor Area

    Question 18
    CBSEENMA11014370
    Wired Faculty App

    Solve the following inequality and represent the solution graphically on the number line. 

    5 (2x – 7) – 3 (2x + 3) ≤ 0 ,   space space 2 straight x plus 19 space less or equal than space 6 straight x plus 47.
    Solution

    5 (2x – 7) – 3 (2x + 3) ≤ 0 
    rightwards double arrow     10 straight x minus 35 minus 6 straight x minus 9 space less or equal than space 0 space rightwards double arrow space 10 straight x minus 6 straight x less or equal than 9 plus 35 space rightwards double arrow space 4 straight x less or equal than 44 space rightwards double arrow space straight x less or equal than 11            ...(i)
    Also,           2 straight x plus 19 less or equal than 6 straight x plus 47
    rightwards double arrow           2 straight x minus 6 straight x space less or equal than space 47 minus 19 space rightwards double arrow space minus 4 straight x less or equal than space 28 space rightwards double arrow space fraction numerator negative 4 straight x over denominator negative 4 end fraction greater or equal than fraction numerator 28 over denominator negative 4 end fraction space rightwards double arrow space straight x space greater or equal than space minus 7             ...(ii)
    From (i) and (ii), we have space space minus 7 less or equal than straight x less or equal than 11
    ∴   The solution set is [-7, 11]. Graphically, we have
               

    Question 19
    CBSEENMA11014371
    Wired Faculty App

    Solve the following pair of inequality and represent the solution graphically on the number line:

     3x – 7 > 2 (x – 6) , 6 – x > 11 – 2x
    Solution
    3 straight x minus 7 space greater than 2 space left parenthesis straight x minus 6 right parenthesis space rightwards double arrow space 3 straight x minus 7 space greater than 2 straight x minus 12 space rightwards double arrow space 3 straight x minus 2 straight x greater than negative 12 plus 7
    rightwards double arrow                 x> - 5                                                                                             ...(i)
    Also,             6 minus straight x greater than 11 minus 2 straight x rightwards double arrow space minus straight x space plus space 2 straight x greater than 11 minus 6 rightwards double arrow straight x greater than 5                                 ...(ii)
    From (i) and (ii), we have space space straight x greater than 5 space rightwards double arrow space straight x space element of space left parenthesis 5 comma space infinity right parenthesis
    The solution set is space space left parenthesis 5 comma space infinity right parenthesis. Graphically on number line, we have
                     
    Question 20
    CBSEENMA11014372
    Wired Faculty App

    Solve the following pair of inequality and represent the solution graphically on the number line:

    2 (x – 1) < x + 5,   3 (x + 2) > 2 – x
    Solution
    space space space 2 left parenthesis straight x minus 1 right parenthesis less than straight x plus 5 space rightwards double arrow space 2 straight x space minus space 2 less than straight x space plus space 5 space rightwards double arrow space 2 straight x space minus straight x less than 5 plus 2 space rightwards double arrow space straight x space less than space 7                         ...(i)
    Also,  3 space left parenthesis straight x plus 2 right parenthesis greater than 2 minus straight x rightwards double arrow space 3 straight x plus 6 greater than 2 minus straight x rightwards double arrow 3 straight x plus straight x greater than 2 minus 6 rightwards double arrow 4 straight x greater than negative 4 rightwards double arrow straight x greater than negative 1      ...(ii)
    From (i) and (ii), we have -1 < x < 7
    The solution set is (-1, 7)
    Graphically on number line, we have
                                  
                                   
    Question 21
    CBSEENMA11014373
    Wired Faculty App

    Find all pairs of consecutive odd positive integers which are smaller than 10 such that their sum is more than 11.

    Solution

    Let the two consecutive odd positive integers be x and x + 2
    Since both the integers are smaller than 10
    ∴                        x < 10 and x + 2 < 10
    rightwards double arrow                     x < 10   and    x < 8
    rightwards double arrow                      x < 8                                                                                              ...(i)
    Also,  straight x plus left parenthesis straight x plus 2 right parenthesis greater than 11 space rightwards double arrow space 2 straight x plus 2 space greater than 11 space rightwards double arrow space 2 straight x greater than 9 space rightwards double arrow space straight x greater than 9 over 2                                     ...(ii)
    From (i) and (ii), we have
                 9 over 2 less than straight x less than 8 comma                   space space space straight x space element of space straight Z space rightwards double arrow space straight x space equals space 5 comma space 6 comma space 7
    But x is odd.
    rightwards double arrow x = 5, 7 and x + 2 = 7, 9
    Hence, the pair of odd numbers are satisfying the given conditions are 5, 7 and 7, 9.

    Question 22
    CBSEENMA11014374
    Wired Faculty App

    Find pairs of consecutive even positive integers which are larger than 5 and are such that their sum is less than 20.

    Solution

    Let the consecutive even positive integers be x and x + 2
    rightwards double arrow                            space space straight x greater than 5 comma space straight x plus 2 greater than 5 space rightwards double arrow space space straight x greater than 5 comma space straight x greater than 3 space rightwards double arrow space straight x greater than 5                         ...(i)
    Their sum is less than 20.
    rightwards double arrow           x + (x + 20) < 20 rightwards double arrow 2x + 2 < 20 rightwards double arrow  2x < 18 rightwards double arrow x < 9                        ...(ii)
    From (i) and (ii), we have 5 < x < 9,  straight x space element of space straight Z space rightwards double arrow space straight x space equals space 6 comma 7 comma space 8
    But, x is given,            ∴  x = 6,  8
    When                          x = 6,  x + 2 = 8
    When                          x = 8,  x + 2 = 10
    Hence, the two pairs that satisfy the given conditions are: 6, 8 and 10.

    Question 23
    CBSEENMA11014375
    Wired Faculty App

    The marks obtained by a student of class XI in first terminal and second terminal examination are 62 and 48 respectively. Find the number of minimum marks he should get in the annual examination to have an average of at least 60 marks.

    Solution

    Let marks obtained by the student in annual examination be x.
    ∴     average of marks = fraction numerator 62 plus 48 plus straight x over denominator 3 end fraction. But, minimum average marks = 60
    rightwards double arrow            fraction numerator 62 plus 48 plus straight x over denominator 3 end fraction greater or equal than 60 space rightwards double arrow space 62 plus 48 plus straight x greater or equal than 180 space rightwards double arrow space straight x greater or equal than 180 minus 62 minus 48 space rightwards double arrow space straight x space greater or equal than 70
    Hence, the student must obtain a minimum of 70 marks to get an average of at least 60 marks.

    Question 24
    CBSEENMA11014376
    Wired Faculty App

    The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61cm, find the minimum length of the shortest side.

    Solution

    Let the shortest side of the triangle be of length x cm
    rightwards double arrow  The length of the longest side = 3x cm
          The length of the third side = (3x - 2) cm
          It is given that the perimeter is at least 61 cm
    rightwards double arrow  Perimeter <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>
    Hence, the minimum length of the shortest side = 9 cm.

    Question 25
    CBSEENMA11014377
    Wired Faculty App

    A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second ?

    Solution

    Let the length of the first piece (or the shortest piece) be x cm
    rightwards double arrow   Length of the second piece = (x + 3) cm
    Also, the length of the third piece = 2x cm. The maximum length of the board available = 91 cm.
    rightwards double arrow straight x plus left parenthesis straight x plus 3 right parenthesis plus 2 straight x less or equal than 91 space rightwards double arrow space 4 straight x plus 3 less or equal than 91 space space rightwards double arrow space 4 straight x less or equal than 88 space rightwards double arrow space straight x less or equal than 22                            ...(i)
    Also, the third piece is to be at least 5 cm longer than the second.
     rightwards double arrow                      2 straight x minus left parenthesis straight x plus 3 right parenthesis space greater or equal than space 5 space space rightwards double arrow space space straight x minus 3 greater or equal than 5 space rightwards double arrow space straight x space greater or equal than 8                                    ...(ii)
    From (i) and (ii), we have 8 less or equal than straight x less or equal than 22
    Hence, the shortest board (or the first board) can have a minimum length of 8 cm and a maximum length of 22 cm.

    Question 26
    CBSEENMA11014378
    Wired Faculty App

    How many litres of water will have to be added to 1125 litres of the 45%, solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

    Solution

    Quantity of 45% acid solution = 1125 litres
    Quantity of acid = fraction numerator 1125 cross times 45 over denominator 100 end fraction litres space equals space 2025 over 4 space litres
    Suppose x litres of water is added
    ∴          Quantity of acid = 2025 over 4 space litres
    Total quantity of mixture = (1125 + x) litres
                 % of acid content = space space fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100
    According to the problem,
                         space space space space space 25 less or equal than fraction numerator 2025 over denominator 4 space left parenthesis 1125 plus straight x right parenthesis end fraction cross times 100 less or equal than 30
    rightwards double arrow                     25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30
    Now,                 25 less or equal than fraction numerator 50625 over denominator 1125 plus straight x end fraction space space rightwards double arrow space space 1 less or equal than fraction numerator 2025 over denominator 1125 plus straight x end fraction space rightwards double arrow space 1125 plus straight x space less or equal than space 2025 space rightwards double arrow space straight x space less or equal than space 900     ...(i)
    Also,        fraction numerator 50625 over denominator 1125 plus straight x end fraction less or equal than 30 space rightwards double arrow space fraction numerator 10125 over denominator 1125 plus straight x end fraction less or equal than 6 space space rightwards double arrow space 10125 space less or equal than space 6750 space plus space 6 straight x space rightwards double arrow space 3375 less or equal than 6 straight x
    rightwards double arrow                         6 straight x greater or equal than 3375
    rightwards double arrow                          2 straight x greater or equal than 1125 space rightwards double arrow space straight x greater or equal than 562.5                                                                        ...(ii)
    From (i) and (ii), we have 562.5 less or equal than straight x less or equal than 900
    Hence, a minimum of 562.5 litres and a maximum of 900 litres of water be added so as to move a solution with 25% to 30% of acid content.


    Question 27
    CBSEENMA11014379
    Wired Faculty App

    In drilling world’s deepest hole, it was found that the temperature T in degree celsius, x km below the surface of the earth, was given by

    straight T space equals space 30 plus space 25 left parenthesis straight x minus 3 right parenthesis comma space space space 3 less than straight x less than 15

    At what depth will the temperature be between 200 degree straight C and 300 degree straight C?

    Solution

    We have
                       space space straight T space equals space 30 plus 25 left parenthesis straight x minus 3 right parenthesis comma space 3 less than straight x less than 15
    It is given that 200 < T < 300
    or                                 200 < 30 + 25(x - 3) < 300
    or                                  170 < 25 (x - 3) < 270
    or                              170 over 25 less than left parenthesis straight x minus 3 right parenthesis less than 270 over 25
    or                          6.8 + 3 < x < 10.8 + 3
    or                                 9.8 < x < 13.8
     Hence, the depth will be between 9.8 km and 13.8 km.
                  

    Question 28
    CBSEENMA11014380
    Wired Faculty App

     IQ of a person is given by the formula:

                            <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 MA is mental age and CA is chronological age. If 80 less or equal than IQ less or equal than 140. For a group of 12 year children, find the range of their mental age.

    Solution

    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>                                                                       ...(i)
    Now,         CA = 12 and 80 less or equal than IQ less or equal than 140 (given)
    rightwards double arrow              80 less or equal than MA over 12 cross times 100 less or equal than 140
    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             960 over 100 less or equal than MA less or equal than fraction numerator 140 cross times 12 over denominator 100 end fraction
    rightwards double arrow                space space 9.6 less or equal than MA less or equal than 16.8
    Hence,  MA is at least 9.6 but not more than 16.8.

    Question 29
    CBSEENMA11014381
    Wired Faculty App

    In an experiment, a solution of a hydrochloric acid is to be kept between 30° and 35° Celsius. What is the range of the temperature in degree Fahrenheit, if conversion formula 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>

    where C and F represent temperature in degree Celsius and degree Fahrenheit, respectively.
                  

    Solution

    It is given that 30 degree less than straight C less than 35 degree
    Putting               <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
                                30 less than 5 over 9 left parenthesis straight F minus 32 right parenthesis less than 35
    or            <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>
    or                      <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>
    or                      86 < F < 95
    Thus, the required range of temperature is between 86 degree straight F space and space 95 degree straight F.

    Question 30
    CBSEENMA11014382
    Wired Faculty App

    To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in the first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

    Solution

    Let Sunita's marks in the fith examination be x out of 100 marks. Sunita's marks in the first four examinations are 87, 92, 94 and 95
    Average marks obtained by Sunita = <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>
    To receive grade 'A' in a course Sunita has to get an average of 90 marks or more
    ∴              fraction numerator 368 plus straight x over denominator 5 end fraction space greater or equal than space 90                   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                <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 x space greater or equal than space 82
    Hence, Sunita must obtain a minimum of 82 marks in the fifth examination to get grade 'A' in the course.     

    Sponsor Area

    Question 40
    CBSEENMA11014392
    Wired Faculty App

    Solve the following inequality:

    30 straight x space less or equal than space 200 space space For space left parenthesis straight i right parenthesis space straight x space element of space straight N space left parenthesis ii right parenthesis space straight x space element of space straight Z space left parenthesis iii right parenthesis space straight x space element of space straight R
    Solution

    Solution not provided.
    Ans. (i) open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 close curly brackets (ii) open curly brackets..... comma space minus 3 comma space minus 2 comma space minus 1 comma space 0 comma space 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma close curly brackets (iii) left parenthesis negative infinity comma space 20 over 3 right square bracket

    Question 41
    CBSEENMA11014393
    Wired Faculty App

    Solve the following inequality:

    5 straight x minus 3 less than 3 straight x plus 1  For (i) straight x space element of space straight Z   (ii)  <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

    Solution not provided.
    Ans. (i) <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>   (ii) space space left parenthesis negative infinity comma space 2 right parenthesis

    Question 42
    CBSEENMA11014394
    Wired Faculty App

    Solve the following inequality:

    4x + 3 < 6x + 7, For (i) straight x space element of space straight N (ii) straight x space element of space straight Z  (iii) <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

    Solution not provided.
    Ans. (i) space space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma........ close curly brackets   (ii) space space space space open curly brackets negative 1 comma space 0 comma space 1 comma space 2 comma space 3 comma....... close curly brackets     (iii) <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 43
    CBSEENMA11014395
    Wired Faculty App

    Solve the following inequality:

    space space fraction numerator 5 minus 2 straight x over denominator 3 end fraction less or equal than straight x over 6 minus 5

    For (i) straight x space element of space straight N   (ii) straight x space element of space straight Z  (iii) "<pre
    Solution

    Solution not provided.
    Ans. (i) {8, 9, 10,.........}   (ii) {8, 9, 10, .....} (iii) left square bracket 8 comma space infinity right parenthesis

    Question 44
    CBSEENMA11014396
    Wired Faculty App

    Solve the following inequality and represent them on the number line:

    3x - 7 > x + 3
    Solution

    Solution not provided.
    Ans. <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 45
    CBSEENMA11014397
    Wired Faculty App

    Solve the following inequality and represent them on the number line:

    4x - 7 < 3 - x
    Solution

    Solution not provided.
    Ans. space left parenthesis negative infinity comma space 2 right parenthesis
                 
                          

    Question 46
    CBSEENMA11014398
    Wired Faculty App

    Solve the following inequality and represent them on the number 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>
    Solution

    Solution not provided.
    Ans. left parenthesis negative infinity comma space minus 2 right square bracket

                           
                              
                 
                          

    Question 47
    CBSEENMA11014399
    Wired Faculty App

    Solve the following inequality and represent them on the number line:

    straight x plus straight x over 2 plus straight x over 3 less than 11
    Solution

    Solution not provided.
    Ans. space space left parenthesis negative infinity comma space 6 right parenthesis
                    
                       
                           
                              
                 
                          

    Question 48
    CBSEENMA11014400
    Wired Faculty App

    Solve the following inequality and represent them on the number line:

    37 minus left parenthesis 3 straight x plus 5 right parenthesis greater or equal than 9 straight x minus 8 left parenthesis straight x minus 3 right parenthesis
    Solution

    Solution not provided.
    Ans. <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 49
    CBSEENMA11014401
    Wired Faculty App

    Solve the following inequality and represent them on the number line:

    2 (2x + 3) - 10 < 6 (x - 2)
    Solution

    Solution not provided.
    Ans. <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 50
    CBSEENMA11014402
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line:

    x + 3 >0;  2x < 14
    Solution

    Solution not provided.
    Ans. (-3, 7)
                 
                       

    Question 51
    CBSEENMA11014403
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line:

    fraction numerator 4 straight x over denominator 3 end fraction minus 9 over 4 less or equal than straight x plus 3 over 4. space space space open parentheses fraction numerator 7 straight x minus 1 over denominator 3 end fraction close parentheses minus open parentheses fraction numerator 7 straight x plus 2 over denominator 6 end fraction close parentheses greater or equal than straight x
    Solution

    Solution not provided.
    Ans. [4, 9]
                      
                 
                       

    Question 52
    CBSEENMA11014404
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line:

    2 space left parenthesis 2 straight x plus 3 right parenthesis space minus space 10 space less than space 6 space left parenthesis straight x space minus space 2 right parenthesis semicolon space space open parentheses fraction numerator straight x minus 2 over denominator 2 end fraction close parentheses plus 13 greater or equal than 2 plus fraction numerator 4 straight x over denominator 3 end fraction
    Solution

    Solution not provided.
    Ans.  (4, 12]
                       
                      
                 
                       

    Question 53
    CBSEENMA11014405
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number 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>
    Solution

    Solution not provided.

    Question 54
    CBSEENMA11014406
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line.

    4 minus 5 straight x greater than negative 11 semicolon space 4 straight x plus 11 less or equal than negative 3
    Solution

    Solution not provided.
    Ans. <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 55
    CBSEENMA11014407
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line.

    7 straight x minus 8 space less than space 4 straight x plus 7 semicolon space minus straight x over 2 greater than 4
    Solution

    Solution not provided.
    Ans. left parenthesis negative infinity comma space minus 8 right parenthesis

                     

    Question 56
    CBSEENMA11014408
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line.

    space space 5 straight x minus 7 less than 3 space left parenthesis straight x plus 3 right parenthesis semicolon space 1 minus fraction numerator 3 straight x over denominator 2 end fraction greater or equal than straight x minus 4
    Solution

    Solution not provided.
    Ans. <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 57
    CBSEENMA11014409
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line:

    2 less or equal than 3 straight x minus 4 less or equal than 5
    Solution

    Solution not provided.
    Ans.  [2, 3]
       

    Question 58
    CBSEENMA11014410
    Wired Faculty App

    Solve the following inequality and show the solution graphically on the number line:

    7 less than 4 minus fraction numerator 3 straight x over denominator negative 5 end fraction less or equal than 10
    Solution

    Solution not provided.
    Ans.  (5, 10]
       

    Question 59
    CBSEENMA11014411
    Wired Faculty App

    Solve the following inequality for (i) straight x space element of space straight N comma (ii) space space space space straight x space element of space straight Z comma   (iii) <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 show that the graphs of the solutions on the number line. : 3x - 2 < 2x + 1
    Solution

    3x - 2 < 2x + 1 rightwards double arrow 3x - 2x < 1 + 2 rightwards double arrow x < 3
    (i) Solution for straight x space element of space straight N space rightwards double arrow space straight x space equals space 1 comma space 2
    The solution set is open curly brackets 1 comma space 2 close curly brackets. The graph on the number line is:
                           
    (ii)         <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                             x = 2, 1, 0, -1, -2, -3, ............
    ∴ The solution set is open curly brackets..... comma space minus 3 comma space minus 2 comma space minus 1 comma space 0 comma space 1 comma space 2 close curly brackets. The graph on the number line is:
                       
    (iii)     straight x less than 3 comma space straight x space element of space straight R space rightwards double arrow space straight x space element of space left parenthesis negative infinity comma space 3 right parenthesis.  The graph on the number line is:
                       

    Question 60
    CBSEENMA11014412
    Wired Faculty App

    Solve the following inequality for (i) straight x space element of space straight N comma (ii) space space space space straight x space element of space straight Z comma   (iii) <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 show that the
    graphs of the solutions on the number line. : 5 straight x minus 3 greater or equal than 3 straight x minus 5.
    Solution
    space space 5 straight x minus 3 greater or equal than 3 straight x minus 5 rightwards double arrow space 5 straight x minus 3 straight x greater or equal than negative 5 plus 3 space rightwards double arrow space 2 straight x greater or equal than negative 2 space rightwards double arrow space space straight x greater or equal than negative 1
    (i) Solution set for straight x space element of space straight N

    rightwards double arrow                        x = 1, 2, 3, 4..........
    ∴  The solution set is space space space space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space....... close curly brackets.  The graph on the number line is:
                            
    (ii) Solution for space space straight x space element of space straight Z comma space straight x greater or equal than negative 1
    rightwards double arrow                       x = -1, 0, 1, 2, 3, 4, ............
    ∴  The solution set 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>  The graph on the number line is:
                      
    (iii) Solution for <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 square bracket negative 1 comma space infinity right parenthesis. The graph on the number line is:
                                    
    Question 61
    CBSEENMA11015066
    Wired Faculty App

    In each of the following, find the equation of circle with centre (-2, 3) and radius 4

    Solution

    Centre (h, k) left right arrow (-2, 3)
    rightwards double arrow                       h = -2, k = 3, Radius, r = 4
    ∴ The equation of 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>
    or      left parenthesis straight x plus 2 right parenthesis squared plus left parenthesis straight y minus 3 right parenthesis squared space equals space left parenthesis 4 right parenthesis squared space or space straight x squared plus straight y squared plus 4 straight x minus 6 straight y minus 3 equals 0

    Question 62
    CBSEENMA11015067
    Wired Faculty App

    In each of the following, find the equation of circle with centre open parentheses 1 half comma space 1 fourth close parentheses and radius 1 over 12

    Solution

    Centre (h, k) left right arrow open parentheses 1 half comma space 1 fourth close parentheses rightwards double arrowspace space space space space space space space space space space space space space straight h equals 1 half comma space straight k space equals space 1 fourth space space Radius comma space space straight r equals 1 over 12
    ∴ Equation of 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>
    or open parentheses straight x minus 1 half close parentheses squared plus open parentheses straight y minus 1 fourth close parentheses squared space equals space open parentheses 1 over 12 close parentheses squared space or space straight x squared plus straight y squared minus straight x minus 1 half straight y plus 1 fourth plus 1 over 16 equals 1 over 144
    or  144 straight x squared plus 144 straight y squared minus 144 straight x minus 72 straight y plus 36 plus 9 equals 1
    or  144 straight x squared plus 144 straight y squared minus 144 straight x minus 72 straight y plus 44 equals 0
    or  36 straight x squared plus 36 straight y squared minus 36 straight x minus 18 straight y plus 11 equals 0
    Hence, the equation of circle is: 36 straight x squared plus 36 straight y squared minus 36 straight x minus 18 straight y plus 11 equals 0

    Question 63
    CBSEENMA11015068
    Wired Faculty App

    In each of the following, find the equation of circle with centre (-a, -b) and radiusspace space space space square root of straight a squared minus straight b squared end root.

    Solution

    Centre (h, k) left right arrow(-a, -b) rightwards double arrow h = -a, k = -b
         Diameter of circle = 2 square root of straight a squared minus straight b squared end root space space rightwards double arrow space Radius comma space straight r space equals space square root of straight a squared minus straight b squared end root
    ∴ Equation of 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>
    or     left parenthesis straight x plus straight a right parenthesis squared plus left parenthesis straight y plus straight b right parenthesis squared equals left parenthesis square root of straight a squared minus straight b squared end root right parenthesis squared
    rightwards double arrow    left parenthesis straight x plus straight a right parenthesis squared plus left parenthesis straight y plus straight b right parenthesis squared space equals space straight a squared minus straight b squared space rightwards double arrow space straight x squared plus straight y squared plus 2 ax plus 2 by plus straight a squared plus straight b squared equals straight a squared minus straight b squared
    Hence, the equation of circle is: straight x squared plus straight y squared plus 2 ax plus 2 by plus 2 straight b squared equals 0

    Question 64
    CBSEENMA11015069
    Wired Faculty App

    Find the equation of a circle with centre at (2, 2) and passes through the point (4,5).

