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Mathematics Chapter 2 Polynomials

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    NCERT Solution For Class 9 About 2.html

    Polynomials Here is the CBSE About 2.html Chapter