    Solution

    The centre of circle is C (h, k) left right arrow (2, 2) rightwards double arrow h = 2, k =2
    Since the circle passes through the point A (4, 5).
    ∴ Radius of the circle is : r = CA
    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>
    ∴ Equation of circle is: space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared
    or   left parenthesis straight x minus 2 right parenthesis squared plus left parenthesis straight y minus 2 right parenthesis squared space equals space left parenthesis square root of 13 right parenthesis squared space or space straight x squared plus straight y squared minus 4 straight x minus 4 straight y plus 4 plus 4 equals 13
    Hence, the equation of 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 65
    CBSEENMA11015070
    Wired Faculty App

    Find the equation of a circle which pass through the points O (0, 0), A (4, 0) and B (0, 3).

    Solution

    Let the equation of the circle 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>                       ...(i)
    Since it passes through (0, 0)
    ∴       left parenthesis 0 minus straight h right parenthesis squared plus left parenthesis 0 minus straight k right parenthesis squared space equals space straight r squared space rightwards double arrow space straight h squared plus straight k squared equals straight r squared                                  ...(ii)
    Since it passes through (4, 0)
    ∴        space space space space space left parenthesis 4 minus straight h right parenthesis squared plus left parenthesis 0 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space 16 plus straight h squared minus 8 straight h plus straight k squared equals straight r squared               ...(iii)
    But, from (ii),          <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>
    ∴ From (iii), we have 16 minus 8 straight h plus straight r squared equals straight r squared space rightwards double arrow space 8 straight h equals 16 space rightwards double arrow space straight h equals 2
    Since it passes through (0, 3)
    ∴      left parenthesis 0 minus straight h right parenthesis squared plus left parenthesis 3 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared minus 6 straight k plus 9 equals straight r squared rightwards double arrow straight r squared minus 6 straight k plus 9 equals straight r squared rightwards double arrow 6 straight k equals 9 rightwards double arrow straight k equals 3 divided by 2
    ∴ From (ii), 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>
    or              left parenthesis 4 right parenthesis plus open parentheses 9 over 4 close parentheses equals straight r squared space space or space space straight r squared equals 25 over 4 space or space straight r equals 5 over 2
    Hence, the equation of 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>
    or  space space left parenthesis straight x minus 2 right parenthesis squared plus open parentheses straight y minus 3 over 2 close parentheses squared equals 25 over 4 space or space straight x squared plus straight y squared minus 4 straight x minus 3 straight y plus 4 plus 9 over 4 equals 25 over 4
    or     <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 66
    CBSEENMA11015071
    Wired Faculty App

    Find the equation of the circle passes through (0, 0) and making intercepts a and b on the co-ordinate axes. 

    Solution

    The circle passes through the origin and makes intercepts a with x-axis and b with y-axis.
    rightwards double arrow                          OA = a, OB = b
    ∴ The co-ordinates of A are (a, 0) and of B are (0, b)
    Now, we have to find the equation of a circle which passes through the points O (0, 0), A (a, 0), B (0, b).
                                    
    Let the equation of the circle
                       <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)
    Since it passes through (0, 0)
    ∴                 left parenthesis 0 minus straight h right parenthesis squared plus left parenthesis 0 minus straight k right parenthesis squared equals straight r 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>                                   ...(ii)
    Since it passes through A (a, 0)
    ∴        left parenthesis straight a minus straight h right parenthesis squared plus left parenthesis 0 minus straight k right parenthesis squared space equals space straight r squared
    rightwards double arrow    straight h squared plus straight k squared minus 2 ah plus straight a squared equals straight r squared                                      ...(iii)
    But, from (ii),  <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>
    ∴ from (iii), 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>
    Since the circle passes through (0, b)
    ∴    left parenthesis 0 minus straight h right parenthesis squared plus left parenthesis straight b minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared minus 2 bk plus straight b squared equals straight r squared rightwards double arrow straight r squared minus 2 bk plus straight b squared equals straight r squared
    rightwards double arrow                       straight b squared equals 2 bk space rightwards double arrow space straight k equals straight b over 2
    ∴ from (ii), we have space space space straight a squared over 4 plus straight b squared over 4 space equals space straight r squared space rightwards double arrow space straight r squared space equals space fraction numerator straight a squared plus straight b squared over denominator 4 end fraction
    ∴ The equation of 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>
    or    space space space open parentheses straight x minus straight a over 2 close parentheses squared plus open parentheses straight y minus straight b over 2 close parentheses squared space equals space fraction numerator straight a squared plus straight b squared over denominator 4 end fraction space rightwards double arrow space straight x squared plus straight y squared minus ax minus by plus straight a squared over 4 plus straight b squared over 4 equals straight a squared over 4 plus straight b squared over 4
    Hence, the equation of circle is : space space space space straight x squared plus straight y squared minus ax minus by space equals space 0.

    Question 67
    CBSEENMA11015072
    Wired Faculty App

    Find the equation of a circle whose centre is (3, –2) and has an area of 154 square units.

    Solution

    The co-ordinates of the centre of the circle are (3, -2)
    rightwards double arrow                                       h = 3, k = -2                                                        ...(i)
    Let r be the radius of the circle
                         Area of the circle =<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>                                                 (Given)
    rightwards double arrow                            straight r squared equals 154 over straight pi equals 154 over 22 cross times 7 equals 49
    ∴     Radius,                              r = 7                                                                ...(ii)
    Equation of the circle in the standard form 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>
    or  straight x squared plus straight y squared minus 6 straight x plus 4 straight y plus 9 plus 4 equals 49 space rightwards double arrow space straight x squared plus straight y squared minus 6 straight x plus 4 straight y minus 36 equals 0
    which is the required equation of the circle.

    Question 68
    CBSEENMA11015073
    Wired Faculty App

    Find the equation of a circle having radius 5 units and the centre as the point of intersection of the straight lines 2x – y – 5 = 0 and 3x + 2y = 4.

    Solution

    Radius, r of the circle = 5                                                               ...(i)
    The centre is the point of intersection of straight lines
                        2x - y - 5=0  and 3x + 2y = 4
    Solving these two equations simultaneously, we have
                         4x - 2y = 10
                         3x + 2y = 4
    Adding, we get 7x = 14 or x = 2
    rightwards double arrow                 4 - y - 5 = 0           or   y = -1
    ∴ The co-ordinates of the centre are (2, -1)
    rightwards double arrow                              h = 2, k = -1 and r = 5
    The equation of the circle in the standard form is:
                                 left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared space space space space space space space or space space left parenthesis straight x minus 2 right parenthesis space plus space left parenthesis straight y plus 1 right parenthesis squared space equals space left parenthesis 5 right parenthesis squared
    or       <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>
    which is the required equation of the circle.

    Question 69
    CBSEENMA11015074
    Wired Faculty App

    Find the equation of a circle, two of whose diameters are along the straight lines x – 2y = 7 and x + 2y + 1 = 0 and which passes through the point (1, 4).

    Solution

    Let AB and EF be the two diameters of the circle and have equations 
             x - 2y = 7 and x + 2y = -1 respectively.
    The centre C of the circle is the point of intersection of the two diameters
    ∴  For C,
                          x - 2y = 7                                                       ...(i)
                          x + 2y = -1                                                     ...(ii)
    Adding, we get           2x = 6                rightwards double arrow       x = 3
    and                      3 - 2y = 7                rightwards double arrow       y = -2
    ∴ The co-ordinates of centre C of the circle are (3, -2)
    Also, the circle passes through point P(1, 4)
    ∴ Radius of the circle, r = CP = space space square root of left parenthesis 1 minus 3 right parenthesis squared plus left parenthesis 4 plus 2 right parenthesis squared end root equals space square root of 4 plus 36 end root space equals space square root of 40
    rightwards double arrow                            h = 3, k = -2, space space straight r equals square root of 40
    The equation 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>
    or     <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>
    which is the required equation of the circle.
            

    Question 70
    CBSEENMA11015075
    Wired Faculty App

    Find the equation of a circle having centre at point C(–2, 3) and touching the straight line 3x – 4y – 2 = 0.

    Solution

    Centre of the required circle is C (-2, 3)
    Let line AB with equation 3x - 4y - 2 = 0 touches the circle at point L
    ∴                       p = r
    where p, is length of perpendicular CL and r is the radius
    rightwards double arrow                           open vertical bar fraction numerator 3 left parenthesis negative 2 right parenthesis minus 4 left parenthesis 3 right parenthesis minus 2 over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction close vertical bar space equals space straight r space space space space rightwards double arrow space space straight r equals open vertical bar fraction numerator negative 20 over denominator 5 end fraction close vertical bar space equals space 4
    The equation of circle is:
                                      left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared space or space left parenthesis straight x plus 2 right parenthesis squared space plus space left parenthesis straight y minus 3 right parenthesis squared space equals space left parenthesis 4 right parenthesis squared
    or       space space straight x squared plus straight y squared plus 4 straight x minus 6 straight y minus 3 equals 0
    Hence, the equation of circle is: space space straight x squared plus straight y squared plus 4 straight x minus 6 straight y minus 3 equals 0

    Question 71
    CBSEENMA11015076
    Wired Faculty App

    Find the equation of a circle, the two ends of whose diameter are A(4, –5) and R (–2, –1).

    Solution

    Let C be the centre of the circle
    ∴  C is the mid-point of AB.
    ∴  Co-ordinates of C, (h, k) left right arrow open parentheses fraction numerator 4 minus 2 over denominator 2 end fraction comma space fraction numerator negative 5 minus 1 over denominator 2 end fraction close parentheses
    rightwards double arrow                          (h, k) left right arrow (1, -3)
    ∴                                   h = 1, k = -3
                                                 
    Also,                       radius, r = CB
    rightwards double arrow                                    straight r equals square root of left parenthesis 1 plus 2 right parenthesis squared plus left parenthesis negative 3 plus 1 right parenthesis squared end root space equals space square root of 9 plus 4 end root space equals space square root of 13
    ∴ The equation of circle is: left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared space or space left parenthesis straight x minus 1 right parenthesis squared plus left parenthesis straight y plus 3 right parenthesis squared space equals space left parenthesis square root of 13 right parenthesis squared
    or    <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>
    Hence, the equation of the circle is : space space straight x squared plus straight y squared minus 2 straight x plus 6 straight y minus 3 equals 0

    Question 72
    CBSEENMA11015077
    Wired Faculty App

    Find the equation of a circle which passes through the points (2, 3) and (–1, 1) and whose centre lies on the straight line x – 3y – 11 = 0.

    Solution

    Let the equation of the circle be space space space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared                                                                       ...(i)
    It passes through (2, 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>                             ...(ii)
    It passes through (-1, 1) rightwards double arrow    left parenthesis negative 1 minus straight h right parenthesis squared plus left parenthesis 1 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2 equals straight r squared                            ...(iii)
    From (ii) and (iii), we have
                  straight h squared plus straight k squared minus 4 straight h minus 6 straight k plus 13 equals straight h squared plus straight k squared plus 2 straight h minus 2 straight k plus 2
    rightwards double arrow                  negative 6 straight h minus 4 straight k plus 11 equals 0 space rightwards double arrow space 6 straight h plus 4 straight k minus 11 equals 0                                                                        ...(iv)
    Also, centre (h, k) lies on the line x - 3y -11 = 0 rightwards double arrow h - 3k - 11 = 0                                                           ...(v)
    Multiplying (v) by 6, we get 
                                6h - 18k - 66 =0                                                                                                        ...(vi)
    Subtracting (vi) from (iv), we get space space space space 22 straight k plus 55 equals 0 space rightwards double arrow space straight k equals fraction numerator negative 5 over denominator 2 end fraction
    ∴ from (v), space space space straight h minus 3 open parentheses negative 5 over 2 close parentheses minus 11 equals 0 space rightwards double arrow space straight h equals fraction numerator negative 15 over denominator 2 end fraction plus 11 equals 7 over 2
    Also, from (ii),   space space space 49 over 4 plus 25 over 4 minus 4 open parentheses 7 over 2 close parentheses minus 6 open parentheses negative 5 over 2 close parentheses plus 13 equals straight r squared
    rightwards double arrow                            straight r squared equals 49 over 4 plus 25 over 4 minus 14 plus 15 plus 13 space rightwards double arrow space space straight r squared equals 49 over 4 plus 25 over 4 plus 14
    rightwards double arrow                             straight r squared equals fraction numerator 49 plus 25 plus 56 over denominator 4 end fraction equals 130 over 4 equals 65 over 2
    Hence, the equation of circle is: open parentheses straight x minus 7 over 2 close parentheses squared plus open parentheses straight y plus 5 over 2 close parentheses squared space equals space 65 over 2
    or   straight x squared plus straight y squared minus 7 straight x plus 5 straight y plus 49 over 4 plus 25 over 4 minus 65 over 2 equals 0 or <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>
    or  space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 56 over 4 equals 0
    Hence, the equation of circle is: space space straight x squared plus straight y squared minus 7 straight x plus 5 straight y minus 14 equals 0

    Question 73
    CBSEENMA11015078
    Wired Faculty App

    Find the equation of the circle passing through the points A (5, 7), B (6, 6) and C (2, –2).

    Solution

    Let the equation of the circle be space space space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared                                                                       ...(i)
    The circle passes through A (5, 7)
    rightwards double arrow  space space space left parenthesis 5 minus straight h right parenthesis squared plus left parenthesis 7 minus straight k right parenthesis squared space equals space straight r squared space rightwards double arrow space straight h squared plus straight k squared minus 10 straight h minus 14 straight k plus 25 plus 49 equals straight r squared                     
     rightwards double arrow             space space space minus 10 straight h minus 14 straight k plus 74 space equals space straight r squared minus straight h squared minus straight k squared                                                                                    ...(ii)
    The circle passes through B (6, 6)
    rightwards double arrowleft parenthesis 6 minus straight h right parenthesis squared plus left parenthesis 6 minus straight k right parenthesis squared space equals space straight r squared space rightwards double arrow space straight h squared plus straight k squared minus 12 straight h minus 12 straight k plus 36 plus 36 equals straight r squared                             
    rightwards double arrow               negative 12 straight h minus 12 straight k plus 72 space equals space straight r squared minus straight h squared minus straight k squared                                                                                       ...(iii)
    The circle passes through point C (2, -2)
    rightwards double arrow    space space space left parenthesis 2 minus straight h right parenthesis squared plus left parenthesis negative 2 minus straight k right parenthesis squared equals straight r squared space rightwards double arrow space straight h squared plus straight k squared minus 4 straight h plus 4 straight k plus 4 plus 4 equals straight r squared
    rightwards double arrow             negative 4 straight h plus 4 straight k plus 8 equals straight r squared minus straight h squared minus straight k squared                                                                                                  ...(iv)
    Equating (ii) and (iii), we have -10h - 14k + 74 = -12h - 12k + 72
    rightwards double arrow            2h - 2k + 2 = 0 rightwards double arrow  h - k + 1 = 0                                                                                             ...(v)
    Equating (iii) and (iv), we have
           -12h - 12k + 72 = -4h + 4k + 8 rightwards double arrow 8h + 16k - 64 = 0 rightwards double arrow h + 2k - 8 =0                                               ...(vi)
    Subtracting (v) from (vi), we get
                    3k - 9 = 0  rightwards double arrow  k = 3
    Also, from (vi), h + 2 (3) - 8 = 0 rightwards double arrow h = 2
    So, the centre of the circle is : left parenthesis straight h comma space straight k right parenthesis space left right arrow space left parenthesis 2 comma space 3 right parenthesis
    It passes through point A (5, 7)
    rightwards double arrow r = distance between the centre and the point.
    rightwards double arrow r = <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>
    Hence, the equation of the circle is: space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared
    or    left parenthesis straight x minus 2 right parenthesis squared plus left parenthesis straight y minus 3 right parenthesis squared space equals space 5 squared space or space straight x squared plus straight y squared minus 4 straight x minus 6 straight y plus 4 plus 9 equals 25
    or   straight x squared plus straight y squared minus 4 straight x minus 6 straight y minus 12 equals 0
      

    Question 74
    CBSEENMA11015079
    Wired Faculty App

    Find the equation of the circle with radius 5, whose centre lies on x-axis and passing through the point (2, 3).

    Solution



    The centre of the circle lies on x-axis.
    rightwards double arrow  Its co-ordinates can be supposed as (h, 0)
    The circle passes through point A (2, 3) and has a radius 5.
    rightwards double arrow                             CA = 5 <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 h squared minus 4 straight h minus 12 equals 0 space space rightwards double arrow space left parenthesis straight h minus 6 right parenthesis space left parenthesis straight h plus 2 right parenthesis space equals space 0 space rightwards double arrow space straight h space equals space 6 space space or space straight h space equals space minus 2
    When h = 6, the centre of circle is left parenthesis straight h comma space 0 right parenthesis space left right arrow space left parenthesis 6 comma space 0 right parenthesis comma space radius of circle = 5
    ∴  The equation of circle is: left parenthesis straight x minus 6 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared space equals space 5 squared
    or   straight x squared plus straight y squared minus 12 straight x plus 36 minus 25 equals 0 space space or space space straight x squared plus straight y squared minus 12 straight x plus 11 equals 0
    When h = -2, the centre of circle is (-2, 0) and the radius is 5
    ∴ the equation of circle is: space space left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared space equals space straight r squared
    or        <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>
    or          straight x squared plus straight y squared plus 4 straight x plus 4 equals 25 space space or space straight x squared plus straight y squared plus 4 straight x minus 21 equals 0
    Hence, the equation of circle is: straight x squared plus straight y squared minus 12 straight x plus 11 equals 0 space or space straight x squared plus straight y squared plus 4 straight x minus 21 equals 0
    Question 75
    CBSEENMA11015080
    Wired Faculty App

    Find the centre and radii of the following circle:
    open parentheses straight x plus 3 over 2 close parentheses squared plus open parentheses straight y minus 5 over 2 close parentheses squared equals 289 over 4

    Solution

    The given equation of the circle is: open parentheses straight x plus 3 over 2 close parentheses squared plus open parentheses straight y minus 5 over 2 close parentheses squared equals 289 over 4
    Comparing it with left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma space we space get
                                   <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 centre of circle is left parenthesis straight h comma space straight k right parenthesis space left right arrow space open parentheses negative 3 over 2 comma space 5 over 2 close parentheses space and space the space radius space is space 17 over 2
    The radius of circle is straight r equals 17 over 2

    Question 76
    CBSEENMA11015081
    Wired Faculty App

    Find the centre and radii of the following circle:
    left parenthesis straight x plus 5 right parenthesis squared plus left parenthesis straight y minus 3 right parenthesis squared space equals space 36

    Solution

    The given equation of 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>
    Comparing it with the standard equation left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we have
                                      straight h equals negative 5 comma space space straight k equals 3 comma space straight r squared space equals space 36 space rightwards double arrow space straight r space equals space 6
    ∴ The centre of 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> and radius is 6.

    Question 77
    CBSEENMA11015082
    Wired Faculty App

    Find the centre and radii of the following circle:
    straight x squared plus straight y squared equals 49

    Solution

    The given equation of the circle is: straight x squared plus straight y squared equals 49
    or           left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared equals 49
    Comparing it with the standard equation of circle: left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we have
                                            straight h equals 0 comma space straight k equals 0 comma space straight r squared equals 49 space rightwards double arrow space straight r space equals space 7
    Hence, the centre of 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> and the radius is 7.

    Question 78
    CBSEENMA11015083
    Wired Faculty App

    Find the centre and radii of the following circle:
    straight x squared plus straight y squared minus 8 straight x plus 10 straight y minus 12 equals 0

    Solution

    The given equation of the circle is: straight x squared plus straight y squared minus 8 straight x plus 10 straight y minus 12 equals 0
    rightwards double arrow      left parenthesis straight x squared minus 8 straight x right parenthesis plus left parenthesis straight y squared plus 10 straight y right parenthesis equals 12 space space rightwards double arrow space space left square bracket straight x squared minus 8 straight x plus left parenthesis 4 right parenthesis squared right square bracket plus left square bracket straight y squared plus 10 straight y plus left parenthesis 5 right parenthesis squared right square bracket space equals space 12 plus left parenthesis 4 right parenthesis squared plus left parenthesis 5 right parenthesis squared
    rightwards double arrow           space space space left parenthesis straight x minus 4 right parenthesis squared plus left parenthesis straight y plus 5 right parenthesis squared space equals space 53 space rightwards double arrow space left parenthesis straight x minus 4 right parenthesis squared plus left parenthesis straight y plus 5 right parenthesis squared space equals space left parenthesis square root of 53 right parenthesis squared
    Comparing it with the standard equation of circle, left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we get
                                         straight h equals 4 comma space straight k equals negative 5 comma space straight r equals square root of 53
    Hence, the co-ordinates of the centre are left parenthesis straight h comma space straight k right parenthesis space left right arrow space left parenthesis 4 comma space minus 5 right parenthesis comma space radius comma space straight r space equals space square root of 53

    Question 79
    CBSEENMA11015084
    Wired Faculty App

    Find the centre and radii of the following circle:
    space space 2 straight x squared plus 2 straight y squared minus straight x equals 0

    Solution

    The given equation 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> (Note this step)
    rightwards double arrow           open parentheses straight x squared minus straight x over 2 close parentheses plus straight y squared space equals space 0 space rightwards double arrow space open square brackets straight x squared minus straight x over 2 plus open parentheses 1 fourth close parentheses squared close square brackets plus straight y squared equals 0 plus open parentheses 1 fourth close parentheses squared space rightwards double arrow space open parentheses straight x minus 1 fourth close parentheses squared plus left parenthesis straight y minus 0 right parenthesis squared space equals space open parentheses 1 fourth close parentheses squared
    Comparing it with the standard equation of the circle, left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we have
                                                  space space space straight h equals 1 fourth comma space space straight k equals 0 space space and space straight r equals 1 fourth
    ∴ the centre of the circle is left parenthesis straight h comma space straight k right parenthesis space left right arrow space open parentheses 1 fourth comma space 0 close parentheses space radius space is space 1 fourth

    Sponsor Area

    Question 80
    CBSEENMA11015085
    Wired Faculty App

    Find the co-ordinates of the centre and the radius of the circle
    straight x squared plus straight y squared minus 2 ax space cosθ space minus space 2 ay space sinθ space minus space 8 straight a squared equals 0

    Solution

    The given equation is : straight x squared plus straight y squared minus 2 ax space cos space straight theta space minus space 2 ay space sinθ space minus space 8 straight a squared equals 0                   ...(i)
    It can be written as
             space space left parenthesis straight x squared minus 2 ax space cosθ right parenthesis space plus space left parenthesis straight y squared minus 2 ay space sinθ right parenthesis space equals space 8 straight a squared
    Completing the squares in each bracket, we haveleft parenthesis straight x squared minus 2 ax space cosθ space plus space straight a squared space cos squared straight theta right parenthesis space plus space left parenthesis straight y squared minus 2 ay space sinθ space plus space straight a squared sin squared straight theta right parenthesis space equals space 8 straight a squared plus straight a squared cos squared straight theta plus straight a squared sin squared straight theta
    or    <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>
    or    left parenthesis straight x minus straight a space cosθ right parenthesis squared plus left parenthesis straight y minus straight a space sinθ right parenthesis squared space equals space 9 straight a squared space equals space left parenthesis 3 straight a right parenthesis squared
    Comparing it with the standard form left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we  get
                                      straight h equals straight a space cosθ comma space straight k equals straight a space sinθ comma space straight r equals 3 straight a
    Hence, the centre of the circle is left parenthesis straight a space cosθ comma space straight a space sinθ right parenthesis and its radius is 3a.

    Question 81
    CBSEENMA11015086
    Wired Faculty App

    Does the point (-2.5, 3.5) lie inside, outside or on the circle straight x squared plus straight y squared equals 25?
    Solution

    The equation of circle is straight x squared plus straight y squared equals 25
    rightwards double arrow      left parenthesis straight x minus 0 right parenthesis squared plus left parenthesis straight y minus 0 right parenthesis squared space equals space 5 squared
    Comparing it with the standard equation of circle, left parenthesis straight x minus straight h right parenthesis squared plus left parenthesis straight y minus straight k right parenthesis squared equals straight r squared comma we have
    h = 0, k=0 and r = 5
    ∴ Centre of circle is O (0, 0) radius = 5
    Distance of point A (-2.5, 3.5) or straight A space open parentheses negative 5 over 2 comma space 7 over 2 close parentheses from the centre is OA space equals space square root of open parentheses 0 plus 5 over 2 close parentheses squared plus open parentheses 0 minus 7 over 2 close parentheses squared end root
                                              space space space equals square root of 25 over 4 plus 49 over 4 end root space equals space fraction numerator square root of 74 over denominator 2 end fraction less than 5
    rightwards double arrow Distance of point from centre is less than the radius.
    Hence, the point lies inside the circle.

    Question 82
    CBSEENMA11015087
    Wired Faculty App

    Does the point (2, -1) lie inside, or on the circle straight x squared plus straight y squared plus 6 straight x minus 2 straight y minus 6 equals 0?
    Solution

    The equation of the circle is: straight x squared plus straight y squared plus 6 straight x minus 2 straight y minus 6 equals 0
    rightwards double arrow   left parenthesis straight x squared plus 6 straight x right parenthesis plus left parenthesis straight y squared minus 2 straight y right parenthesis space equals space 6 space rightwards double arrow space left parenthesis straight x squared plus 6 straight x plus 9 right parenthesis plus left parenthesis straight y squared minus 2 straight y plus 1 right parenthesis space equals space 6 plus 9 plus 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>
    Comparing it with the standard form of circle <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
                                           straight h equals negative 3 comma space straight k equals 1 space and space straight r space equals space 4
    ∴    The centre is C (-3, 1) and radius is r = 4
    Distance of point A (2, -1) from the centre.
                                  CA space equals space square root of left parenthesis negative 3 minus 2 right parenthesis squared plus left parenthesis 1 plus 1 right parenthesis squared end root space equals space square root of 25 plus 4 end root space equals space square root of 29 greater than 4
    ∴ Distance of point from the centre is greater than the radius.
    Hence, the point lies outside the circle.
                                    

    Question 83
    CBSEENMA11015088
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre (-3, -2), radius 7
    Solution

    Solution not provided.
    Ans. space space space straight x squared plus straight y squared plus 6 straight x plus 4 straight y minus 36 equals 0

    Question 84
    CBSEENMA11015089
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre (a, a), radius space square root of 2 straight a
    Solution

    Solution not provided.
    Ans. space space straight x squared plus straight y squared minus 2 ax minus 2 ay equals 0

    Question 85
    CBSEENMA11015090
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre open parentheses 1 half comma space 1 half close parentheses comma space radius space fraction numerator 1 over denominator square root of 2 end fraction
    Solution

    Solution not provided.
    Ans.  space space straight x squared plus straight y squared minus straight x minus straight y equals 0

    Question 86
    CBSEENMA11015091
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre (-a, -b), radius square root of straight a squared plus straight b squared end root

    Solution

    Solution not provided.
    Ans. straight x squared plus straight y squared plus 2 ax plus 2 by equals 0

    Question 87
    CBSEENMA11015092
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre at origin, radius 2 over 3 straight a 

    Solution

    Solution not provided.
    Ans. <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 88
    CBSEENMA11015093
    Wired Faculty App

    Find the equation of the circle whose centre and radii are:
    Centre <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

    Solution not provided.
    Ans. straight x squared plus straight y squared minus 2 ax space cosα space minus space 2 ay space sinα space plus space straight a squared minus straight b squared equals 0

    Question 94
    CBSEENMA11015099
    Wired Faculty App

    Find the equation of a circle whose centre is (2, - 3) and has an area of 38.5 sq. units.

    Solution

    Solution not provided.
    Ans. space space 4 straight x squared plus 4 straight y squared minus 16 straight x plus 24 straight y plus 3 equals 0

    Question 98
    CBSEENMA11015103
    Wired Faculty App

    Find the equation of circle passing through the origin and points (4, 0) and (0, 5).

    Solution

    Solution not provided.
    Ans.  space space straight x squared plus straight y squared minus 4 straight x minus 5 straight y equals 0

    Question 101
    CBSEENMA11015106
    Wired Faculty App

    Find the co-ordinates of the centre and radii of the following circle:
    <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

    Solution not provided.
    Ans.  (a, a) ; 2a

    Question 102
    CBSEENMA11015107
    Wired Faculty App

    Find the co-ordinates of the centre and radii of the following circle:
    straight x squared plus straight y squared minus 4 straight x minus 8 straight y minus 45 equals 0

    Solution

    Solution not provided.
    Ans. (2, 4) ;  square root of 65

    Question 103
    CBSEENMA11015108
    Wired Faculty App

    Find the co-ordinates of the centre and radii of the following circle:
    2 straight x squared plus 2 straight y squared minus straight y equals 0

    Solution

    Solution not provided.
    Ans. open parentheses 0 comma space 1 fourth close parentheses semicolon space 1 fourth

    Question 104
    CBSEENMA11015109
    Wired Faculty App

    Find the co-ordinates of the centre and radii of the following circle:
    2 straight x squared plus 2 straight y squared minus 4 straight x minus 8 straight y minus 17 equals 0

    Solution

    Solution not provided.
    Ans. left parenthesis 1 comma space 2 right parenthesis space semicolon space square root of 27 over 2 end root

    Question 105
    CBSEENMA11015110
    Wired Faculty App

    Find the co-ordinates of the centre and radii of the following circle:
    space space straight x squared plus straight y squared equals straight r squared

    Solution

    Solution not provided.
    Ans. (0, 0) ; r

    Question 106
    CBSEENMA11015111
    Wired Faculty App

    Does the indicated point lie inside, on, or outside the corresponding circles?
    (1,3), space space straight x squared plus straight y squared plus 2 straight x plus 6 straight y plus 1 equals 0

    Solution

    Solution not provided.
    Ans. outside, d>r.

    Question 107
    CBSEENMA11015112
    Wired Faculty App

    Does the indicated point lie inside, on, or outside the corresponding circles?
    (-3, 4), straight x squared plus straight y squared equals 25

    Solution

    Solution not provided.
    Ans. on, d=r.

    Question 108
    CBSEENMA11015113
    Wired Faculty App

    Does the indicated point lie inside, on, or outside the corresponding circles?
    (2, -1), space space space straight x squared plus straight y squared plus 4 straight x plus 2 straight y minus 20 equals 0

    Solution

    Solution not provided.
    Ans. inside, d<r

    Question 109
    CBSEENMA11015114
    Wired Faculty App

    In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.
    straight y squared equals 10 straight x

    Solution
    straight y squared equals 10 straight x
    It is the equation of parabola in the standard form straight y squared equals 4 ax comma space straight a greater than 0 and open towards the right.
    Comparing with straight y squared space equals space 4 ax comma we have 4a = 10 rightwards double arrow space space straight a equals 5 over 2
    ∴  The co-ordinates of the focus S are (a, 0) left right arrow open parentheses 5 over 2 comma space 0 close parentheses
         The equation of the axis of the parabola is y = 0 (x - axis).
         The equation of directrix is x + a = 0 or space space straight x plus 5 over 2 equals 0 space or space 2 straight x plus 5 equals 0
          The length of latus rectum = <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 110
    CBSEENMA11015115
    Wired Faculty App

    In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.
    straight y squared equals negative 8 straight x

    Solution
    space space straight y squared equals negative 8 straight x
    It is the equation of a parabola in the standard form straight y squared equals negative 4 ax comma space straight a greater than 0 and open towards the left.
    Comparing 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>
    The focus is straight S space left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 2 comma space 0 right parenthesis
    The equation of axis of parabola is y = 0 (x-axis).
    The equation of directrix is x - a = 0 or x - 2 = 0
    The length of latus rectum = 4a = 4 space cross times space 2 equals 8
    Question 111
    CBSEENMA11015116
    Wired Faculty App

    In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.
    space space space space straight x squared equals 6 straight y

    Solution
    space space space space straight x squared equals 6 straight y
    It is the equation of a parabola in the standard form straight x squared equals 4 ay comma space straight a greater than 0 and open upwards.
    Comparing it 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>
    ∴    (i) co-ordinates of the focus are S(0, a) left right arrow open parentheses 0 comma space 3 over 2 close parentheses
          (ii) equation of axis of parabola is x = 0 (y-axis)
          (iii) equation of tangent at the origin is y = 0 (x-axis)
          (iv) equation of directrix is y + a = 0 or straight y plus 3 over 2 equals 0 space or space 2 straight y plus 3 equals 0
           (v) length of the latus rectum = <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 112
    CBSEENMA11015117
    Wired Faculty App

    In the following, find the co-ordinates of the focus, axis of the parabola, the equation of the directrix and length of the latus rectum.
    straight x squared space equals negative 9 straight y

    Solution
    straight x squared space equals negative 9 straight y space or space straight x squared plus 9 straight y equals 0
    It is equation of parabola in the standard form space space space space straight x squared equals negative 4 ay comma space straight a greater than 0 and it open towards.
    Comparing the equation 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> we have negative 4 straight a space equals negative 9 space or space straight a equals 9 over 4
    ∴ (i) the focus is S(0, -a) left right arrow open parentheses 0 comma space fraction numerator negative 9 over denominator 4 end fraction close parentheses
       (ii) the equation of axis is x = 0 (y-axis).
       (iii) the equation of tangent at the origin is y = 0 (x-axis)
       (iv) the equation of directrix is y - a = 0 or straight y minus 9 over 4 space equals space 0 space or space 4 straight y minus 9 space equals 0
       (v) the length of the latus rectum = 4a = 4 cross times 9 over 4 equals 9
       
    Question 113
    CBSEENMA11015118
    Wired Faculty App

    Find: the co-ordinates of the focus and the vertex, the equations of axis, tangent at the vertex and directrix, the length of latus rectum of the parabola.
    2 straight y squared minus 9 straight x equals 0

    Solution
    2 straight y squared minus 9 straight x equals 0 space rightwards double arrow space straight y squared equals 9 over 2 straight x
    It is equation of a parabola in the standard form straight y squared equals 4 ax comma space straight a greater than 0 and open to the right.
    Comparing it 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> we have space space 4 straight a equals 9 over 2 space rightwards double arrow space straight a space equals space 9 over 8
    ∴       Co-ordinate of focus are S (a, 0) left right arrow open parentheses 9 over 8 comma space 0 close parentheses
              Co-ordinates of vertex are (0, 0)
              Equation of axis of parabola is y = 0 (x-axis).
              Equation of tangent at the origin is x = 0 (y-axis).
              Equation of directrix is x + a = 0 or space space space straight x plus 9 over 8 equals 0 space or space 8 straight x plus 9 equals 0
          Latus rectum = 4a = 4 cross times 9 over 8 equals 9 over 2
    Question 114
    CBSEENMA11015119
    Wired Faculty App

    Find: the co-ordinates of the focus and the vertex, the equations of axis, tangent at the vertex and directrix, the length of latus rectum of the parabola.
    4 straight x squared plus 9 straight y equals 0

    Solution
    4 straight x squared plus 9 straight y equals 0 space space rightwards double arrow space straight x squared space equals space minus 9 over 4 straight y
    It is equation of a parabola in the standard form space straight x squared equals negative 4 ay comma space straight a greater than 0 and open towards
    Comparing with straight x squared equals negative 4 ay comma space we space have space minus 4 straight a space equals space minus 9 over 4 space rightwards double arrow space straight a equals 9 over 16
          Co-ordinates of focus are (0, -a) = open parentheses 0 comma space fraction numerator negative 9 over denominator 16 end fraction close parentheses
          Co-ordinates of vertex are (0, 0)
          Equation of axis is x = 0 (y-axis)
          Equation of tangent at origin is y = 0 (x-axis)
          Equation of directrix is y - a = 0 or <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>
          Length of latus -rectum = 4a = 4 cross times 9 over 16 equals 9 over 4
            
    Question 115
    CBSEENMA11015120
    Wired Faculty App

    Find the equation of the parabola whose focus is at point S (2, 5) and the directrix is line 3x + 4y + 1 = 0.

    Solution

    Let line l be the directrix with equation
    3x + 4y + 1 = 0 and S (2, 5) be the focus.
    Take a point straight P space left parenthesis straight alpha comma space straight beta right parenthesis on the parabola
    ∴ By definition, PS = 'PM   rightwards double arrow d = p
    rightwards double arrow             square root of left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared end root space equals space open vertical bar fraction numerator 3 straight alpha plus 4 straight beta plus 1 over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root end fraction close vertical bar
    rightwards double arrow               left parenthesis straight alpha minus 2 right parenthesis squared plus left parenthesis straight beta minus 5 right parenthesis squared space equals space fraction numerator left parenthesis 3 straight alpha plus 4 straight beta plus 1 right parenthesis squared over denominator 25 end fraction
    rightwards double arrow          25 straight alpha squared plus 25 straight beta squared minus 100 straight alpha minus 250 straight beta plus 725 equals 9 straight alpha squared plus 16 straight beta squared plus 24 αβ plus 6 straight alpha plus 8 straight beta plus 1
    rightwards double arrow              space space space space space space 16 straight alpha squared plus 9 straight beta squared minus 24 αβ minus 106 straight alpha minus 258 straight beta plus 724 equals 0
    Hence, the locus of P or the equation of parabola is :
    16 straight x squared plus 9 straight y squared minus 24 xy minus 106 straight x minus 258 straight y plus 724 equals 0
                                              

    Question 116
    CBSEENMA11015121
    Wired Faculty App

    Find the equation of the parabola whose vertex is at (0, 0) and the focus is at point S (5, 0).

    Solution
    The vertex of the parabola is at O (0, 0) and focus is at S (5, 0).
    rightwards double arrow                y = 0 for both points
    rightwards double arrow                The axis of the parabola is along y = 0 or x-axis.
    ∴                   The parabola is of the standard form straight y squared equals 4 ax                       ...(i)
    Now, the focus is at (a, 0) left right arrow (5, 0) rightwards double arrow a = 5
    Hence, the equation of the parabola 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 117
    CBSEENMA11015122
    Wired Faculty App

    Find the equation of the parabola whose vertex is at (0, 0) and the focus is at point (-3, 0).

    Solution

    The vertex is at (0, 0) and the focus is at (-3, 0). For both the points y = 0
    ∴  Axis of the parabola is y = 0 or the x-axis.
    rightwards double arrow   Equation of the parabola in the standard form is straight y squared equals 4 ax
    Also,      focus straight S space left parenthesis straight a comma space 0 right parenthesis space left right arrow space straight S space left parenthesis negative 3 comma space 0 right parenthesis space rightwards double arrow space straight a space equals space minus 3
    Hence, the equation of the parabola is straight y squared equals 4 left parenthesis negative 3 right parenthesis straight x space or space straight y squared equals negative 12 straight x

    Question 118
    CBSEENMA11015123
    Wired Faculty App

    Find the equation of the parabola whose vertex is at (0, 0) and focus is at (0, – 2).

    Solution

    The vertex of the parabola is at O(0, 0) and the focus at S (0, -2). For both the points, x = 0.
    rightwards double arrow     The x-axis of the parabola is along x = 0 or the y-axis.
    rightwards double arrow     The equation of the parabola in the standard form 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>    ...(i)
    Also, the focus straight S space left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space minus 2 right parenthesis space rightwards double arrow space straight a space equals space minus 2
    Hence, the equation of the parabola is: straight x squared equals 4 left parenthesis negative 2 right parenthesis space straight y space or space straight x squared equals negative 8 straight y

    Question 119
    CBSEENMA11015124
    Wired Faculty App

    Find the equation of the parabola that satisfying the following condition:
    Vertex (0,0) passing through (2,3) and axis is along x-axis.

    Solution

    The vertex of the parabola is O (0, 0) and the axis is along x-axis (y = 0).
    rightwards double arrow      The equation of the parabola is straight y squared equals 4 ax                                                               ...(i)
    Since it passes through the point (2, 3)
    ∴     3 squared space equals space 4 straight a left parenthesis 2 right parenthesis space rightwards double arrow space 9 space equals space 8 straight a space space space space rightwards double arrow space space straight a equals 9 over 8
    Hence, from (i), the equation of the parabola 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>

    Sponsor Area

    Question 120
    CBSEENMA11015125
    Wired Faculty App

    Find the equation of the parabola that satisfying the following condition:
    Vertex at (0,0), symmetrical about y-axis and passing through the point (2, –3)

    Solution

    The vertex of the parabola is at (0, 0) and it is symmetrical about the y-axis.
    ∴    The axis of the parabola is y-axis (x = 0)
    rightwards double arrow  The equation of the parabola 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>                                                       ...(i)
    Since it passes through the points (2, -3)
    ∴    2 squared space equals space 4 straight a left parenthesis negative 3 right parenthesis space rightwards double arrow space 4 space equals space minus 12 straight a space rightwards double arrow space straight a equals negative 1 third
    Hence, from (i), the equation of the parabola is: straight x squared space equals space 4 open parentheses negative 1 third close parentheses straight y space or space 3 straight x squared space equals space minus 4 straight y

    Question 121
    CBSEENMA11015126
    Wired Faculty App

    Find the equation of the parabola that satisfying the following condition:
    Vertex at (0,0), focus on the positive x-axis and length of latus rectum 16 over 9.

    Solution

    The vertex is (0, 0) and the focus on positive x-axis is S (a, 0), a>0
    ∴  Equation of parabola is straight y squared equals 4 ax                                                      ...(i)
    Latus rectum = 4a = 16 over 9                                                         (∵ a > 0)
    Hence, from (i), the equation of parabola is: straight y squared equals 16 over 9 straight x space or space 9 straight y squared equals 16 straight x

    Question 122
    CBSEENMA11015127
    Wired Faculty App

    Find the equation of the parabola that satisfying the following condition:
    Vertex at (0, 0) focus on the negative side of y-axis and latus rectum equal to it.

    Solution

    The vertex is at (0, 0) and the focus is on negative y-axis is S(0, -a), a>0
    ∴ The equation of the parabola is space space space space straight x squared equals negative 4 ay comma space straight a greater than 0                    ...(i)
                         Latus rectum = 4a = 11
    Hence, from (i), the equation of parabola is: straight x squared space equals negative 11 straight y

    Question 123
    CBSEENMA11015128
    Wired Faculty App

    Find the equation of a parabola that satisfies the given condition:
    Focus (6, 0), directrix is x = – 6

    Solution

    The focus is S(6, 0) which lies on x-axis.
    The directrix is x + 6 = 0 which is a line parallel to y-axis i.e., perpendicular to x-axis.
    ∴   The parabola is of the standard form straight y squared equals 4 ax.
    Also, focus (a, 0) left right arrow (6, 0) directrix x + a = 0 is x + 6 = 0 rightwards double arrow a = 6
    Hence, the equation of the parabola is straight y squared equals 4 space left parenthesis 6 right parenthesis space straight x space or space straight y squared equals 24 straight x
    Alternative method:
    Let line l be the directrix with equation x + 6 = 0
    The focus is S (6, 0). Take a point straight P space left parenthesis straight alpha comma space straight beta right parenthesis on the parabola.
    From P, draw PM perpendicular on directrix l and join PS. By definition of parabola, PS = PM
    rightwards double arrow                                      d = p
    rightwards double arrow      square root of left parenthesis straight alpha minus 6 right parenthesis squared plus left parenthesis straight beta minus 0 right parenthesis squared end root space equals space open vertical bar fraction numerator straight alpha plus 6 over denominator square root of left parenthesis 1 right parenthesis squared plus 0 squared end root end fraction close vertical bar
                                               equals square root of left parenthesis straight alpha minus 6 right parenthesis squared plus straight beta squared end root space equals space open vertical bar straight a plus 6 close vertical bar
    Squaring both sides, we get
    space space space space space space straight alpha squared minus 12 straight alpha plus 36 plus straight beta squared space equals space straight alpha squared plus 12 straight alpha plus 36 space rightwards double arrow space straight beta squared minus 24 straight alpha space equals space 0
    Hence, the locus of P i.e., the equation of parabola 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 124
    CBSEENMA11015129
    Wired Faculty App

    Find the equation of a parabola that satisfies the given condition:
    Focus (0 – 3); directrix y = 3

    Solution

    The focus of parabola is (0, -3) which lies on y-axis. Directrix of the parabola is y - 3 = 0 which is parallel to x-axis.
    ∴ The equation of the parabola is of the standard form straight x squared equals negative 4 ay                 ...(i)
    Focus is (0, -a) left right arrow (0, -3) and directrix y - a = 0 is y - 3 = 0   rightwards double arrow  a = 3
    Hence, form (i), the equation of the parabola 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>
    Alternative method:
    Let l be the directrix with equation y - 3 = 0.
    S (0, -3) is the focus.
    Take a point straight P space left parenthesis straight alpha comma space straight beta right parenthesis on the parabola. From P, draw PM perpendicular on the directrix l and join PS. By definition of parabola, PS = PM
    rightwards double arrow           square root of left parenthesis straight alpha minus 0 right parenthesis squared plus left parenthesis straight beta plus 3 right parenthesis squared end root space equals space open vertical bar fraction numerator straight beta minus 3 over denominator 0 squared plus left parenthesis 1 right parenthesis squared end fraction close vertical bar
    rightwards double arrow            square root of straight alpha squared plus straight beta squared plus 6 straight beta plus 9 end root space equals space open vertical bar straight beta minus 3 close vertical bar
    rightwards double arrow            straight alpha squared plus straight beta squared plus 6 straight beta plus 9 space equals space straight beta squared plus 9 minus 6 straight beta space rightwards double arrow space straight alpha squared plus 12 straight beta space equals space 0
    Hence, the equation of locus of P i.e, equation of parabola is straight x squared plus 12 straight y equals 0
                                                  

    Question 125
    CBSEENMA11015130
    Wired Faculty App

    Find the co-ordinates of a point on the parabola y2 = l8x, where the ordinate is 3 times the abscissa.

    Solution

    Let the point be straight P space left parenthesis straight alpha comma space straight beta right parenthesis
    ∴     straight beta equals 3 straight alpha                                                                          [∵   Ordinate is 3 times the abscissa]
    space space space space space straight P left parenthesis straight alpha comma space straight beta right parenthesis lies on the parabola
    rightwards double arrow               straight beta squared equals 18 straight alpha space space rightwards double arrow space space left parenthesis 3 straight alpha right parenthesis squared space equals space 18 straight alpha space space space rightwards double arrow space space straight a squared space equals space 2 straight alpha
    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>
    When space space space straight alpha equals 0 comma space straight beta equals 0 space and space when space straight alpha equals 2 comma space straight beta equals 6
    Hence, the coordinates of the point are (2, 6).

    Question 126
    CBSEENMA11015131
    Wired Faculty App

    Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of the latus rectum. 

    Solution

    The equation of the parabola is straight x squared equals 12 straight y.
    rightwards double arrow                    4a = 12   or   a = 3
    Focus S is (0, a) left right arrow (0, 3)
    rightwards double arrow                 OS = 3
                          LL' = length of the latus rectum
                               = 4a = 12
    Hence, area of ΔOLL apostrophe space equals space 1 half left parenthesis LL apostrophe right parenthesis space cross times space OS
                                     equals 1 half cross times 12 cross times 3 space equals space 18 space square space units.

    Question 127
    CBSEENMA11015132
    Wired Faculty App

    An equilateral triangle is inscribed in the parabola straight y squared equals 4 ax., where one vertex is at the vertex of the parabola. Find (a)  the length of the side of the triangle, (b) area of triangle ABC.
    Solution

    Let OAB be the triangle with side p.
    In straight capital delta space OLA comma space OA space equals space straight p
    Now,               OL over straight p space equals space cos space 30 degree space equals space fraction numerator square root of 3 over denominator 2 end fraction space rightwards double arrow space OL space equals space fraction numerator square root of 3 over denominator 2 end fraction straight p
    and   AL over straight p space equals space sin space 30 degree space equals space 1 half space rightwards double arrow space AL space equals space 1 half straight p
    ∴ Point A is open parentheses fraction numerator square root of 3 over denominator 2 end fraction straight p comma space 1 half straight p close parentheses
    Since it lies on straight y squared equals 4 ax
    ∴    open parentheses 1 half straight p close parentheses squared space equals space 4 straight a open parentheses fraction numerator square root of 3 over denominator 2 end fraction straight p close parentheses space rightwards double arrow space 1 fourth straight p squared space equals space 2 straight a square root of 3 straight p space rightwards double arrow space straight p equals 8 square root of 3 straight a
    Hence, (i) each side of space space space space space space ΔABC space equals straight p equals space 8 square root of 3 straight a,    (ii) area of ΔABC space equals space fraction numerator square root of 3 straight p squared over denominator 4 end fraction equals fraction numerator square root of 3 over denominator 4 end fraction left parenthesis 64 cross times 3 straight a squared right parenthesis space equals space 48 square root of 3 straight a squared
                                                

    Question 128
    CBSEENMA11015133
    Wired Faculty App

    If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus. 

    Solution

    A parabolic reflection with diameter PR = 20 cm and OQ = 5 cm is shown below:
    Here, vertex of the parabola is (0, 0)
    Let the focus be S (a, 0)
    Let the equation of the parabola be straight y squared equals 4 ax.
    Now,     PR = 20 cm rightwards double arrow PQ = 10 cm
    Also,                   OQ = 5 cm
    ∴   Point P is (5, 10)
    Since it lies on the parabola straight y squared equals 4 ax
    ∴   10 squared space equals 4 space left parenthesis straight a right parenthesis space 5 space rightwards double arrow space 100 space equals space 20 straight a space rightwards double arrow space straight a space equals space 5
    ∴    Focus is S (5, 0) which is same as point Q.

    Question 129
    CBSEENMA11015134
    Wired Faculty App

    LL' is the latus rectum of a parabola, y2 = 4ax, a > 0. Find the co-ordinates of points L and L'.

    Solution

    Equation of parabola is y2 = 4ax.
    rightwards double arrow   The focus is at S (a, 0).
    ∴     OS = a
    Let coordinates of point L be left parenthesis straight alpha comma space straight beta right parenthesis and of L' bespace space left parenthesis straight alpha comma space minus straight beta right parenthesis with straight beta greater than 0
    ∴            straight beta squared equals 4 straight a left parenthesis straight a right parenthesis space equals space 4 straight a squared space rightwards double arrow space straight beta space equals space 2 straight a
    ∴            Coordinates of L are (a, 2a) and of L' are (a, -2a).
                                        

    Question 130
    CBSEENMA11015135
    Wired Faculty App

    LL' is the latus rectum of a parabola, x2 = -8y. Find the co-ordinates of points L and L'.

    Solution

    Equation of parabola is straight x squared equals negative 8 straight y
    rightwards double arrow       4a = 8 rightwards double arrow  a = 2  rightwards double arrow Focus is at point (0, -2).
    rightwards double arrow        OS = -2
    Let the co-ordinates of L be left parenthesis straight alpha comma space minus 2 right parenthesis and of L' be left parenthesis negative straight alpha comma space minus 2 right parenthesis space with space straight alpha greater than 0
    Since L left parenthesis straight alpha comma space minus 2 right parenthesis lies on straight x squared equals negative 8 straight y
    rightwards double arrow          straight alpha squared space equals negative 8 space left parenthesis negative 2 right parenthesis space equals space 16 space rightwards double arrow space straight alpha space equals space 4
    ∴       Coordinates of L are (4, -2) and of L' are (-4, -2)
                                 

    Question 131
    CBSEENMA11015136
    Wired Faculty App

    An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2m from the vertex of the parabola? 

    Solution

    Let AB be the parabolic arch having O at the vertex and the vertical line OY as the axis.
    The parabola open upwards
    ∴   Its equation is of the form <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)
         Width of the arch, LM = 5 m
           rightwards double arrow                    OM = 2.5 m
          Height of the arch, BM = 10 m
    ∴ Co-ordinates of point B are (2.5, 10)
    Since point B lies on the parabola <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>
    ∴     left parenthesis 2.5 right parenthesis squared space equals space 4 straight a left parenthesis 10 right parenthesis space rightwards double arrow space 25 over 10 cross times 25 over 10 space equals space 40 straight a rightwards double arrow space straight a space equals space 25 over 160 space equals space 5 over 32
    ∴     From (i), the equation of the parabola 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>
    or                       straight x squared equals 5 over 8 straight y                                                           ...(ii)
    We have to find the width PQ of the arch at a distance ON = 2 m from the vertex.
    Let                       PQ = d rightwards double arrow NQ = <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>
    ∴    Co-ordinates of point Q are open parentheses straight d over 2 comma space 2 close parentheses
    Putting it in (ii), we get straight d squared over 4 space equals space 5 over 8 left parenthesis 2 right parenthesis space rightwards double arrow space straight d squared space equals space 5
    Hence, the width of the arch = d = square root of 5 space straight m or 2.23 (approx).
                                                        

    Question 138
    CBSEENMA11015143
    Wired Faculty App

    Find the area of the triangle formed by joining the vertex of parabola y2 = 8x and the two ends of the latus rectum.

    Solution

    Solution not provided.
    Ans. 8 square units.

    Question 139
    CBSEENMA11015144
    Wired Faculty App

    Find the area of the triangle formed by joining the vertex of the parabola x2 = 64y to the two ends of its latus rectum.

    Solution

    Solution not provided.
    Ans. 512 square units.

    Question 141
    CBSEENMA11015146
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    <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

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - (-3, 0)
           Axis - (y = 0)
           Tangent at vertex - (x = 0)
           directrix - (x - 3 = 0)
           latus rectum = 12

    Question 142
    CBSEENMA11015147
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    space space straight x squared space equals space 36 straight y
    Solution

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - (0, 9)
           Axis - (x = 0)
           Tangent at vertex - (y = 0)
           directrix - (y + 9 = 0)
           latus rectum - (36)

    Question 143
    CBSEENMA11015148
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    straight x squared plus 9 straight y equals 0
    Solution

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus -space space open parentheses 0 comma space minus 9 over 4 close parentheses
           Axis - (x = 0)
           Tangent at vertex - (y = 0)
           directrix - straight y minus 9 over 4 equals 0 space or space 4 straight y minus 9 equals 0
           latus rectum - (9)

    Question 144
    CBSEENMA11015149
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    9 straight y squared plus 25 straight x equals 0
    Solution

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - space space open parentheses negative 25 over 36 comma space 0 close parentheses
           Axis - (y = 0)
           Tangent at vertex - (x = 0)
           directrix - space space space space straight x minus 25 over 36 equals 0 space or space 36 straight x minus 25 equals 0
           latus rectum - space space space 25 over 9

    Question 145
    CBSEENMA11015150
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    straight x squared equals negative 9 ay comma space straight a greater than 0
    Solution

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - <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>
           Axis - (x = 0)
           Tangent at vertex - (y = 0)
           directrix - straight y minus 9 over 4 straight a equals 0 space or space 4 straight y minus 9 straight a equals 0
           latus rectum - (9a)

    Question 146
    CBSEENMA11015151
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    <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

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - (2, 0)
           Axis - (y = 0)
           Tangent at vertex - (x = 0)
           directrix - (x + 2 = 0)
           latus rectum - (8)

    Question 147
    CBSEENMA11015152
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    <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

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - open parentheses 0 comma space 25 over 16 close parentheses
           Axis - (x = 0)
           Tangent at vertex - (y = 0)
           directrix - <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>
           latus rectum - 25 over 4

    Question 148
    CBSEENMA11015153
    Wired Faculty App

    Find the coordinates of vertex, focus, equations of axis, tangent at vertex, directrix, latus rectum of the following equation:
    straight y squared plus 4 straight x equals 0
    Solution

    Solution not provided.
    Ans. Vertex - (0, 0)
           Focus - (-1, 0)
           Axis - (y = 0)
           Tangent at vertex - (x = 0)
           directrix - (x - 1 = 0)
           latus rectum - (4)

    Question 149
    CBSEENMA11015154
    Wired Faculty App

    Find equation of parabola for Vertex at origin, focus (6, 0), directrix x = – 6

    Solution

    Solution not provided.
    Ans. straight y squared equals 24 straight x

    Question 150
    CBSEENMA11015155
    Wired Faculty App

    Find equation of parabola for vertex at origin, focus (0, 4), directrix y = – 4

    Solution

    Solution not provided.
    Ans. straight x squared equals 16 straight y

    Question 151
    CBSEENMA11015156
    Wired Faculty App

    Find equation of parabola for vertex at origin, focus (0, –2), directrix y = 2.

    Solution

    Solution not provided.
    Ans.  straight x squared equals negative 8 straight y

    Question 152
    CBSEENMA11015157
    Wired Faculty App

    Find equation of parabola for vertex at origin, focus (–3, 0), directrix x = 3.

    Solution

    Solution not provided.
    Ans.  <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 155
    CBSEENMA11015160
    Wired Faculty App

    The cable of a uniformly loaded suspension bridge hang in the form of a parabola.
    The roadway which is horizontal and 100 m long, is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to a roadway 18 m from the middle.
    Solution

    Let AOB be the cable of a uniformly loaded suspension bridge as shown in the figure.
    Let AL and BM be the longest wires of length 30 m each. Let OC be the shortest wire of length 6 m and LM be the road way as shown in the figure.
                         
    The longest supporting wire = AL = BM = 30 m
    The shortest supporting cable = OC = 6 m
              Length of the roadway  = LM = 100 m
    ∴                           LC = CM = 50 m
    Let O be the vertex and OY be the axis of the parabola
    ∴    Equation of the parabola 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>                                 ...(i)
                                ON = CM = 50
                                BN = BM - MN = BM - OC = 30 - 6 = 24
    ∴   Co-ordinates of point B are (50, 24)
    Since it lies on the parabola (i)
    ∴                               left parenthesis 50 right parenthesis squared equals 4 left parenthesis straight a right parenthesis space left parenthesis 24 right parenthesis space or space straight a equals 2500 over 96 equals 625 over 24
    ∴   From (i), the equation of the parabola is: straight x squared equals 4 open parentheses 625 over 24 close parentheses straight y
    or                            straight x squared equals 625 over 6 straight y                                          ...(ii)
    Let the length of the supporting wire PQ at a distance of 18 m be h.
    ∴                             OR = CQ = 18 m
                                    PR = PQ - RQ = PQ - OC = h - 6
    rightwards double arrow    Co-ordinates of point P are (18, h - 6)
    Since it lies on parabola (ii)
    ∴                        space space open parentheses 18 close parentheses squared space equals space 625 over 6 left parenthesis straight h minus 6 right parenthesis
    rightwards double arrow                       space space space space 324 cross times 6 space equals space 625 space straight h space minus space 3750
    rightwards double arrow                       1944 + 3750 = 625 h
    rightwards double arrow                                         straight h equals 5694 over 625 equals 9.11 space straight m     

    Question 156
    CBSEENMA11015161
    Wired Faculty App

    Verify that the locus of a point P which moves so that the sum of its distance from points S1 (– 4,0) and S2 (4,0) is 10, is an ellipse.

    Solution

    Let point P be left parenthesis straight alpha comma space straight beta right parenthesis.
    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>
    Givent 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>
    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>
    Squaring both sides, we get
                  straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 space equals space 100 plus left parenthesis straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 right parenthesis minus 20 square root of straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 end root
    rightwards double arrow                     space space space space space space space 16 straight alpha minus 100 space equals space minus 20 square root of straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 end root space rightwards double arrow space 4 straight alpha minus 25 space equals space minus 5 square root of straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 end root
    Squaring again, we get <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                <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>
    Hence, the locus of point P is straight x squared over 25 plus straight y squared over 9 equals 1
    which is the equation of an ellipse in the standard form space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1

    Question 157
    CBSEENMA11015162
    Wired Faculty App

    In the following, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
    straight x squared over 36 plus straight y squared over 16 equals 1

    Solution

    Equation of ellipse is straight x squared over 36 plus straight y squared over 16 equals 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>
    ∴ Equation of ellipse is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1
    Now,                         space space space space straight a squared space equals space 36 space rightwards double arrow space space space straight a space equals space 6 space space and space space straight b squared equals 16 space rightwards double arrow space straight b space equals space 4
    Co-ordinates of foci
    Focus,          straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of straight a squared minus straight b squared end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 36 minus 16 end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 20 comma space 0 right parenthesis space left right arrow space left parenthesis negative 2 square root of 5 comma space 0 right parenthesis
    Focus,            straight S subscript 2 space left parenthesis straight c comma space 0 right parenthesis left right arrow left parenthesis 2 square root of 5 comma space 0 right parenthesis
    Co-ordinates of vertices
                           straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 6 comma space 0 right parenthesis
                         straight A apostrophe space left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 6 comma space 0 right parenthesis
    Length of major axis = 2a = 2 (6) = 12
    Length of minor axis = 2b = 2 x 4 = 8
    Let the eccentricity be e. Now, c = ae rightwards double arrow 2 square root of 5 space equals space 6 straight e space rightwards double arrow space straight e space equals space fraction numerator square root of 5 over denominator 3 end fraction
    ∴                             straight e equals fraction numerator square root of 5 over denominator 3 end fraction
    Length of latus rectum, l equals fraction numerator 2 straight b squared over denominator straight a end fraction space rightwards double arrow space l equals fraction numerator 2 cross times 16 over denominator 6 end fraction equals 16 over 3

    Question 158
    CBSEENMA11015163
    Wired Faculty App

    In the following, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
    <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

    Equation of ellipse 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 space space 100 greater than 25 space rightwards double arrow space space space straight a squared equals 100 comma space straight b squared equals 25
    ∴ Equation of ellipse in the standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1
    Co-ordinates of foci
    Focus,    straight S subscript 1 left right arrow left parenthesis 0 comma space minus straight c right parenthesis space left right arrow space left parenthesis 0 comma space minus square root of straight a squared minus straight b squared end root right parenthesis space left right arrow space left parenthesis 0 comma space minus square root of 100 minus 25 end root right parenthesis space left right arrow space left parenthesis 0 comma space minus square root of 75 right parenthesis space left right arrow space left parenthesis 0 comma space minus 5 square root of 3 right parenthesis
    Focus,    <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>
    Co-ordinates of vertices
    Vertex, straight A space left right arrow space left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space 10 right parenthesis
    Vertex,                              <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>
    Length of major axis = 2a = 2 x 10 = 20
    Length of minor axis = 2b = 2 x 5 = 10
    Let e be the eccentricity. Now, c = ae rightwards double arrow 5 square root of 3 space equals space 10 straight e space rightwards double arrow space straight e space equals space fraction numerator 5 square root of 3 over denominator 10 end fraction
    ∴                         straight e equals fraction numerator square root of 3 over denominator 2 end fraction
    Latus rectum,  l equals fraction numerator 2 straight b squared over denominator straight a end fraction equals fraction numerator 2 cross times 25 over denominator 10 end fraction equals 5

    Question 159
    CBSEENMA11015164
    Wired Faculty App

    In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.
    16 straight x squared plus straight y squared equals 16

    Solution

    The equation of ellipse is : 16 straight x squared plus straight y squared equals 16
    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,                           16 greater than 1 space rightwards double arrow space straight a squared equals 16 comma space straight b squared equals 1
    ∴      Equation of ellipse in standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1
                                            straight a squared equals 16 space space rightwards double arrow space straight a space equals space 4 space semicolon space straight b squared space equals space 1 space rightwards double arrow space straight b space equals space 1
    Focus,      <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>
    Focus,         <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>
    Vertex,        straight A left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space 4 right parenthesis
    Vertex,      <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>
          Length of major axis = 2a = 2(4) = 8
          Length of minor axis = 2b = 2(1) = 2
    Let the eccentricity be e.
    Now,                    c = ae rightwards double arrow square root of 15 space equals space 4 straight e  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>
       Length of latus rectum =  fraction numerator 2 straight b squared over denominator straight a end fraction equals fraction numerator 2 cross times 1 over denominator 4 end fraction equals 1 half

    Question 160
    CBSEENMA11015165
    Wired Faculty App

    In the following equation, Find the co-ordinates of foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of latus rectum of the ellipse.
    <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

    Equation of ellipse 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>
    rightwards double arrow               fraction numerator 4 straight x squared over denominator 36 end fraction plus fraction numerator 9 straight y squared over denominator 36 end fraction equals 1 space space space rightwards double arrow space space straight x squared over 9 plus straight y squared over 4 equals 1
    Now, 9 greater than 4 space rightwards double arrow space straight a squared equals 9 comma space straight b squared equals 4 space rightwards double arrow space straight a equals 3 comma space straight b equals 2
    ∴    Equation of ellipse in the standard form is space space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1
    (i) Focus,      straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative square root of straight a squared minus straight b squared end root comma space 0 close parentheses space left right arrow space left parenthesis negative square root of 9 minus 4 end root comma space 0 right parenthesis space left right arrow space left parenthesis negative square root of 5 comma space 0 right parenthesis
    Focus,               straight S subscript 2 space left parenthesis straight c comma space 0 right parenthesis space left right arrow space left parenthesis square root of 5 comma space 0 right parenthesis
    (ii) Vertex,           straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 3 comma space 0 right parenthesis
    Vertex,            straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 3 comma space 0 right parenthesis
    (iii) Length of major axis = 2a = 2(3) = 6
          Length of minor axis = 2b = 2(2) = 4
    (iv)  Let the eccentricity be e.
    Now,                           c = ae,  rightwards double arrow  square root of 5 space equals space 3 straight e   ∴   straight e equals fraction numerator square root of 5 over denominator 3 end fraction
    (v) Length of latus rectum equals space fraction numerator 2 straight b squared over denominator straight a end fraction space equals space fraction numerator 2 left parenthesis 4 right parenthesis over denominator 3 end fraction equals 8 over 3

    Question 161
    CBSEENMA11015166
    Wired Faculty App

    Find: (i) the co-ordinates of foci.
    (ii) the co-ordinates of vertices.
    (iii) the length of major axis and minor axis.
    (iv) the eccentricity.
    (v) latus rectum of ellipse.
                4 straight x squared plus 9 straight y squared equals 1
    Solution

    Equation of ellipse is : space space space 4 straight x squared plus 9 straight y squared equals 1 space space space rightwards double arrow space space fraction numerator straight x squared over denominator open parentheses begin display style 1 fourth end style close parentheses end fraction plus fraction numerator straight y squared over denominator open parentheses begin display style 1 over 9 end style close parentheses end fraction equals 1
    Now,              1 fourth greater than 1 over 9 space rightwards double arrow space space straight a squared equals 1 fourth comma space straight b squared equals 1 over 9
    The equation of the ellipse in standard form is space space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1
                                      straight a squared equals 1 fourth space rightwards double arrow space straight a equals 1 half space and space straight b squared equals 1 over 9 space space rightwards double arrow space straight b equals 1 third
    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>
    (i) Focus,                   straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative fraction numerator square root of 5 over denominator 6 end fraction comma 0 close parentheses
    Focus,                         straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space open parentheses fraction numerator square root of 5 over denominator 6 end fraction comma 0 close parentheses
    (ii) Vertex,               straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space open parentheses 1 half comma space 0 close parentheses
    Vertex,                  straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space open parentheses negative 1 half comma space 0 close parentheses
    (iii) Length of major axis = 2a = 2 open parentheses 1 half close parentheses space equals space 1
          Length of minor axis = 2b = 2 open parentheses 1 third close parentheses equals 2 over 3
    (iv) Let the eccentricity be e
    Now,                       c = ae rightwards double arrow  fraction numerator square root of 5 over denominator 6 end fraction equals 1 half straight e space rightwards double arrow space space space straight e equals fraction numerator square root of 5 over denominator 6 end fraction cross times 2 over 1 space rightwards double arrow space straight e equals fraction numerator square root of 5 over denominator 3 end fraction
    (v) Length of latus rectum = 2 straight b squared over straight a equals fraction numerator 2 cross times 1 cross times 2 over denominator 9 cross times 1 end fraction equals 4 over 9

    Question 162
    CBSEENMA11015167
    Wired Faculty App

    Find: (i) the co-ordinates of foci.
    (ii) the co-ordinates of vertices.
    (iii) the length of major axis and minor axis.
    (iv) the eccentricity.
    (v) latus rectum of ellipse.
    <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

    Equation of ellipse is:  16 straight x squared plus 9 straight y squared equals 1 space space rightwards double arrow space space fraction numerator straight x squared over denominator open parentheses begin display style 1 over 16 end style close parentheses end fraction plus fraction numerator straight y squared over denominator open parentheses begin display style 1 over 9 end style close parentheses end fraction equals 1
    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>
    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>
    Equation of the ellipse in standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1
    (i)   Focus,   straight S subscript 1 space left parenthesis 0 comma space minus straight c right parenthesis space rightwards arrow space open parentheses 0 comma space minus fraction numerator square root of 7 over denominator 12 end fraction close parentheses
          Focus,   <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>
    (ii)  Vertex,   straight A space left parenthesis 0 comma space straight a right parenthesis space left right arrow space open parentheses 0 comma space 1 third close parentheses
           space space Vertex comma space straight A apostrophe space left parenthesis 0 comma space minus straight a right parenthesis space left right arrow space open parentheses 0 comma space minus 1 third close parentheses
    (iii) Length of major axis = 2a = 2 open parentheses 1 third close parentheses equals 2 over 3
          Length of minor axis = 2b = <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) Let the eccentricity be e
    Now,                          straight c equals ae comma space rightwards double arrow space fraction numerator square root of 7 over denominator 12 end fraction equals 1 third straight e space rightwards double arrow space straight e equals fraction numerator square root of 7 over denominator 12 end fraction cross times 3 over 1 space rightwards double arrow space straight e equals fraction numerator square root of 7 over denominator 4 end fraction
    (v) Length of latus rectum = fraction numerator 2 straight b squared over denominator straight a end fraction equals 2 cross times 1 over 16 cross times 3 over 1 equals 3 over 8
          

    Question 163
    CBSEENMA11015168
    Wired Faculty App

    Find the equation of the ellipse that satisfies the following condition:
    Vertices (± 6, 0), foci (± 4, 0).

    Solution

    The foci (± 4, 0) lie on x-axis.
    rightwards double arrow  Equation of ellipse in standard form is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1                        ...(i)
    Foci:                       left parenthesis plus-or-minus straight c comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus 4 comma space 0 right parenthesis space rightwards double arrow space straight c space equals space 4
    Vertices:               space space left parenthesis plus-or-minus straight a comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus 6 comma space 0 right parenthesis space rightwards double arrow space straight a equals 6 space or space straight a squared equals 36
    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>
    ∴  From (i), equation of ellipse is straight x squared over 36 plus straight y squared over 20 equals 1

    Question 164
    CBSEENMA11015169
    Wired Faculty App

    Find the equation of the ellipse that satisfies the following condition:
    Vertices (0, ± 13), foci (0, ± 5).

    Solution

    The foci (0, ± 5) lie on y-axis.
    ∴ Equation of ellipse in standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1                                     ...(i)
    Foci:                     left parenthesis 0 comma space plus-or-minus 5 right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus straight c right parenthesis space space rightwards double arrow space space straight c space equals space 5
    Vertices:              space space left parenthesis 0 comma space plus-or-minus straight a right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus 13 right parenthesis space rightwards double arrow space straight a space equals space 13               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 c squared equals straight a squared minus straight b squared space space space space space space rightwards double arrow space space 25 space equals space 169 minus straight b squared space space space rightwards double arrow space space straight b squared equals 144
    Hence, from (i), the equation of the ellipse is straight y squared over 169 plus straight x squared over 144 equals 1

    Question 165
    CBSEENMA11015170
    Wired Faculty App

    Find the equation of an ellipse that satisfies the following conditions
    Ends of major axis at <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>, ends of minor axis at left parenthesis 0 comma space plus-or-minus 2 right parenthesis

    Solution

    Ends of major axis <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> lie on x-axis.
    ∴  Equation of ellipse in standard form is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1                                ...(i)
    Ends of major axis (vertices) are open parentheses plus-or-minus straight a comma space 0 close parentheses space left right arrow space left parenthesis plus-or-minus 5 comma space 0 right parenthesis space rightwards double arrow space straight a space equals space 5
    Ends of minor axis are left parenthesis 0 comma space plus-or-minus space straight b right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus space 2 right parenthesis space rightwards double arrow space straight b equals 2
    Also,                          straight c squared equals straight a squared minus straight b squared equals 5 squared minus 2 squared equals 21 space rightwards double arrow space space space straight c equals square root of 21
    ∴ from (i), equation of ellipse is straight x squared over 5 squared plus straight y squared over 2 squared equals 1 space space or space space straight x squared over 25 plus straight y squared over 4 equals 1                 ...(ii)
    The foci are <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 166
    CBSEENMA11015171
    Wired Faculty App

    Find the equation of an ellipse that satisfies the following conditions
    Ends of major axis at space left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis comma ends of minor axis at space space left parenthesis plus-or-minus 2 comma space 0 right parenthesis.  Also find the co-ordinate of foci.

    Solution

    Ends of major axis left parenthesis 0 comma space plus-or-minus square root of 13 right parenthesis lie on y-axis.
    ∴ Equation of ellipse in the standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1
    Ends of major axis are <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>
    Ends of minor axis are left parenthesis plus-or-minus straight b comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus 2 comma space 0 right parenthesis space rightwards double arrow space straight b space equals space 2
    Hence, from (i), the equation of ellipse is space space fraction numerator straight y squared over denominator left parenthesis square root of 13 right parenthesis squared end fraction plus straight x squared over 2 squared equals 1 space or space straight y squared over 13 plus straight x squared over 4 equals 1
    Also,                          straight c squared equals straight a squared minus straight b squared equals 13 minus 14 space rightwards double arrow space straight c squared equals 9 space space or space straight c equals 3
    ∴ The foci are left parenthesis 0 comma space plus-or-minus straight c right parenthesis space or space left parenthesis 0 comma space plus-or-minus 3 right parenthesis.

    Question 167
    CBSEENMA11015172
    Wired Faculty App

    Find the equation of an ellipse that satisfy the following condition:
    Length of major axis 26, foci (± 5, 0).

    Solution

    The foci (± 5, 0) are on x-axis.
    ∴  The equation of ellipse in standard form is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1                   ...(i)
    Foci are               left parenthesis plus-or-minus 5 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight c comma space 0 right parenthesis space rightwards double arrow space space straight c space equals space 5
       Length of major axis 2a = 26 rightwards double arrow   a = 13 or  <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>
    Also,                      straight c squared equals straight a squared minus straight b squared space rightwards double arrow space 5 squared equals 13 squared minus straight b squared space rightwards double arrow space 25 space equals space 169 minus straight b squared space rightwards double arrow space straight b squared equals 169 minus 25 space equals space 144
    ∴ From (i), equation of ellipse is : straight x squared over 169 plus straight y squared over 144 equals 1
    Let e be the eccentricity
    ∴                       c = ae  rightwards double arrow 5 = 13e rightwards double arrow  straight e equals 15 over 13

    Question 168
    CBSEENMA11015173
    Wired Faculty App

    Find the equation of an ellipse that satisfy the following condition:
    Length of minor axis 16, foci (0, ± 6). Find also, the eccentricity .

    Solution

    The foci (0, ± 6) lie on y-axis.
    ∴  The equation of ellipse in standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1                   ...(i)
    The Foci are        left parenthesis 0 comma space plus-or-minus 6 right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus straight c right parenthesis space rightwards double arrow space straight c space equals space 6       
       Length of minor axis 2b = 16 rightwards double arrow   b = 8 or  <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>
    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>
    ∴ From (i), equation of ellipse is : straight y squared over 100 plus straight x squared over 64 equals 1
    Let e be the eccentricity
    ∴                       c = ae  rightwards double arrow 6 = 10e 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 169
    CBSEENMA11015174
    Wired Faculty App

    Find the equation of an ellipse whose major axis is along the x-axis and which passes through the points (6, 3) and (12, 0).

    Solution

    The major axis of the ellipse is along the x-axis.
     ∴ The equation of ellipse is space straight x squared over straight a squared plus straight y squared over straight b squared equals 1                                                  ...(i)
    It passes through (6, 3) rightwards double arrow   6 squared over straight a squared plus 3 squared over straight b squared equals 1 space space rightwards double arrow space 36 over straight a squared plus 9 over straight b squared equals 1                    ...(ii)
    It passes through (12, 0) space space space rightwards double arrow space    12 squared over straight a squared plus 0 squared over straight b squared equals 1 space rightwards double arrow space 144 over straight a squared equals 1 space rightwards double arrow space space straight a squared equals 144
    ∴ From (ii), 36 over 144 plus 9 over straight b squared equals 1 space space rightwards double arrow space 1 fourth plus 9 over straight b squared equals 1 space space rightwards double arrow space 9 over straight b squared equals 3 over 4 space space space space rightwards double arrow space space straight b squared equals 12
    Hence, the equation of ellipse, from (i), 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 170
    CBSEENMA11015175
    Wired Faculty App

    Find the equation of an ellipse whose major axis is along the y-axis and which passes through the points P (2, 2) and Q (1, 4).

    Solution

    The major axis is along the y-axis.
    ∴ The equation of ellipse in the standard form is space space straight y squared over straight a squared plus straight x squared over straight b squared equals 1                  ...(i)
    It passes through (2, 2) rightwards double arrow  space space 4 over straight a squared plus 4 over straight b squared equals 1                                               ...(ii)
    It passes through (1, 4) rightwards double arrow     16 over straight a squared plus 1 over straight b squared equals 1                                             ...(iii)
    Multiplying both sides of (ii) by 4, we get
                                        <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)
    Subtracting (iv) from (iii), we get fraction numerator negative 15 over denominator straight b squared end fraction equals negative 3 space space rightwards double arrow space space straight b squared equals 5                          
    ∴ From (iii),   16 over straight a squared plus 1 fifth equals 1 space space rightwards double arrow space space 16 over straight a squared equals 1 minus 1 fifth equals 4 over 5 space rightwards double arrow space straight a squared equals 16 cross times 5 over 4 equals 20
    Hence, from (i), the equation of ellipse 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 171
    CBSEENMA11015176
    Wired Faculty App

    Find the equation of an ellipse for which foci are at (± 3, 0), a = 4 and centre is at origin. Also, find the eccentricity of the ellipse.

    Solution

    The foci left parenthesis plus-or-minus 3 comma space 0 right parenthesis lie on x-axis.
    ∴ The equation of ellipse in standard form is straight x squared over straight a squared plus straight b squared over straight b squared equals 1
    Foci left parenthesis plus-or-minus 3 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight c comma space 0 right parenthesis space rightwards double arrow space straight c space equals space 3
    Also,         a = 4 (given)
    Using,        straight c squared equals straight a squared minus straight b squared comma space we space get space 9 space equals space 16 minus straight b squared space rightwards double arrow space straight b squared equals 7
    Hence, from (i), the equation of ellipse is : straight x squared over 16 plus straight y squared over 7 equals 1
    Also,                       c = ae rightwards double arrow   3 = 4e rightwards double arrow  straight e equals 3 over 4

    Question 172
    CBSEENMA11015177
    Wired Faculty App

    Find the equation of an ellipse for which c = 4, b = 3, centre is at the origin and foci on x-axis. Determine the eccentricity of the ellipse.

    Solution

    Equation of ellipse in the standard form is space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1    [∵ foci lie on x-axis]
    Here,      c = 4, b = 3
    ∴             straight c squared equals straight a squared minus straight b squared space rightwards double arrow space 16 space equals space straight a squared minus 9 space rightwards double arrow space straight a squared equals 25
    Hence, from (i), equation of ellipse is : straight x squared over 25 plus straight y squared over 9 equals 1
    Let the eccentricity of the ellipse be e.
    ∴                        c = ae rightwards double arrow 4 = 5e rightwards double arrow   straight e equals 4 over 5

    Question 173
    CBSEENMA11015178
    Wired Faculty App

    Let P be any point on the ellipse space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 with the two foci at space straight S subscript 1 and straight S subscript 2. Verify 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>
    Solution

    Let point P be (x, y). Focus space space straight S subscript 1 is (-c, 0),  <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 (c, 0)
    rightwards double arrow                       PS subscript 1 equals square root of left parenthesis straight x plus straight c right parenthesis squared plus straight y squared end root comma space PS subscript 2 equals square root of left parenthesis straight x minus straight c right parenthesis squared plus straight y squared end root
    ∴        space space PS subscript 1 plus PS subscript 2 space equals space square root of left parenthesis straight x plus straight c right parenthesis squared plus straight y squared end root space plus space square root of left parenthesis straight x minus straight c right parenthesis squared plus straight y squared end root
    Now,          straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space space space rightwards double arrow space straight y squared over straight b squared equals 1 minus straight x squared over straight a squared equals fraction numerator straight a squared minus straight x squared over denominator straight a squared end fraction rightwards double arrow space straight y squared equals straight b squared open parentheses fraction numerator straight a squared minus straight x squared over denominator straight a squared end fraction close parentheses
    But,                          straight c squared equals straight a squared minus straight b squared space rightwards double arrow space straight b squared equals straight a squared minus straight c squared
    ∴                             straight y squared equals open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis
    ∴   PS subscript 1 plus PS subscript 2 space equals space square root of left parenthesis straight x plus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared right parenthesis end root plus square root of left parenthesis straight x minus straight c right parenthesis squared plus open parentheses fraction numerator straight a squared minus straight c squared over denominator straight a squared end fraction close parentheses left parenthesis straight a squared minus straight x squared 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>
       equals space 1 over straight a open square brackets square root of straight a to the power of 4 plus 2 straight a squared xc plus straight c squared straight x squared end root plus square root of straight a to the power of 4 minus 2 straight a squared xc plus straight c squared straight x squared end root close square brackets space equals space 1 over straight a open square brackets square root of left parenthesis straight a squared plus cx right parenthesis squared end root plus square root of left parenthesis straight a squared minus cx right parenthesis squared end root close square brackets
        equals space 1 over straight a left square bracket straight a squared plus cx plus straight a squared minus cx right square bracket space equals space 1 half left square bracket 2 straight a squared right square bracket space equals space 2 straight a
    Hence, PS subscript 1 plus PS subscript 2 space equals space 2 straight a, which verifies the ellipse property, space space PS subscript 1 plus PS subscript 2 equals 2 straight a

    Question 174
    CBSEENMA11015179
    Wired Faculty App

    For ellipse <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>  verify PS subscript 1 plus PS subscript 2 equals 10 comma where space straight S 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 foci of the ellipse and P is any point on it.
    Solution

    Let P(x, y) be any point on the ellipse straight x squared over 25 plus straight y squared over 9 equals 1                                          ...(i)
    rightwards double arrow                space space space space straight y squared over 9 space equals space 1 minus straight x squared over 25 space rightwards double arrow space straight y squared equals 9 open parentheses fraction numerator 25 minus straight x squared over denominator 25 end fraction close parentheses                                       ...(ii)
    Focus space space straight S subscript 1  is (-c, 0), Focus <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 (c, 0)
    From (i), straight a squared equals 25 comma space straight b squared equals 9
    Now,                         straight c squared equals straight a squared minus straight b squared space rightwards double arrow space straight c squared equals 25 minus 9 space equals space 16 space rightwards double arrow space straight c space equals space 4
    ∴       Focus <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>  Focus straight S subscript 2 space left right arrow space left parenthesis 4 comma space 0 right parenthesis
    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>
    ∴         <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 space square root of straight x squared plus straight y squared plus 8 straight x plus 16 end root space plus space square root of straight x squared plus straight y squared minus 8 straight x plus 16 end root
                             space space space space space space space equals space square root of straight x squared plus 8 straight x plus 16 plus fraction numerator 9 left parenthesis 25 minus straight x squared right parenthesis over denominator 25 end fraction end root plus square root of straight x squared minus 8 straight x plus 16 plus fraction numerator 9 left parenthesis 25 minus straight x squared right parenthesis over denominator 25 end fraction end root
                                 space space equals space square root of 25 plus 8 straight x plus fraction numerator 16 straight x squared over denominator 25 end fraction end root plus square root of 25 minus 8 straight x plus fraction numerator 16 straight x squared over denominator 25 end fraction end root
                                   equals square root of open parentheses 5 plus fraction numerator 4 straight x over denominator 5 end fraction close parentheses squared end root space plus space square root of open parentheses 5 minus fraction numerator 4 straight x over denominator 5 end fraction close parentheses squared end root space equals space 5 plus fraction numerator 4 straight x over denominator 5 end fraction plus 5 minus fraction numerator 4 straight x over denominator 5 end fraction equals 10
    Hence,  <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 175
    CBSEENMA11015180
    Wired Faculty App

    A rod of length 9 cm moves with its ends always touching the co-ordinate axes. Show that the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis, is an ellipse.

    Solution
    Let AB be the rod of length 9 cm with end A touching the y-axis and end B touching the x-axis.
                                            
    P is a point on the rod, such that BP = 3 cm
    ∴ AP = 9 - 3 = 6 cm.
    We have to find the locus of point P
    Let the co-ordinate of point P be left parenthesis straight alpha comma space straight beta right parenthesis
    From P, draw PL and PM perpendicular to x-axis and y-axis respectively.
    ∴                           MP equals OL equals straight alpha space and space PL equals OM equals straight beta
    Now, from similar triangles PLB and AOB,
                                   AO over PL equals AB over PB space rightwards double arrow space AO over straight beta equals 9 over 3 space space rightwards double arrow space AO space equals space 3 straight beta
    Also, from similar triangles AMP and AOB,
                                     OB over PM space equals space AB over AP space space rightwards double arrow space OB over straight alpha equals 9 over 6 space rightwards double arrow space OB equals 3 over 2 straight alpha
    Using Pythagoras theorem, we get
                               OA squared plus OB squared equals AB 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                straight alpha squared over 36 plus straight beta squared over 9 equals 1
    Hence, the equation of the locus of point space space straight P left parenthesis straight alpha comma space straight beta right parenthesis space is colon space straight x squared over 36 plus straight y squared over 9 equals 1
    which is the equation of the ellipse in standard form straight x squared over straight a squared plus straight y squared over straight b squared equals 1 comma space straight a squared greater than straight b squared
    Question 176
    CBSEENMA11015181
    Wired Faculty App

    A man running a racecourse notes that the sum of the distances from two flag posts is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

    Solution

    Let straight F subscript 1 space and space straight F subscript 2 be the two points where the flag posts are fixed on the ground.
    Take the line joining points <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> as the x-axis, the mid-point of line segment straight F subscript 1 straight F subscript 2 as the origin O and the line through O, perpendicular to straight F subscript 1 straight F subscript 2 as y-axis.
                                            <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                                     <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>
    ∴     Co-ordinates of points straight F subscript 1 space and space straight F subscript 2 are (-4, 0) and (4, 0) respectively.

    Take any point space space space straight P left parenthesis straight alpha comma space straight beta right parenthesis on the track.
    ∴                                        <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                 square root of left parenthesis straight alpha plus 4 right parenthesis squared plus left parenthesis straight beta minus 0 right parenthesis squared end root plus square root of left parenthesis straight alpha minus 4 right parenthesis squared plus left parenthesis straight beta minus 0 right parenthesis squared end root equals 10
    rightwards double arrow                         square root of left parenthesis straight alpha plus 4 right parenthesis squared plus straight beta squared end root space equals space 10 minus square root of left parenthesis straight alpha minus 4 right parenthesis squared plus straight beta squared end root
    Squaring both sides, we get
                          straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 space equals space 100 plus straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 minus 20 square root of straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 end root
    rightwards double arrow          negative 16 straight alpha plus 100 space equals space 20 square root of straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 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>
    Squaring both sides, we get
                       16 straight alpha squared plus 625 minus 200 straight alpha space equals space 25 straight alpha squared plus 25 straight beta squared minus 200 straight alpha plus 400
    rightwards double arrow                          9 straight alpha squared plus 25 straight beta squared space equals space 225
    rightwards double arrow                           straight alpha squared over 25 plus straight beta squared over 9 equals 1
    Hence, the equation of  the locus of straight P left parenthesis straight alpha comma space straight beta right parenthesis space is : straight x squared over 25 plus straight y squared over 9 equals 1
    which is the equation of the ellipse in standard form space space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 comma space left parenthesis straight a squared greater than straight b squared right parenthesis

    Question 177
    CBSEENMA11015182
    Wired Faculty App

    The arch given below is in form of a semi-ellipse. It is 8 m wide and its height at the centre is 2 m. Find the height of the arch at a distance of 1.5 m from one end.
                                          

    Solution

    Width of the elliptical arch = 8 m   rightwards double arrow  AA' = 8m rightwards double arrow 2a = 8m rightwards double arrow a = 4
    Height at the centre = 2m  rightwards double arrow      OB = 2m rightwards double arrow   b = 2
    ∴ the equation of ellipse is straight x squared over 16 plus straight y squared over 4 equals 1                                     ...(i)
                                       AP space equals space 1.5 space straight m space rightwards double arrow space OP space equals space OA minus AP space equals space 4 minus 1.5 equals 2.5 space straight m
    Let the height at P be h.
    ∴   Coordinates of L are (2.5, h)
    It lies on ellipse (i)     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 h squared equals fraction numerator 16 minus 6.25 over denominator 4 end fraction equals fraction numerator 9.75 over denominator 4 end fraction equals 2.44 space left parenthesis Approx right parenthesis
    Hence, straight h equals square root of 2.44 end root space equals space 1.56 space straight m space left parenthesis Approx. right parenthesis

    Question 179
    CBSEENMA11015184
    Wired Faculty App

    In the following, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
    straight x squared over 9 plus straight y squared over 16 equals 1
    Solution

    Solution not provided.
    Ans. (i) <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>    (ii)  space left parenthesis 0 comma space plus-or-minus 4 right parenthesis     (iii) 8, 6      (iv) <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>      (v) 9 over 2

    Question 181
    CBSEENMA11015186
    Wired Faculty App

    In the following, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
    9 straight x squared plus straight y squared equals 9
    Solution

    Solution not provided.
    Ans. (i) left parenthesis 0 comma space plus-or-minus 2 square root of 2 right parenthesis      (ii) <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>     (iii) 6, 2     (iv) fraction numerator 2 square root of 2 over denominator 3 end fraction   (v) 2 over 3

    Question 183
    CBSEENMA11015188
    Wired Faculty App

    In the following, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
    space space space space 9 straight x squared plus 16 straight y squared equals 144
    Solution

    Solution not provided.
    Ans. (i) left parenthesis plus-or-minus square root of 7 comma space 0 right parenthesis       (ii) left parenthesis plus-or-minus 4 comma space 0 right parenthesis        (iii) 8, 6         (iv) <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>          (v) 9 over 2

    Question 184
    CBSEENMA11015189
    Wired Faculty App

    Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the lotus rectum of the ellipse.
    <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

    Solution not provided.
    Ans. (i) left parenthesis 0 space plus-or-minus space 10 square root of 3 right parenthesis  (ii) <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> (iii) 40, 20   (iv) fraction numerator square root of 3 over denominator 2 end fraction   (v) 10

    Tips: -

     
    Question 185
    CBSEENMA11015190
    Wired Faculty App

    Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the lotus rectum of the ellipse.
    space space straight x squared over 49 plus straight y squared over 36 equals 1

    Solution

    Solution not provided.
    Ans. (i) left parenthesis plus-or-minus square root of 13 comma space 0 right parenthesis  (ii) left parenthesis plus-or-minus 7 comma space 0 right parenthesis (iii) 14, 12  (iv) fraction numerator square root of 13 over denominator 7 end fraction  (v) <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 186
    CBSEENMA11015191
    Wired Faculty App

    Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the lotus rectum of the ellipse.
    space space straight x squared plus 4 straight y squared equals 100

    Solution

    Solution not provided.
    Ans. (i) left parenthesis plus-or-minus 5 square root of 3 comma space 0 right parenthesis    (ii) <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>   (iii) 20, 10   (iv) fraction numerator square root of 3 over denominator 2 end fraction   (v) 5

    Question 187
    CBSEENMA11015192
    Wired Faculty App

    Find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the lotus rectum of the ellipse.
    space space straight x squared over 25 plus straight y squared over 169 equals 1

    Solution

    Solution not provided.
    Ans. (i) left parenthesis 0 space plus-or-minus space 12 right parenthesis   (ii) left parenthesis 0 space plus-or-minus space 13 right parenthesis    (iii) 26, 10  (iv) <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>    (v) 50 over 13

    Question 188
    CBSEENMA11015193
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Vertices <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>, foci left parenthesis plus-or-minus 4 comma space 0 right parenthesis
    Solution

    Solution not provided.
    Ans.  straight x squared over 25 plus straight y squared over 9 equals 1

    Question 189
    CBSEENMA11015194
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Vertices <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>, foci left parenthesis 0 comma space plus-or-minus 3 right parenthesis
    Solution

    Solution not provided.
    Ans.  straight y squared over 49 plus straight x squared over 40 equals 1

    Question 192
    CBSEENMA11015197
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Ends of major axis 18, foci left parenthesis plus-or-minus 3 comma space 0 right parenthesis
    Solution

    Solution not provided.
    Ans.  straight x squared over 81 plus straight y squared over 72 equals 1

    Question 193
    CBSEENMA11015198
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Length of minor axis 8, foci left parenthesis 0 comma space plus-or-minus 3 right parenthesis
    Solution

    Solution not provided.
    Ans.  straight y squared over 25 plus straight x squared over 16 equals 1

    Question 194
    CBSEENMA11015199
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Foci, left parenthesis 0 comma space plus-or-minus 3 right parenthesis comma space straight a space equals space 5
    Solution

    Solution not provided.
    Ans.  straight y squared over 25 plus straight x squared over 16 equals 1

    Question 195
    CBSEENMA11015200
    Wired Faculty App

    Find the equation for the ellipse that satisfies the given condition:
    Foci space space left parenthesis plus-or-minus 5 comma space 0 right parenthesis comma space space straight a equals 8
    Solution

    Solution not provided.
    Ans.  straight x squared over 64 plus straight y squared over 39 equals 1

    Question 200
    CBSEENMA11015205
    Wired Faculty App

    Find the eccentricity of an ellipse if its latus rectum is 1 half of its minor axis.
    Solution

    Solution not provided.
    Ans.  fraction numerator square root of 3 over denominator 2 end fraction

    Question 205
    CBSEENMA11015210
    Wired Faculty App

    Verify that the locus of a point which moves so that the difference of its distances from points A (4, 0) and B (– 4, 0) is always equal to 2, is a hyperbola in the standard form <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 point be straight P space left parenthesis straight alpha comma space straight beta right parenthesis.
    The fixed points are A (4, 0), B (-4, 0). The given condition is :
                               PA - PB = 2
    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>
    Squaring both sides, we get
                       space space straight alpha squared plus straight beta squared minus 8 straight alpha plus 16 space equals space 4 plus straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 plus 4 square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root
    rightwards double arrow                              negative 16 straight alpha minus 4 space equals space 4 square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root space rightwards double arrow space minus left parenthesis 4 straight alpha plus 1 right parenthesis space equals space square root of straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 end root
    Squaring again, we get
                            16 straight alpha squared plus 8 straight alpha plus 1 space equals space straight alpha squared plus straight beta squared plus 8 straight alpha plus 16 space rightwards double arrow space 15 straight alpha squared minus straight beta squared space equals space 15 space rightwards double arrow space straight alpha squared over 1 minus straight beta squared over 15 equals 1
    Hence, the locus of point P is straight x squared over 1 minus straight y squared over 15 equals 1
    which is the standard form of hyperbola in the form straight x squared over straight a squared minus straight y squared over straight b squared equals 1.

    Question 206
    CBSEENMA11015211
    Wired Faculty App

    Verify that the locus of a point which moves so that the difference of its distances from points A (0, –5) and B (0,5) is always 6, is a hyperbola in the standard form 
                                       space space straight y squared over straight a squared minus straight x squared over straight b squared equals 1.

    Solution

    Let the point be straight P space left parenthesis straight alpha comma space straight beta right parenthesis. It is 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>
    rightwards double arrow    square root of straight alpha squared plus straight beta squared plus 10 straight beta plus 25 end root space equals space 6 plus square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root
    Squaring both sides, we get
    straight alpha squared plus straight beta squared plus 10 straight beta plus 25 equals 36 plus straight alpha squared plus straight beta squared minus 10 straight beta plus 25 plus 12 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root
    rightwards double arrow            20 straight beta minus 36 space equals space 12 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root space space rightwards double arrow space 5 straight beta minus 9 space equals space 3 square root of straight alpha squared plus straight beta squared minus 10 straight beta plus 25 end root
    Squaring again, we get
                  <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               9 straight alpha squared minus 16 straight beta squared equals negative 144 space rightwards double arrow space 16 straight beta squared minus 9 straight alpha squared equals 144 space rightwards double arrow space fraction numerator 16 straight beta squared over denominator 144 end fraction minus fraction numerator 9 straight alpha squared over denominator 144 end fraction equals 1 space rightwards double arrow space straight beta squared over 9 minus straight alpha squared over 16 equals 1
    Hence, the locus of straight P space left parenthesis straight alpha comma space straight beta right parenthesis is straight y squared over 9 minus straight x squared over 16 equals 1
    which is the equation of a hyperbola in the standard form space straight y squared over straight a squared minus straight x squared over straight b squared equals 1.

    Question 207
    CBSEENMA11015212
    Wired Faculty App

    Find the co-ordinates of foci, the vertices, the lenghts of transverse and conjugate axis, the eccentricity the length of latus rectum of hyperbola
    16 straight x squared minus 9 straight y squared equals 576

    Solution

    The equation of hyperbola is : 16 straight x squared minus 9 straight y squared equals 576
    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> which is of the form straight x squared over straight a squared minus straight y squared over straight b squared equals 1
    ∴ The foci and vertices lie on x-axis.
    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>
                                                straight b squared equals 64 space rightwards double arrow space straight b equals 8
    ∴                                        space space straight c squared equals straight a squared plus straight b squared space rightwards double arrow space straight c squared equals 36 plus 64 equals 100 space rightwards double arrow space straight c space equals space 10
    The co-ordinates of foci are:
                              straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space left parenthesis negative 10 comma space 0 right parenthesis space space straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space left parenthesis 10 comma space 0 right parenthesis
    The co-ordinates of the vertices are:
                                straight A left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 6 comma space 0 right parenthesis
                            straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 6 comma space 0 right parenthesis
    The length of transverse axis = 2a = 2 x 6 = 12
    The length of conjugate axis = 2b = 2 x 8 = 16
    Let the eccentricity be e, we know that c = ae rightwards double arrow 10 = 6e rightwards double arrow straight e equals 10 over 6 equals 5 over 3
    The length of latus rectum <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 208
    CBSEENMA11015213
    Wired Faculty App

    Find the co-ordinates of foci, the vertices, the lenghts of transverse and conjugate axis, the eccentricity the length of latus rectum of hyperbola
    49 straight y squared minus 16 straight x squared equals 784

    Solution

    The equation of hyperbola 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>
    which is the equation of hyperbola in the standard form straight y squared over straight a squared minus straight x squared over straight b squared equals 1
    ∴ The foci and vertices lie on y-axis.
    So,         space space straight a squared equals 16 space rightwards double arrow space straight a space equals space 4 space and space straight b squared equals 49 space rightwards double arrow space straight b equals 7
    Also,         straight c squared equals straight a squared plus straight b squared equals 16 plus 49 equals 65 space rightwards double arrow space straight c equals square root of 65
    The co-ordinates of foci are:
                              straight S subscript 1 left parenthesis 0 comma space minus straight c right parenthesis space left right arrow space left parenthesis 0 comma space minus square root of 65 right parenthesis space space space straight S subscript 2 left parenthesis 0 comma space straight c right parenthesis space left right arrow space left parenthesis 0 comma space square root of 65 right parenthesis
    The co-ordinates of vetices are:
                          straight A space left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space 4 right parenthesis
                        <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 length of transverse axis = 2a = 2 x 4 = 8
    The length of conjugate axis = 2b = 2 x 7 = 14
    Let the eccentricity be  e
    ∴                              c = ae rightwards double arrow space square root of 65 space equals space 4 straight e space rightwards double arrow space straight e space equals space fraction numerator square root of 65 over denominator 4 end fraction
    Length of latus rectum  = fraction numerator 2 straight b squared over denominator straight a end fraction space equals space fraction numerator 2 cross times 49 over denominator 4 end fraction equals 49 over 2

    Question 209
    CBSEENMA11015214
    Wired Faculty App

    Find the equation of a hyperbola whose foci are left parenthesis 0 comma space plus-or-minus square root of 17 right parenthesis comma space vertices are <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

    Since the foci left parenthesis 0 comma space plus-or-minus square root of 17 right parenthesis and the vertices <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> lie on y-axis.
    ∴ The equation of hyperbola in the standard form is straight y squared over straight a squared minus straight x squared over straight b squared equals 1                         ...(i)
    Comparing foci: space space space space left parenthesis 0 comma space plus-or-minus square root of 17 right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus straight c right parenthesis space rightwards double arrow space straight c equals square root of 17
    Comparing vertices:    left parenthesis 0 comma space plus-or-minus space 4 right parenthesis space left right arrow space left parenthesis 0 comma space plus-or-minus space straight a right parenthesis space rightwards double arrow space straight a space equals space 4
    Using straight c squared equals straight a squared plus straight b squared comma space we have 17 equals 16 plus straight b squared space rightwards double arrow space straight b squared equals 1
    ∴  From (i), the equation fo hyperbola is straight y squared over 16 minus straight x squared over 1 equals 1 space or space straight y squared minus 16 straight x squared equals 16

    Question 210
    CBSEENMA11015215
    Wired Faculty App

    Find the equation of a hyperbola whose foci are <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 vertices are left parenthesis plus-or-minus 3 comma space 0 right parenthesis.

    Solution

    Since foci <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 the vertices left parenthesis plus-or-minus 3 comma space 0 right parenthesis lie on the x-axis.
    ∴  The equation of the hyperbola is straight x squared over straight a squared minus straight y squared over straight b squared equals 1                                            ...(i)
    Comparing foci : left parenthesis plus-or-minus 5 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight c comma space 0 right parenthesis space rightwards double arrow space straight c equals space 5
    Comparing vertices : left parenthesis plus-or-minus 3 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight a comma space 0 right parenthesis space rightwards double arrow space straight a space equals space 3
    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>
    Hence, from (i), the equation of hyperbola is straight x squared over 9 minus straight y squared over 16 equals 1 space or space 16 straight x squared minus 9 straight y squared equals 144

    Question 211
    CBSEENMA11015216
    Wired Faculty App

    Find the equation of a hyperbola, the foci on x-axis, centre is at origin, the distance between the foci equal to 10 and the length of transverse axis is 8.

    Solution

    The foci lie on x-axis.
    ∴ The equation of the hyperbola in standard form is straight x squared over straight a squared minus straight y squared over straight b squared equals 1                        ...(i)
    The foci are <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 the two foci = 2c = 10 (given)
    rightwards double arrow                            c = 5
    Also, the length of transverse axis = 2a = 8 rightwards double arrow  a = 4
    Now,                         straight c squared equals straight a squared plus straight b squared space rightwards double arrow space 25 space equals space 16 plus straight b squared space rightwards double arrow space straight b squared equals 9
    Hence, from (i), the equation of hyperbola 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 212
    CBSEENMA11015217
    Wired Faculty App

    Find the equation of a hyperbola whose vertices lie on y-axis, centre is at the origin, the distance between the foci is 16 and eccentricity 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 vertices of the hyperbola lie on the y-axis.
    ∴ The equation of the hyperbola in standard form is straight y squared over straight a squared minus straight x squared over straight b squared equals 1                                  ...(i)
    The foci are straight S subscript 1 left parenthesis 0 comma space minus straight c right parenthesis space and space straight S subscript 2 left parenthesis 0 comma space straight c right parenthesis
    The distance between them is 2c = 16 (given)
    rightwards double arrow                                    c = 8
    Eccentricity, e = square root of 2
    ∴                             c = 8 rightwards double arrow ae = 8 rightwards double arrow straight a left parenthesis square root of 2 right parenthesis space equals space 8 space rightwards double arrow space straight a space equals space 4 square root of 2 or space space space straight a squared equals 32
    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>
    Hence, from (i), the equation of the hyperbola 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>  or  straight y squared minus straight x squared equals 32

    Question 213
    CBSEENMA11015218
    Wired Faculty App

    Find the co-ordinates of foci, the vertices, the length of transverse axis, the length of conjugate axis, the eccentricity, the latus rectum of the hyperbola:
    <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 hyperbola 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> which is of the form straight x squared over straight a squared minus straight y squared over straight b squared equals 1
    ∴  Its foci and vertices lie on x-axis.
                                      <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 b squared equals 9 space rightwards double arrow space straight b 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>
    The foci are : straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space left parenthesis negative 5 comma space 0 right parenthesis
                                space space space space straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space left parenthesis 5 comma space 0 right parenthesis
    The vertices are: straight A space left parenthesis straight a comma space 0 right parenthesis space left right arrow space left parenthesis 4 comma space 0 right parenthesis
                           straight A apostrophe left parenthesis negative straight a comma space 0 right parenthesis space left right arrow space left parenthesis negative 4 comma space 0 right parenthesis
    The length of transverse axis = 2a = 2 x 4 = 8
    The length of conjugate axis  = 2b = 2 x 3 = 6
    Let the eccentricity be e. We know that c = ae
    rightwards double arrow                                       5 = 4e rightwards double arrow straight e equals 5 over 4
    Length of latus rectum = fraction numerator 2 straight b squared over denominator straight a end fraction equals fraction numerator 2 cross times 9 over denominator 4 end fraction equals 9 over 2

    Question 214
    CBSEENMA11015219
    Wired Faculty App

    Find the co-ordinates of foci, the vertices, the length of transverse axis, the length of conjugate axis, the eccentricity, the latus rectum of the hyperbola:
      <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 equation of hyperbola 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> which is of the form straight y squared over straight a squared minus straight x squared over straight b squared equals 1
    ∴ The foci and vertices lie on  the y-axis.   rightwards double arrow   straight a squared equals 9 space rightwards double arrow space straight a 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>
    Also,                                 straight c squared equals straight a squared plus straight b squared equals 9 plus 27 equals 36 space rightwards double arrow space straight c equals 6
    The co-ordinates of foci are : <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 S subscript 2 left parenthesis 0 comma space straight c right parenthesis space left right arrow space left parenthesis 0 comma space 6 right parenthesis
    The co-ordinates of vertices are: straight A left parenthesis 0 comma space straight a right parenthesis space left right arrow space left parenthesis 0 comma space 3 right parenthesis
                                  straight A apostrophe left parenthesis 0 comma space minus straight a right parenthesis space left right arrow space left parenthesis 0 comma space minus 3 right parenthesis
    The length of transverse axis = 2a = 2 x 3 = 6
    The length of conjugate axis = 2b = <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 eccentricity be e. We know that c = ae rightwards double arrow 6 = 3(e) rightwards double arrow e = 2
    Length of latus rectum = fraction numerator 2 straight b squared over denominator straight a end fraction equals fraction numerator 2 cross times 27 over denominator 3 end fraction equals 18

    Question 215
    CBSEENMA11015220
    Wired Faculty App

    Find the equation of the hyperbola satisfying the given conditions.
    Foci <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 the latus rectum is of length 8.

    Solution

    The foci <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> lie on the x-axis.
    ∴ The standard equation of the hyperbola is of the form : straight x squared over straight a squared minus straight y squared over straight b squared equals 1                 ...(i)
       The foci are      left parenthesis plus-or-minus 3 square root of 5 comma space 0 right parenthesis space left right arrow space left parenthesis plus-or-minus straight c comma space 0 right parenthesis space rightwards double arrow space straight c space equals space 3 square root of 5
                The latus rectum equals 8 space rightwards double arrow space fraction numerator 2 straight b squared over denominator straight a end fraction equals 8 rightwards double arrow space straight b squared equals 4 straight a                                         ...(ii)
    Now,                           straight c squared equals straight a squared plus straight b squared space rightwards double arrow space left parenthesis 3 square root of 5 right parenthesis squared space equals space straight a squared plus straight b squared space rightwards double arrow space 45 space equals space straight a squared plus 4 straight a     (∵ space space straight b squared equals 4 straight a )
    rightwards double arrow             straight a squared plus 4 straight a minus 45 equals 0 space rightwards double arrow space left parenthesis straight a plus 9 right parenthesis space left parenthesis straight a minus 5 right parenthesis space equals space 0 space rightwards double arrow space straight a space equals space 5    (∵ a = -9 is not possible)
    ∴                       space space straight a squared equals 25
    Also,                    straight c squared equals straight a squared plus straight b squared space rightwards double arrow space 45 space equals space 25 plus straight b squared space rightwards double arrow space straight b squared equals 20
    Hence, from (i), the equation of the hyperbola 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 216
    CBSEENMA11015221
    Wired Faculty App

    Find the equation of the hyperbola satisfying the given condition.
    Foci and the latus rectum is of length 12.

    Solution

    The foci are <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> of the hyperbola lie on the y-axis.
    ∴ The standard equation of the hyperbola is: straight y squared over straight a squared minus straight x squared over straight b squared equals 1                            ...(i)
          The foci are left parenthesis 0 comma plus-or-minus space straight c right parenthesis space left right arrow space left parenthesis 0 comma plus-or-minus space 4 right parenthesis space space rightwards double arrow space space space straight c space equals space 4
    Also, the latus rectum 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>             rightwards double arrow  straight b squared equals 6 straight a                                ...(ii)
    Now,                             space space straight c squared space equals space straight a squared plus straight b squared space rightwards double arrow space 16 space equals space straight a squared plus straight b squared space rightwards double arrow space 16 space equals space straight a squared plus 6 straight a         (∵ straight b squared equals 6 straight a)
    rightwards double arrow              space space space straight a squared plus 6 straight a minus 16 equals 0 space rightwards double arrow space left parenthesis straight a plus 8 right parenthesis space left parenthesis straight a minus 2 right parenthesis space equals space 0 space rightwards double arrow space straight a space equals space 2 space rightwards double arrow space straight a squared space equals space 4 (∵ a = -8 is not possible)
    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>
    Hence, from (i), the equation of the hyperbola is space space straight y squared over 4 minus straight x squared over 12 equals 1  or  3 straight y squared minus straight x squared equals 12.

    Question 217
    CBSEENMA11015222
    Wired Faculty App

    Find the equation of the hyperbola satisfying the given condition.
    vertices left parenthesis plus-or-minus 7 comma space 0 right parenthesis and eccentricity 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 vertices left parenthesis plus-or-minus 7 comma space 0 right parenthesis of the hyperbola lie on x-axis.
    ∴ The standard equation of the hyperbola is straight x squared over straight a squared minus straight y squared over straight b squared equals 1                                          ...(i)
    The vertices are <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>
    Also,               eccentricity  straight e equals 4 over 3
    ∴                            c = ae = 7 open parentheses 4 over 3 close parentheses equals 28 over 3
    Now,                       straight c squared space equals space straight a squared plus straight b squared space rightwards double arrow space open parentheses 28 over 3 close parentheses squared space equals space 49 plus straight b squared space rightwards double arrow space straight b squared equals 784 over 9 minus 49 space equals space fraction numerator 784 minus 441 over denominator 9 end fraction
    rightwards double arrow                          straight b squared equals 343 over 9
    Hence, from (i), the equation of hyperbola is : straight x squared over 49 minus fraction numerator 9 straight y squared over denominator 343 end fraction equals 1 space or space 7 straight x squared minus 9 straight y squared equals 343.

    Question 218
    CBSEENMA11015223
    Wired Faculty App

    Find the equation of the hyperbola satisfying the given condition.
    vertices are left parenthesis 0 comma space plus-or-minus space 5 right parenthesis comma space straight e space equals space 5 over 2.

    Solution

    The vertices left parenthesis 0 comma plus-or-minus 5 right parenthesis of the hyperbola lie on the y-axis.
    ∴ The standard equation of the hyperbola is straight y squared over straight a squared minus straight x squared over straight b squared equals 1                          ...(i)
    Vertices are <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>
    Also, eccentricity straight e equals 5 over 2
    Now,                            straight c equals ae equals 5 open parentheses 5 over 2 close parentheses equals 25 over 2 space rightwards double arrow space straight c squared equals open parentheses 25 over 2 close parentheses squared space equals space 625 over 4
    Also,                          space space straight c squared equals straight a squared plus straight b squared space space rightwards double arrow space 625 over 4 space equals space left parenthesis 5 right parenthesis squared plus straight b squared space rightwards double arrow space straight b squared space equals space 625 over 4 minus 25 equals 525 over 4  
    Hence, from (i), the equation of the hyperbola 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>
    or              straight y squared over 25 minus fraction numerator 4 straight x squared over denominator 525 end fraction equals 1 space or space 21 straight y squared minus 4 straight x squared equals 525

    Question 219
    CBSEENMA11015224
    Wired Faculty App

    Find the equation of a hyperbola whose foci are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis end root and which pass through point (2, 3).
    Solution

    The foci of the hyperbola are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis comma which lie on y-axis.
    ∴ The standard equation of the hyperbola is straight y squared over straight a squared minus straight x squared over straight b squared equals 1                                                  ...(i)
    It passes through (2, 3) rightwards double arrow    9 over straight a squared minus 4 over straight b squared equals 1                                                                       ...(ii)
    The foci are left parenthesis 0 comma plus-or-minus square root of 10 right parenthesis space left right arrow space left parenthesis 0 comma plus-or-minus straight c right parenthesis space rightwards double arrow space straight c space equals space square root of 10
    But,                   straight c squared equals straight a squared plus straight b squared space rightwards double arrow space left parenthesis square root of 10 right parenthesis squared space equals space straight a squared plus straight b squared space rightwards double arrow space straight a squared plus straight b squared equals 10 space or space straight b squared equals 10 minus straight a squared          ...(iii)
    Using (iii) in (ii), we have 9 over straight a squared minus fraction numerator 4 over denominator 10 minus straight a squared end fraction equals 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           <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>
    When space straight a squared equals 18 from (iii),  straight b squared equals 10 minus 18 space space rightwards double arrow space straight b squared equals negative 8 (not possible)
    When straight a squared equals 5 comma space from space left parenthesis iii right parenthesis comma space straight b squared equals 10 minus 5 equals 5
    ∴                          straight a squared equals 5 comma space straight b squared equals 5
    Hence, from (i), the equation of the hyperbola is: straight y squared over 5 minus straight x squared over 5 equals 1 or straight y squared minus straight x squared equals 5.

    Question 220
    CBSEENMA11015225
    Wired Faculty App

    For the hyperbola 3X2 – 2y2 = 1.

    Find: (i) the lengths of the transverse and conjugate axes.

    (ii) the co-ordinates of foci.

    (iii) the co-ordinates of vertices.

    (iv) the eccentricity.

    (v) the length of latus rectum.
    Solution

    Equation of the hyperbola is 3 straight x squared minus 2 straight y squared equals 1 space rightwards double arrow space space fraction numerator straight x squared over denominator open parentheses begin display style 1 third end style close parentheses end fraction minus fraction numerator straight y squared over denominator open parentheses begin display style 1 half end style close parentheses end fraction equals 1
    which is of the standard form straight x squared over straight a squared minus straight y squared over straight b squared equals 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>
    Also,                            straight c squared equals straight a squared plus straight b squared space rightwards double arrow space straight c squared equals 1 third plus 1 half equals 5 over 6 space rightwards double arrow space straight c space equals space square root of 5 over 6 end root
    (i) Length of transverse axis equals 2 straight a space equals space 2 cross times fraction numerator 1 over denominator square root of 3 end fraction equals fraction numerator 2 over denominator square root of 3 end fraction equals fraction numerator 2 square root of 3 over denominator 3 end fraction
    Length of conjugate axis = 2b = 2 cross times fraction numerator 1 over denominator square root of 2 end fraction equals square root of 2
    (ii) The co-ordinates of foci are:
                          space space straight S subscript 1 left parenthesis negative straight c comma space 0 right parenthesis space left right arrow space open parentheses negative square root of 5 over 6 comma end root 0 close parentheses space space and space space straight S subscript 2 left parenthesis straight c comma space 0 right parenthesis space left right arrow space open parentheses square root of 5 over 6 end root comma space 0 close parentheses
    (iii) The co-ordinates of vertices are:
                            <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) Let the eccentricity be 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 e equals square root of 5 over 2 end root
    (v) Length of latus rectum  = fraction numerator 2 straight b squared over denominator straight a end fraction equals 2 cross times 1 half cross times fraction numerator square root of 3 over denominator 1 end fraction equals square root of 3

    Question 236
    CBSEENMA11015241
    Wired Faculty App

    Find the equation of the hyperbola under the following condition:
    Foci <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> eccentricity e = 2.

    Solution

    Solution not provided.
    Ans.  straight x squared over 4 minus straight y squared over 12 equals 1

    Question 237
    CBSEENMA11015242
    Wired Faculty App

    Find the equation of a hyperbola whose latus rectum is 8 and eccentricity is space fraction numerator 3 over denominator square root of 5 end fraction.
    Solution

    Solution not provided.
    Ans.  <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 244
    CBSEENMA11015249
    Wired Faculty App

    Find the equation of the circle passing through points (0, 2), (3, 0) and (3, 2).

    Solution

    Solution not provided.
    Ans.  <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 246
    CBSEENMA11015251
    Wired Faculty App

    Find (i) the co-ordinates of the focus.
    (ii) the equation of directrix
    (iii) the latus rectum for the parabola
    <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

    Solution not provided.
    Ans.  (i) (0, -4) ; (ii) y - 4 = 0;  (iii) 16

    Question 247
    CBSEENMA11015252
    Wired Faculty App

    Find (i) the co-ordinates of the focus.
    (ii) the equation of directrix.
    (iii) the latus rectum for the parabola.
    <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

    Solution not provided.
    Ans.(i) (-2, 0)
          (ii) x - 2 =0
          (iii) 8

    Question 248
    CBSEENMA11015253
    Wired Faculty App

    Find (i) the co-ordinates of the focus.
    (ii) the equation of directrix.
    (iii) the latus rectum for the parabola.
    2 straight y squared plus 5 straight x equals 0
    Solution

    Solution not provided.
    Ans. (i) open parentheses negative 5 over 8 comma space 0 close parentheses
            (ii) straight x minus 5 over 8 equals 0
             (iii) 5 over 2

    Question 249
    CBSEENMA11015254
    Wired Faculty App

    Find (i) the co-ordinates of the focus.
    (ii) the equation of directrix.
    (iii) the latus rectum for the parabola.
    space space space straight x squared equals 6 straight y
    Solution

    Solution not provided.
    Ans.   (i) open parentheses 0 comma space 3 over 2 close parentheses
             (ii)  straight y plus 3 over 2 equals 0
             (iii) 6

    Question 251
    CBSEENMA11015256
    Wired Faculty App

    Check whether the points (2, 2), (4, 4) and left parenthesis 1 comma space 2 square root of 2 right parenthesis lie inside, outside or on the cirlcle straight x squared plus straight y squared equals 9.
    Solution

    The given equation of the circle is straight x squared plus straight y squared equals 9
    The centreo of the circle is (0, 0) and radius of circle is 3
    For point (2, 2).
    Distance of point (2, 2) from centre (0, 0) = square root of left parenthesis 2 minus 0 right parenthesis squared plus left parenthesis 2 minus 0 right parenthesis squared end root space equals space 2 square root of 2 space less than 3 (radius of circle)
    ∴ Point (2, 2) lies inside the circle
    For point (4, 4),
    Distance of point (4, 4) from centre (0, 0) = <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> (radius of circle)
    ∴ Point (4, 4) lies outside the circle
    For point left parenthesis 1 comma space 2 square root of 2 right parenthesis comma
    Distance of point left parenthesis 1 comma space 2 square root of 2 right parenthesis from the centre of the circle (0, 0).
                                                 equals square root of left parenthesis 1 minus 0 right parenthesis squared plus left parenthesis 2 square root of 2 minus 0 right parenthesis squared end root equals square root of 1 plus 8 end root space equals space square root of 9 space equals space 3 (radius of the circle).
    ∴ Point left parenthesis 1 comma space 2 square root of 2 right parenthesis lies on the circle.

    Question 252
    CBSEENMA11015257
    Wired Faculty App

    Find the equation of a circle, the two ends of whose diameter AB are A (3,–1) and B (5, 3).

    Solution

    Mid-point of AB is open parentheses fraction numerator 3 plus 5 over denominator 2 end fraction comma space fraction numerator negative 1 plus 3 over denominator 2 end fraction close parentheses
    ∴ Centre C of the circle is left parenthesis straight h comma space straight k right parenthesis space left right arrow space left parenthesis 4 comma space 1 right parenthesis
          Radius of the circle = CA
    rightwards double arrow     straight r equals square root of left parenthesis 4 minus 3 right parenthesis squared plus left parenthesis 1 plus 1 right parenthesis squared end root space equals space square root of 1 plus 4 end root space equals space square root of 5
    Hence, the equation of 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>
    rightwards double arrow            left parenthesis straight x minus 4 right parenthesis squared plus left parenthesis straight y minus 1 right parenthesis squared equals left parenthesis square root of 5 right parenthesis squared space space or space space space left parenthesis straight x minus 4 right parenthesis squared plus left parenthesis straight y minus 1 right parenthesis squared equals 5

    Question 253
    CBSEENMA11015258
    Wired Faculty App

    Find the equation of a circle whose centre is C (2, –3) and which touches the straight line 3x – 4y – 3 = 0.

    Solution

    The centre of the circle is (2, -3)
    rightwards double arrow               left parenthesis straight h comma space straight k right parenthesis space left right arrow space left parenthesis 2 comma space minus 3 right parenthesis space space space rightwards double arrow space space straight h space equals space 2 comma space straight k equals negative 3
    The circle touches the straight line 3x - 4y - 3 = 0
    rightwards double arrow                           space space straight r equals open vertical bar fraction numerator 3 straight h minus 4 straight k minus 3 over denominator square root of 9 plus 16 end root end fraction close vertical bar space space rightwards double arrow space straight r equals open vertical bar fraction numerator 3 left parenthesis 2 right parenthesis minus 4 left parenthesis negative 3 right parenthesis minus 3 over denominator 5 end fraction close vertical bar space rightwards double arrow space straight r space equals space open vertical bar fraction numerator 6 plus 12 minus 3 over denominator 5 end fraction close vertical bar equals 3
    Hence, the equation 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 254
    CBSEENMA11015365
    Wired Faculty App

    Represent the following inequality graphically in two dimensional plane:
    2x + 6 > 0

    Solution

    2x + 6 > 0 rightwards double arrow 2x > -6 rightwards double arrow x > - 3
    The graph of x = -3, is obtained by drawing a vertical line at x = -3 on the x-axis and dotted line is drawn.
    (0, 0) substituted in the inequality gives 0 > -3 which is true.
    ∴ The region is the one that contains (0, 0)
    (or alternatively the inequality is x > -3, we shall mark the region towards the right of the vertical line)
    The graph is:


    Question 255
    CBSEENMA11015366
    Wired Faculty App

    Represent the following inequality graphically in two dimensional plane:
    3y + 6 < 0

    Solution

    3y + 6 < 0  rightwards double arrow   3y < - 6  or y < - 2
    The line drawn will be at y = -2 on y-axis and it will be horizontal.
    The inequality is y<-2, so dotted line is drawn O (0, 0) substituted in y > -2
    rightwards double arrow                     0 < -2 which is false.
    ∴  The region represented is the one which does not contain (0, 0).
    (or alternatively y < -2 is towards the lower side of the horizontal line)
    The graph of inequality is :

    Question 256
    CBSEENMA11015367
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    <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

    First, we draw the graph of straight line 2x + 3y = 0

    Now, let us test a point B(1, 1)
    2 left parenthesis 1 right parenthesis space plus space 3 left parenthesis 1 right parenthesis space less or equal than space 0 space or space 5 space less or equal than space 0 which is not true.
    From these obseravtions, the graph is given below in which the shaded region represents the solution of given inequality which does not contain point B(1, 1).

    Question 257
    CBSEENMA11015368
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    4 straight x minus 3 straight y greater or equal than 0

    Solution

    First, we draw the graph of straight line 4x - 3y = 0
    x     0        3
    y     0        4
    Now, let us take a point B(1, 1)
    ∴  4(1) - 3(1) greater or equal than 0 or 1 greater or equal than 0 which is true.
    From these observations, the required solution is represented by the shaded portion of the graph given below which contains point B(1, 1).

    Question 258
    CBSEENMA11015369
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    x + y < 5

    Solution

    First, we draw the dotted line to represent the graph of x + y = 5
    x       0        5
    y       5        0
    Now, let us take a point O(0, 0)
    Substitute in given inequality, 0 + 0 < 5, i.e., 0 < 5 which is true.
    Therefore, origin O is a solution of given inequality.
    From these observations, the graph is given below, in which the shaded portion represents the solution of the given inequality which contains O(0, 0).

    Question 259
    CBSEENMA11015370
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    space space straight x space minus space straight y space greater or equal than space 2

    Solution

    First, we draw the dark line to represent the graph of x - y = 2
    x         0          2
    y         -2         0
    Now, let us take a point O(0, 0)
    Substitute in given inequality <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> which is not true.
    Therefore, origin is not the solution of given inequality.
    From these observations, the graph is given below, in which the shaded portion represents the solution of the given inequality.

    Question 260
    CBSEENMA11015371
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    – 3x + 2y ≥ – 6
    Solution

    First, we draw the dark line to represent the graph of straight line -3x + 2y = -6
    x      0         2
    y      -3        0
    Now, let us take a point O(0, 0), i.e., origin.
    Substitute in given inequality, -3(0) + 2(0) ≥ -6, i.e., 0 ≥ -6 which is true. Therefore, origin O(0, 0) is solution of given inequality. From these observations, the graph is given below, in which the shaded region represents the solution of the given inequality.

    Question 261
    CBSEENMA11015372
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    3y – 5x < 30
    Solution

    First, we draw the dotted line to represent the graph of straight line 3y - 5x = 30
    x         0         -6
    y         10         0
    Now, let us take a point O(0, 0) i.e, origin.
    Substitute in given inequality, 3(0) - 5(0) < 30  or 0 < 30 which is true. Therefore, origin O(0, 0) is the solution of given inequality. From these observations, the graph is given ahead in which shaded region represents the solution of the given inequality.

    Question 262
    CBSEENMA11015373
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    2x – 3y > 6
    Solution

    First, we draw a dotted line to represent graph of line 2x - 3y = 6.
    x        0        3
    y       -2        0
    Now, let us take a point O(0, 0). Substitute in given inequality. 2(0) - 3(0) > 6 or 0 > 6 which is not true. Therefore, origin is not the solution of given inequality. From these observations, the graph is given below in which the shaded region represents the solution of the given inequality.

    Question 263
    CBSEENMA11015374
    Wired Faculty App

    Solve the following inequality graphically in two-dimensional plane:
    y + 8 ≥ 2x
    Solution

    First, we draw a dark line to represent the graph of straight line y + 8 = 2x.
    i.e.,       2x - y = 8
    x             0              4
    y            -8              0
    Now, let us take a point O(0, 0), i.e, origin.
    Substitute in given inequality, 0 + 8 ≥ 2(0) or 8 ≥ 0 which is true.
    Therefore, origin is the solution of given inequality.
    From these observations, the graph is given below, in which shaded region represents the solution of the given inequality.

    Question 264
    CBSEENMA11015375
    Wired Faculty App

    Solve the following system of inequality graphically.
    x ≥ 2, y ≤ 3

    Solution

    The given linear inequalities are :
                         x ≥ 2                                                             ...(i)
                         y ≤ 3                                                             ...(ii)
    First, we draw the graph of straight lines x = 2 and y = 3.
    For x ≥ 2, we mark the region towards right of line x = 2.
    For y ≤ 3, we mark the region below the line y = 3.
    Now, we shade the common region.
    Thus, the graphical solution is given below:

    Question 265
    CBSEENMA11015376
    Wired Faculty App

    Solve the following system of inequality graphically.
    x ≥ 3, y ≥ 2

    Solution

    The given linear inequalities are:
                             x ≥ 3                                                    ...(i)
                             y ≥ 2                                                    ...(ii)
    First, we draw the lines x = 3  and y = 2.
    For x ≥ 3, we mark the region on right side of line x = 3.
    For y ≥ 2, we mark the region above the line y = 2.
    Now, we shade the common region.
    Thus, the graphical solution is:

    Question 266
    CBSEENMA11015377
    Wired Faculty App

    Solve the following system of inequality graphically.
    3x + 2y ≤ 12, x ≥ 1, y ≥ 2

    Solution

    First, we draw the straight line 3x + 2y = 12
    x         0           4
    y         6           0
    Now, let us take the point O(0, 0), i.e., origin.
    Substituting O(0, 0) in inequality 
    3x + 2y ≤ 12, we get
    3(0) + 2(0) ≤ 12 or 0 ≤ 12 which is true.
    ∴ origin is solution of inequality 
    3x + 2y ≤ 12.
    Therefore, we mark that region which contains region.
    For x ≥ 1, we mark the region on right side of line x = 1.
    For y ≥ 2, we mark the region above the line y = 2.
    Now, we shade the common region.
    Thus, the graphical solution is the triangular region ABC as shown in the figure.

    Question 267
    CBSEENMA11015378
    Wired Faculty App

    Solve the following system of inequality graphically.
    4x + 5y ≥ -20, x + 1 ≤ 0, y ≤ -1

    Solution

    First, we draw the line 4x + 5y = -20
    x           0             -5
    y          -4             0
    Now, consider the point O(0, 0), i.e, origin.
    Substitute in inequality 4x + 5y ≥ -20,
    i.e, 4(0) + 5(0) ≥ - 20  or 0 ≥ - 20 which is true.
    ∴ origin is solution of inequality 
    4x + 5y ≥ -20
    Therefore, we mark that region which contains origin.
    For x + 1 ≤ 0  or x ≤ -1, we mark the region on left side of line x = -1.
    For y ≤ -1, we mark the region below the line y = -1.
    Now, we shade the common region. The required solution is given by the triangular region ABC as shown below in the graph.

    Question 268
    CBSEENMA11015379
    Wired Faculty App

    Solve the following system of inequality graphically.
    x + y ≤ 9, y ≥ x, x ≥ 0

    Solution

    First, we draw the graph of line x + y = 9
    x           0             9
    y           9             0
    Now, consider the point O(0, 0), i.e, origin.
    Substitute in inequality x + y ≤ 9, i.e., 0 + 0 ≤ 9 or 0 ≤ 9 which is true. Therefore, origin is solution of inequality x + y ≤ 9. Therefore, we mark that region which contains origin.
    Now, we draw the line y = x or y - x = 0
    x    0         1            2
    y    0         1            2
    Let us consider the point (1, 0).
    Substitute it in inequality y ≥ x, i.e., 0 ≥ 1 which is not true.
    ∴ We mark that region which does not contain point (1, 0).
    For x ≥ 0, we mark the region on right side of the line x = 0.
    Now, we shade the common region. Thus, the solution is given by the triangular region OAB as shown in the graph.

    Question 269
    CBSEENMA11015380
    Wired Faculty App

    Solve the following system of inequality graphically.
    3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0.

    Solution

    Clearly, x ≥ 0, y ≥ 0 means that the solution set lies in the first quadrant.
    Now, we draw the graph of line 3x + 4y = 60
    x             0            20
    y             15           0
    Now, we consider the point O(0, 0) i.e., origin.
    Substitute in inequality 3x + 4y ≤ 60, i.e., 3(0) + 4(0) ≤ 60 or 0 ≤ 60 which is true.
    Therefore, we mark that region which contains origin.
    Now, we draw the graph of line x + 3y = 30
    x               0          30
    y               10         0
    Now, we consider the point O(0, 0), i.e, origin.
    Substitute in inequality x + 3y ≤ 30,  0 + 3(0) ≤ 30  or 0 ≤ 30 which is true. Therefore, we mark that region which contains origin.
    Now, we shade the common region. The solution is given by the shaded portion OABC of the graph given below;

    Question 270
    CBSEENMA11015381
    Wired Faculty App

    Solve the following system of inequality graphically.
    x - 2y ≤ 3, 3x + 4y ≤ 12, x ≥ 0, y ≥ 1

    Solution

    First, we draw the graph of line x - 2y = 3
    x           0            3
    y          -3/2         0
    Now, consider a point O(0, 0), i.e, origin.
    Substitute in given inequality x - 2y ≤ 3, i.e, 0 - 2(0) ≤ 3 or  0 ≤ 3 which is true. Therefore, we mark that region which contains origin. Now, we draw the graph of line 3x + 4y = 12.
    x          0          4
    y          3          0
    Now, we consider a point O(0, 0), i.e., origin.
    Substitute in given inequality 3x + 4y ≤ 12, i.e.,  3(0) + 4(0) ≤ 12 or 0 ≤ 12 which is true.
    ∴ we mark that region which contains region.
    For x ≥ 0, we mark the region on right side of line x = 0.
    For y ≥ 1, we mark the region above the line y = 1
    Now, we shade the common region.
    The solution is given by the triangular region ABC as shown in the graph.

    Question 271
    CBSEENMA11015382
    Wired Faculty App

    Solve the following system of inequality graphically.
    3x + 2y ≤ 150, x + 4y ≤ 80,  x ≥ 15, y ≥ 0

    Solution

    First, we draw the graph of line 3x + 2y = 150
    x                0            50
    y                75           0
    Now, we consider a point O(0, 0), i.e,  origin.
    Substitute it in inequality 3x + 2y ≤ 150, i.e., 3(0) + 2(0) ≤ 150 or 0 ≤ 150 which is true.
    ∴ we mark that region which contains origin.
    We draw the graph of line x + 4y = 80
    x            0            80
    y            20           0
    Now, we consider a point O(0, 0), i.e, origin.
    Substituting it in inequality x + 4y ≤ 80, we get
    0 + 4(0) ≤ 80  or 0 ≤ 80 which is true.
    ∴ we mark that region which contains origin.
    For x ≥ 15, we mark the region on right side of the line x = 15
    For y ≥ 0, we mark the region above the line y = 0
    Now, we shade the common region.
    The required solution is given by shaded region ABCD as shown in the graph.
     

    Question 272
    CBSEENMA11015383
    Wired Faculty App

    Solve the following system of inequality graphically.
    2x + y ≥ 6, 3x + 4y ≤ 12

    Solution

    We draw the graph of line 2x + y = 6
    x        0          3
    y        6          0
    Now, we consider a point O(0, 0), i.e, origin.
    Substitute in inequality 2x + y ≥ 6, i.e. 7,  2(0) + 0 ≥ 6 or 0 ≥ 6 which is not true.
    ∴ we mark that region which does not contain origin.
    Now, we draw the graph of line 3x + 4y = 12
    x        0           4
    y        3           0
    Now, we consider a point O(0, 0), i.e., origin.
    Substituting in given inequality, 3x + 4y ≤ 12, we get 3(0) + 4(0) ≤ 12 which is true.
    Therefore, we mark that region which contains origin. Now, we shade the common region. Thus, the required solution set is given by the shaded portion of graph given below.

    Question 273
    CBSEENMA11015384
    Wired Faculty App

    Solve the following system of inequality graphically.
    2x - y > 1,  x - 2y < -1

    Solution

    We draw the dotted line to represent the graph of 2x - y = 1
    x          0            1/2
    y         -1             0
    Now, we consider a point O(0, 0), i.e., origin.
    Substitue it in inequality 2x - y > 1, we get 2 (0) - 0 > 1 or 0 > 1 which is not true.
    Therefore, we mark tht region, which does not contain origin.
    We draw the dotted line to represent the graph of line x - 2y = -1
    x          0         -1
    y         1/2        0
    Now, we consider a point O(0, 0) i.e., origin.
    Substitute it in inequality x - 2y < - 1,
    i.e., 0 - 2(0) < -1 or 0 < -1 which is not true.
    ∴ we mark that region which does not contain origin.
    Now, we shade the common region. Thus, the solution is given by the shaded region of the graph given below;

    Question 274
    CBSEENMA11015385
    Wired Faculty App

    Find the linear inequalities for which the shaded area in the given graph is the solution set:

    Solution

    Consider the line 2x + y = 2. Clearly, the shaded portion and the origin are on the opposite sides of the graph. So, we have its inequality 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>                                              [∵ 2(0) + 0 ≥ 2 is not true]
    Now, consider the line x + 2y = 8.
    Origin and shaded region are on the same side of the line x + 2y = 8.
    So, its inequality is:
                           box enclose straight x plus 2 straight y less or equal than 8 end enclose                                                 [∵ 0 + 2(0) ≤ 8 is true]
    Finally, consider the line x - y = 1
    Origin and shaded region are on the same side.
    So, its inequality is:
                            box enclose straight x minus straight y less or equal than 1 end enclose                                                    [∵ 0 - 0 ≤ 1 is true]
    Also,                   x ≥ 0 and y ≥ 0
    Thus, the linear inequalities are:
    2x + y ≥ 2,  x + 2y ≤ 8, x - y ≤ 1, x ≥ 0 and y ≥ 0

    Question 275
    CBSEENMA11015386
    Wired Faculty App

    Solving the following inequalities. Also represent the solutions on the number line.
    3x - 2 > x + 5
    Solution

    Solution not provided.
    Ans. open parentheses 7 over 2 comma space infinity close parentheses   

    Question 276
    CBSEENMA11015387
    Wired Faculty App

    Solving the following inequalities. Also represent the solutions on the number line.
    -4x + 1 ≤ 2 (1 - x)
    Solution

    Solution not provided.
    Ans.    <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 277
    CBSEENMA11015388
    Wired Faculty App

    Solving the following inequalities. Also represent the solutions on the number 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>
    Solution

    Solution not provided.
    Ans.    left parenthesis negative infinity comma space 5 over 3 right square bracket,            
       
     

    Question 278
    CBSEENMA11015389
    Wired Faculty App

    Solving the following inequality. Also represent the solutions on the number line.
    straight x less than open parentheses fraction numerator 2 straight x plus 1 over denominator 3 end fraction close parentheses minus open parentheses fraction numerator 5 straight x minus 1 over denominator 2 end fraction close parentheses
    Solution

    Solution not provided.
    Ans.   left parenthesis negative infinity comma space 5 over 17 right square bracket

           
        
     

    Question 279
    CBSEENMA11015390
    Wired Faculty App

    Solve x + 1 < 8 - x, where
    (i) straight x space element of space straight N        (ii) straight x space element of space straight Z          (iii) <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

    Solution not provided.
    Ans.   (i) {1, 2, 3}      (ii) {........ -3, -2, -1, 0, 1, 2, 3}   (iii) <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 280
    CBSEENMA11015391
    Wired Faculty App

    Solve the following system of simultaneous linear inequality and represent its solution on the number line.
    2x + 3 < x - 1 and x + 4 > 2x - 3
    Solution

    Solution not provided.
    Ans.   (i) left parenthesis negative infinity comma space minus 4 right parenthesis            

           
        
     

    Question 281
    CBSEENMA11015392
    Wired Faculty App

    Solve the following system of simultaneous linear inequality and represent its solution on the number line.
    -3x + 2 ≥ 0 and 2 - x < 0
    Solution

    Solution not provided.
    Ans.   No solution            

           
        
     

    Question 282
    CBSEENMA11015393
    Wired Faculty App

    Make the region represented by the following inequality:
    2x - y ≥ 1
    Solution

    Solution not provided.
    Ans.  


           
        
     

    Question 283
    CBSEENMA11015394
    Wired Faculty App

    Make the region represented by the following inequality:
    2x + 3y ≤ 6
    Solution

    Solution not provided.
    Ans.  



           
        
     

    Question 284
    CBSEENMA11015395
    Wired Faculty App

    Make the region represented by the following inequality:
    x ≥ 5 
    Solution

    Solution not provided.
    Ans.  

    Question 285
    CBSEENMA11015396
    Wired Faculty App

    Make the region represented by the following inequality:
    y < - 1
    Solution

    Solution not provided.
    Ans.  

    Question 288
    CBSEENMA11015399
    Wired Faculty App

    Find the linear inequalities for which the shaded area in the diagram is the solution set.
    Solution

    Solution not provided.
    Ans.  2x + 3y ≥ 6,  x + 2y ≤ 6, -3x + 2y ≤ 3, x - 2y ≤ 2, x ≥ 0, y ≥ 0

    Question 289
    CBSEENMA11015400
    Wired Faculty App

    Find all pairs of consecutive odd positive integers both of which are smaller than 10, such that their sum is greater than 11.
    Solution

    Solution not provided.
    Ans.  5 and 7,  7 and 9

    Question 290
    CBSEENMA11015401
    Wired Faculty App

    Solve the following system of linear inequality.
    <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

    Solution not provided.
    Ans. <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 291
    CBSEENMA11015402
    Wired Faculty App

    Solve the following system of linear inequality:
    3 minus 7 straight x greater or equal than 10 space and space 2 straight x plus 3 less or equal than 5
    Solution

    Solution not provided.
    Ans. <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 292
    CBSEENMA11015403
    Wired Faculty App

    Solve the following system of linear inequality:
    fraction numerator 2 straight x over denominator 5 end fraction minus 1 third less than straight x plus 1 half space and space 7 straight x plus 1 less than 4 straight x plus 5
    Solution

    Solution not provided.
    Ans.  space space space open parentheses negative 25 over 18 comma space 4 over 3 close parentheses

    Question 293
    CBSEENMA11015404
    Wired Faculty App

    Draw the graph of x + y ≤ 5
    Solution
    Question 294
    CBSEENMA11015405
    Wired Faculty App

    Draw the graph of the following inequalities: 2x + 3y - 6 ≤ 0 and x + y ≥ 1
    Solution

    Solution not provided.
    Ans. 

    Question 295
    CBSEENMA11015406
    Wired Faculty App

    Draw the graph of 2y - x ≥ 0
    Solution

    Solution not provided.
    Ans.  

    Question 297
    CBSEENMA11015408
    Wired Faculty App

    Find the linear inequalities for which the shaded area in the figure given below is the solution.
    Solution

    3x + 2y ≤ 6,  2x + 3y ≤ 6, x ≥ 0, y ≥ 0

    Question 298
    CBSEENMA11015409
    Wired Faculty App

    Give the graphical solution of the following system of inequalitiy:
    x ≤ - 2, y > 3

    Solution

    Solution not provided.
    Ans. 

    Question 299
    CBSEENMA11015410
    Wired Faculty App

    Give the graphical solution of the following system of inequality:
    x < -2, y < -3

    Solution

    Solution not provided.
    Ans. 

    Question 300
    CBSEENMA11015411
    Wired Faculty App

    Give the graphical solution of the following system of inequality:
    x + y ≥ 5,  x - y ≤ 3

    Solution

    Solution not provided.
    Ans. 

    Question 301
    CBSEENMA11015412
    Wired Faculty App

    Give the graphical solution of the following system of inequality:
    x + y > 4,  2x - y < 0

    Solution

    Solution not provided.
    Ans. 
     

    Question 314
    CBSEENMA11015425
    Wired Faculty App

    Give the graphical solution of the following systems of inequality.
    x + y ≤ 6, x + y ≥ 4

    Solution

    Solution not provided.
    Ans. 

    Question 319
    CBSEENMA11015430
    Wired Faculty App

    Find the linear inequalities for which the shaded area in the figure given below is a solution.
    Solution

    4x + 9y ≤ 36,   x + 3y ≥ 6,  3x + y ≥ 6,  4x - 3y ≤ 12

    Question 320
    CBSEENMA11015431
    Wired Faculty App

    Draw the graph of the following inequality:
    x > 3

    Solution
    Question 321
    CBSEENMA11015432
    Wired Faculty App

    Draw the graph of the following inequality:
    x ≤ -2

    Solution
    Question 322
    CBSEENMA11015433
    Wired Faculty App

    Draw the graph of the following inequality:
    y ≥ 2

    Solution
    Question 323
    CBSEENMA11015434
    Wired Faculty App

    Draw the graph of the following inequality:
    y < -3

    Solution
    Question 326
    CBSEENMA11015437
    Wired Faculty App

    Draw the graph of the following inequality:
    x - 3y > 0

    Solution
    Question 328
    CBSEENMA11015439
    Wired Faculty App

    Draw the graph of the following inequality:
    space space straight x over 9 plus straight y over 6 greater or equal than 1

    Solution
    Question 334
    CBSEENMA11015475
    Wired Faculty App

    The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2 =6 on any tangent to it is

    • (x2-y2)2 = 6x2+2y2

    • (x2-y2)2 = 6x2 -2y2

    • (x2+y2)2 = 6x2+2y2

    • (x2+y2)2 = 6x2-2y2
    Solution

    C.

    (x2+y2)2 = 6x2+2y2

    Equation of ellipse is,
    straight x squared space plus 3 straight y squared space equals space 6 space or space straight x squared over 6 space plus straight y squared over 2 space equals space 1 Equaion space of space the space tangent space is space fraction numerator straight x space cos space straight theta over denominator straight a end fraction plus fraction numerator straight y space sin space straight theta over denominator straight b end fraction space equals space 1 Let space left parenthesis straight h comma straight k right parenthesis space be space any space point space of space the space locus therefore space straight h over straight a cos space straight theta space plus straight k over straight b sinθ space equals space 1 space... space left parenthesis straight i right parenthesis Slope space of space the space tangent space line space is space fraction numerator negative straight b over denominator straight a end fraction cot space straight theta.
    The slope of the perpendicular drawn from the centre (0,0) to (h,k) is k/h.
    Since, both the lines are perpendicular,
    therefore space open parentheses straight k over straight h close parentheses space straight x open parentheses negative straight b over straight a space cot space straight theta close parentheses space equals space minus space 1 rightwards double arrow space cos space straight theta space equals space αha comma space sin space straight theta space equals space αkb From space eq space left parenthesis straight i right parenthesis space we space get straight h over straight a left parenthesis αha right parenthesis space plus space straight k over straight b space left parenthesis αkb right parenthesis space equals space 1 rightwards double arrow space straight h squared straight alpha space plus straight k squared straight alpha space equals space 1 rightwards double arrow straight alpha space equals space fraction numerator 1 over denominator straight h squared plus straight k squared end fraction Also comma space sin squared straight theta space plus space Cos squared space straight theta space equals 1 rightwards double arrow space left parenthesis αkb right parenthesis squared space plus space left parenthesis αha right parenthesis squared space equals 1 rightwards double arrow space straight alpha squared straight k squared straight b squared space plus straight alpha squared straight h squared straight a squared space equals 1 rightwards double arrow fraction numerator straight k squared straight b squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator straight h squared straight a squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1 fraction numerator 2 straight k squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator 6 straight h squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1 6 straight x squared plus 2 straight y squared space equals space left parenthesis straight x squared plus straight y squared right parenthesis squared  

    Question 335
    CBSEENMA11015477
    Wired Faculty App

    The slope of the line touching both the parabolas y2 = 4x and x2-32y is 

    • 1/2

    • 3/2

    • 1/8

    • 2/3
    Solution

    A.

    1/2

    For parabola, y2 = 4x
    Let y = mx + 1/m be tangent line and it touches the parabola x2=-32y
    therefore space straight x squared space equals space minus 32 space open parentheses mx plus 1 over straight m close parentheses rightwards double arrow space straight x squared space plus 32 mx space plus 32 over straight m space equals space 0 because space straight D space equals 0 therefore space left parenthesis 32 straight m right parenthesis squared space minus space 4 space open parentheses 32 over straight m close parentheses space equals space 0 rightwards double arrow space straight m cubed space equals space 1 over 8 straight m space equals space 1 half

    Question 336
    CBSEENMA11015489
    Wired Faculty App

    The circle passing through (1,-2) and touching the axis of x at (3,0) also passes through the point

    • (-5,2)

    • (2,-5)

    • (5,-2)

    • (-2,5)
    Solution

    C.

    (5,-2)

    (x − 3)2+ y2+ λy = 0
    The circle passes through (1, − 2)
    ⇒ 4 + 4 − 2λ = 0 ⇒ λ = 4
    (x − 3)2+ y2+ 4y = 0
    ⇒ Clearly (5, − 2) satisfies.

    Question 337
    CBSEENMA11015496
    Wired Faculty App

    Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1 − Tn = 10, then the value of n is

    • 7

    • 5

    • 10

    • 8
    Solution

    B.

    5

    Given,Tn = nC3
    Tn+1 = n+1C3
    therefore,
    Tn+1-Tn = n+1C3-nC3 =10
    ⇒nC2 + nC3-nC3 = 10
    [∵nCr +nCr-1 =n+1Cr]
    nC2 =10
    nC2 =5C2
    n =5

    Question 338
    CBSEENMA11015499
    Wired Faculty App

    Given A circle, 2x2 + 2y2= 5 and parabola, straight y squared space equals space 4 square root of 5 space straight x 
    Statement I An equation of a common tangent to these curves is straight y space equals straight x plus square root of 5
    Statement II If the line straight y space equals space mx space plus fraction numerator space square root of 5 over denominator straight m end fraction space left parenthesis straight m space not equal to 0 right parenthesis is the common tangent, then m satisfies m4-3m2+2 =0

    • Statement I is true,  Statement II is true; Statement II is a correct explanation for statement I

    • Statement I is true, Statement II is true; Statement II is not a correct explanation for statement I

    • Statement I is true, Statement II is false

    • Statement I is false, Statement II is true
    Solution

    B.

    Statement I is true, Statement II is true; Statement II is not a correct explanation for statement I

    Equation of circle can be rewritten as,
    x2 +y2 = 5/2
    Let common tangent be 
    straight y space equals space mx space plus fraction numerator square root of 5 over denominator straight m end fraction space space space left square bracket straight m space not equal to 0 right square bracket
    The perpendicular from centre to the tangent is equal to radius,
    therefore, 
    fraction numerator begin display style fraction numerator square root of 5 over denominator straight m end fraction end style over denominator square root of 1 plus straight m squared end root end fraction space equals space square root of 5 over 2 end root rightwards double arrow space straight m square root of 1 plus straight m squared end root space equals space square root of 2 straight m squared space left parenthesis 1 plus straight m squared right parenthesis space equals space 2 rightwards double arrow space straight m to the power of 4 space plus straight m squared minus 2 space equals space 0 left parenthesis straight m squared plus 2 right parenthesis left parenthesis straight m squared minus 1 right parenthesis space equals space 0 straight m space equals space plus-or-minus 1
    thus, both Statements are correct.

    Question 341
    CBSEENMA11015514
    Wired Faculty App

    If z ≠ 1 and fraction numerator straight z squared over denominator straight z minus 1 end fraction is real, then the point represented by the complex number z lies

    • either on the real axis or on a circle passing through the origin

    • on a circle with centre at the origin

    • either on the real axis or on a circle not passing through the origin

    • on the imaginary axis
    Solution

    A.

    either on the real axis or on a circle passing through the origin

    Let z = x + iy (∴ x ≠ 1 as z ≠ 1)
    z2= (x2 – y2) + i(2xy)
    its imaginary part = 0
    fraction numerator straight z squared over denominator straight z minus 1 end fraction is real
    ⇒ 2xy (x – 1) – y(x2– y2) = 0
    ⇒ y(x2+ y2– 2x) = 0
    ⇒y 0; x2+ y2 – 2x = 0
    ∴ z lies either on real axis or on a circle through origin.

    Question 342
    CBSEENMA11015515
    Wired Faculty App

    The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is

    • 10/3

    • 3/5

    • 6/5

    • 5/3
    Solution

    A.

    10/3


    Let (h, k) be centre. (h – 1)2 + (k – 0)2 = k2
    ⇒ h = 1
    (h – 2)2 + (k – 3)2 = k2
    ⇒ k = 5 /3
    therefore, diameter is 2k = 10/3
    Question 343
    CBSEENMA11015517
    Wired Faculty App

    An ellipse is drawn by taking a diameter of the circle (x–1)2 + y2 = 1 as its semiminor axis and a diameter of the circle x2 + (y – 2)2 = 4 as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

    • 4x2+ y2 = 4

    • x2 +4y2 =8

    • 4x2 +y2 =8

    • x2+4y2 =16
    Solution

    D.

    x2+4y2 =16

    The equation of the ellipse.
    Diameter of circle (x-1)2 +y2 = 1 is 2 and that of circle x2 +(y-2)2 = 4 is 4 units.
    ⇒ Semi-minor axis of ellipse, b =2 units are semi major axis of ellipse, a  4 unite
    Hence, the equation of the ellipse is 
    straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1 rightwards double arrow space straight x squared over 16 plus straight y squared over 4 space equals 1 rightwards double arrow space straight x squared space plus 4 straight y squared space equals space 16

    Question 345
    CBSEENMA11015542
    Wired Faculty App

    hyperbola passes through the point P(√2, √3) and has foci at (± 2, 0). Then the tangent to this hyperbola at P also passes through the point

    • left parenthesis negative square root of 2 comma space minus square root of 3 right parenthesis
    • left parenthesis 3 square root of 2 space comma space 2 square root of 3 right parenthesis
    • left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis
    • left parenthesis square root of 3 comma square root of 2 right parenthesis

    Solution

    C.

    left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis

    Equation of hyperbola is straight x squared over straight a squared minus straight y squared over straight b squared space equals space 1
    foci is (±2, 0) hence ae = 2, ⇒ a2e2 = 4

    b2 = a2(e2 – 1)
    ∴ a2 + b2 = 4
    Hyperbola passes through √2,√3
    therefore space 2 over straight a squared space minus 3 over straight b squared space equals space 1 space... space left parenthesis 2 right parenthesis On space sol space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis straight a squared space equals 8 space left parenthesis is space rejected right parenthesis space and space straight a squared space equals space 1 space and space straight b squared space equals 3 therefore space straight x squared over 1 minus straight y squared over 3 space equals 1 Equation space of space tangent space is space fraction numerator square root of 2 straight x over denominator 1 end fraction minus fraction numerator square root of 3 straight y over denominator 3 end fraction space equals 1 Hence space left parenthesis 2 square root of 2 comma 3 space square root of 3 right parenthesis space satisfy space it.

    Question 346
    CBSEENMA11015543
    Wired Faculty App

    Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such
    that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀x,y ∈ R,then sum from straight n space equals 1 to 10 of space straight f space left parenthesis straight n right parenthesis is equal to

    • 225

    • 330

    • 165

    • 190
    Solution

    B.

    330

    f(x) = ax2 + bx + c
    f(1) = a + b + c = 3
    Now f(x + y) = f(x) + f(y) + xy
    put y = 1
    f(x + 1) = f(x) + f(1) + x
    f(x + 1) = f(x) + x + 3
    Now
    f(2) = 7
    f(3) = 12
    Now
    Sn = 3 + 7 + 12 + ..... tn ...(1)
    Sn = 3 + 7 + ......tn – 1 + tn ...(2)
    On subtracting (2) from (1)
    tn = 3 + 4 + 5 + ....... upto n termsstraight t subscript straight n space equals space fraction numerator left parenthesis straight n squared space plus 5 straight n right parenthesis over denominator 2 end fraction straight S subscript straight n space equals space straight capital sigma space straight t subscript straight n space equals space space stack sum space fraction numerator left parenthesis straight n squared space plus space 5 straight n right parenthesis over denominator 2 end fraction with space below and space on top straight S subscript straight n space equals space 1 half space open square brackets fraction numerator straight n space left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus fraction numerator 5 straight n space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets straight S subscript 10 space equals space 330

    Question 347
    CBSEENMA11015548
    Wired Faculty App

    The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directives is x= –4, then the equation of the normal to it at (1,3/2) is

    • x + 2y = 4

    • 2y – x = 2

    • 4x – 2y = 1

    • 4x + 2y =7
    Solution

    C.

    4x – 2y = 1

    Eccentricity of ellipse =1/2

    Now, a/e = -4
    ⇒ a = 4 x (1/2) = 2
    therefore, b2 = a2(1-e2)
    = a2 (1-(1/4)) = 3
    straight x squared over 4 space plus space straight y squared over 3 space equals space 1 rightwards double arrow space straight x over 2 space plus space fraction numerator 2 straight y over denominator 3 end fraction space straight x space straight y apostrophe space equals space 0 rightwards double arrow space straight y apostrophe space equals space minus fraction numerator 3 straight x over denominator 4 straight y end fraction straight y apostrophe vertical line subscript left parenthesis 1 comma space 3 divided by 2 right parenthesis end subscript space equals space minus 3 over 4 space straight x 2 over 3 space equals space minus 1 half

    Equation of normal at (1, 3/2)
    y-3/2 = 2(x – 1)
    ⇒ 2y – 3 = 4x – 4
    ⇒ 4x – 2y = 1

    Question 348
    CBSEENMA11015575
    Wired Faculty App

    Statement space minus 1 space colon space sum from straight r space equals space 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space left parenthesis straight n plus 2 right parenthesis 2 to the power of straight n minus 1 end exponent Statement space minus space 2 colon thin space sum from straight r equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space left parenthesis 1 plus straight x right parenthesis to the power of straight n space plus space nx space left parenthesis 1 plus straight x right parenthesis to the power of straight n minus 1 end exponent

    • Statement −1 is false, Statement −2 is true

    • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

    • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

    • Statement − 1 is true, Statement − 2 is false. 
    Solution

    B.

    Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

    sum from straight r space equals 0 to straight n of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space sum from straight r space equals space 0 to straight n of space straight r to the power of straight n straight C subscript straight r space plus space to the power of straight n straight C subscript straight r space equals space sum from straight r space equals space 0 to straight n of space straight r space straight n over straight r space to the power of straight n minus 1 end exponent straight C subscript straight r minus 1 end subscript space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n space equals space 2 to the power of straight n minus 1 end exponent space left parenthesis straight n plus 2 right parenthesis Statement space minus 1 space true sum for space of space left parenthesis straight r plus 1 right parenthesis to the power of straight n straight C subscript straight r space end subscript straight x to the power of straight r space equals space sum for space of straight r to the power of straight n space straight C subscript straight r straight x to the power of straight r space plus space sum for space of to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space straight n space sum from straight r space equals space 0 space to straight n of space to the power of straight n straight C subscript straight r minus 1 end subscript space straight x to the power of straight r space plus space sum from straight r space equals space 0 to straight n of space to the power of straight n straight C subscript straight r straight x to the power of straight r space equals space straight n space straight x space left parenthesis 1 space plus straight x right parenthesis to the power of straight n minus 1 end exponent space plus space left parenthesis 1 plus straight x right parenthesis to the power of straight n substituting space straight x space equals space 1 sum for space of space left parenthesis straight r plus 1 right parenthesis space to the power of straight n straight C subscript straight r space equals space straight n 2 to the power of straight n minus 1 end exponent space plus space 2 to the power of straight n
    Hence Statement −2 is also true and is a correct explanation of Statement −1.
    Question 350
    CBSEENMA11015590
    Wired Faculty App

    The sum of the series 20C0 – 20C1 + 20C2 – 20C3 + …… - ….. + 20C10 is-

    • – 20C10

    • 1 half straight C presuperscript 20 subscript 10

    • 0

    • 20C10

    Solution

    B.

    1 half straight C presuperscript 20 subscript 10

    (1 + x)20 = 20C0 + 20C1x + … + 20C10x10 + …+ 20C20x20
    put x = − 1,
    0 = 20C0 − 20C1 + … − 20C9 + 20C10 − 20C11 + … + 20C20
    0 = 2 (20C0 − 20C1 + … − 20C9) + 20C10
    ⇒ 20C0 − 20C1 + … + 20C10 =1 half straight C presuperscript 20 subscript 10

    Question 351
    CBSEENMA11015612
    Wired Faculty App

    If the expansion in powers of x of the function fraction numerator 1 over denominator left parenthesis 1 minus ax right parenthesis left parenthesis 1 minus bx right parenthesis end fraction is a0 + a1x + a2x2 + a3x3 + … , then an is

    • fraction numerator straight b to the power of straight n minus straight a to the power of straight n over denominator straight b minus straight a end fraction
    • fraction numerator straight a to the power of straight n minus straight b to the power of straight n over denominator straight b minus straight a end fraction
    • fraction numerator straight a to the power of straight n plus 1 end exponent minus straight b to the power of straight n plus 1 end exponent over denominator straight b minus straight a end fraction
    • fraction numerator straight b to the power of straight n plus 1 end exponent minus straight a to the power of straight n plus 1 end exponent over denominator straight b minus straight a end fraction

    Solution

    D.

    fraction numerator straight b to the power of straight n plus 1 end exponent minus straight a to the power of straight n plus 1 end exponent over denominator straight b minus straight a end fraction

    (1-ax)-1(1-bx)-1 = (1+ax+a2x2+.....)(1+bx+b2x2+....)
    therefore coefficient of xn = bn +abn-1 +a2bn-2 +.....+an-1b +an =fraction numerator straight b to the power of straight n plus 1 end exponent minus straight a to the power of straight n plus 1 end exponent over denominator straight b minus straight a end fraction

    Question 353
    CBSEENMA11015614
    Wired Faculty App

    The value of integral subscript 1 superscript straight a left square bracket straight x right square bracket straight f apostrophe left parenthesis straight x right parenthesis space dx comma space straight a space greater than 1,where [x] denotes the greatest integer not exceeding x is

    • af(a) − {f(1) + f(2) + … + f([a])}

    • [a] f(a) − {f(1) + f(2) + … + f([a])}

    • [a] f([a]) − {f(1) + f(2) + … + f(a)}

    • af([a]) − {f(1) + f(2) + … + f(a)}
    Solution

    B.

    [a] f(a) − {f(1) + f(2) + … + f([a])}

    Let a = k + h, where [a] = k and 0 ≤ h < 1
    integral subscript 1 superscript straight a left square bracket straight x right square bracket straight f apostrophe left parenthesis straight x right parenthesis space dx space integral subscript 1 superscript 2 1 straight f apostrophe left parenthesis straight x right parenthesis space dx space plus integral subscript 2 superscript 3 2 straight f apostrophe space left parenthesis straight x right parenthesis dx space plus space....... integral subscript straight k minus 1 end subscript superscript straight k left parenthesis straight k minus 1 right parenthesis space dx space plus integral subscript straight k superscript straight k plus straight h end superscript kf apostrophe left parenthesis straight x right parenthesis space dx
    {f(2) − f(1)} + 2{f(3) − f(2)} + 3{f(4) − f(3)}+…….+ (k−1) – {f(k) − f(k − 1)} + k{f(k + h) − f(k)}
    = − f(1) − f(2) − f(3)……. − f(k) + k f(k + h)
    = [a] f(a) − {f(1) + f(2) + f(3) + …. + f([a])}

    Question 355
    CBSEENMA11015621
    Wired Faculty App

    If xm.yn = (x+y)m+n, then dy/dx is

    • y/x

    • x+y/xy

    • xy

    • x/y
    Solution

    A.

    y/x

    straight x to the power of straight m. straight y to the power of straight n space equals space left parenthesis straight x plus straight y right parenthesis to the power of straight m plus straight n end exponent rightwards double arrow space mln space straight x space plus straight n space lny space equals space left parenthesis straight m plus straight n right parenthesis ln left parenthesis straight x plus straight y right parenthesis therefore space straight m over straight x space plus straight n over straight y dy over dx space equals fraction numerator straight m plus straight n over denominator straight x plus straight y end fraction open parentheses 1 plus dy over dx close parentheses rightwards double arrow space open parentheses straight m over straight x minus fraction numerator straight m plus straight n over denominator straight x plus straight y end fraction close parentheses space equals space open parentheses fraction numerator my minus nx over denominator straight y left parenthesis straight x plus straight y right parenthesis end fraction close parentheses dy over dx rightwards double arrow space dy over dx space equals straight y over straight x
    Question 356
    CBSEENMA11015667
    Wired Faculty App

    The coefficient of the middle term in the binomial expansion in powers of x of (1 +αx)4  and of (1−αx )6  is the same if α equals

    • -5/3

    • 3/5

    • -3/10

    • 10/3
    Solution

    C.

    -3/10

    The coefficient of x in the middle term of expansion of (1+ αx)4= 4C2.α2
    The coefficient of x in middle term of the expansion of 3(1− αx) = 6C3 (−α)3
    According to question
    straight C presuperscript 4 subscript 2 straight alpha squared space equals space to the power of 6 straight C subscript 3 space left parenthesis negative straight alpha right parenthesis cubed fraction numerator 4 factorial over denominator 2 factorial 2 factorial end fraction straight alpha squared space equals space minus space fraction numerator 6 factorial over denominator 3 factorial 3 factorial end fraction straight alpha cubed 6 straight alpha squared space equals space minus space 20 straight alpha cubed straight alpha space equals space minus space 6 over 20 straight alpha space equals space minus 3 over 10

    Question 357
    CBSEENMA11015669
    Wired Faculty App

    If space straight S subscript straight n space equals space sum from straight r equals 0 to straight n of fraction numerator 1 over denominator straight C presuperscript straight n subscript straight r end fraction space and space straight t subscript straight n space equals space sum from straight r equals space 0 to straight n of space straight r over straight C subscript straight r space then comma straight t subscript straight n over straight S subscript straight n space is
    • 1 half straight n
    • 1 half straight n minus 1

    • n-1

    • n
    Solution

    A.

    1 half straight n straight S subscript straight n space equals space sum from straight r space equals 0 to straight n of fraction numerator 1 over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction straight S subscript straight n space equals space sum from straight r space equals 0 to straight n of fraction numerator 1 over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction space left parenthesis because space to the power of straight n straight C subscript straight r space equals straight C presuperscript straight n subscript straight n minus straight r end subscript right parenthesis ns subscript straight n space equals space sum from straight r space equals 0 to straight n of fraction numerator 1 over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction space ns subscript straight n space equals space sum from straight r space equals 0 to straight n of open square brackets fraction numerator straight n minus straight r over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction plus fraction numerator straight r over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction close square brackets ns subscript straight n space equals space sum from straight r space equals 0 to straight n of space fraction numerator straight n minus straight r over denominator straight C presuperscript straight n subscript straight n minus straight r end subscript end fraction space plus space sum from straight r space equals 0 to straight n of space fraction numerator straight r over denominator straight C presuperscript straight n subscript straight r end fraction ns subscript straight n space equals space straight t subscript straight n plus straight t subscript straight n ns subscript straight n space equals space 2 straight t subscript straight n straight t subscript straight n over straight s subscript straight n space equals straight n over 2
    Question 358
    CBSEENMA11015705
    Wired Faculty App

    Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 12 + 2.22 + 32 + 2.42 + 52 + 2.62 + .....
    If B – 2A = 100λ, then λis equal to

    • 496

    • 232

    • 248

    • 464
    Solution

    C.

    248

    A =12 + 2.22 + 32+...2.202 = (12 + 22 + 32+ .....202) + 4(12 +22 + 32 + ......+ 102)= 20 x 21x416 + 4 x 10 x 11x 216 = 2870 + 1540 = 4410B = 12 + 2.22 + 32+...2.402 = (12 + 22 + 32+ ....402) + 4(12 +22 + 32 + ......+ 202) = 40 x 41 x 816 + 4 x 20 x 21 x 416= 22140 + 11480 = 33620⇒  B- 2A = 33620 - 8820 = 24800⇒ 100λ = 24800λ = 248

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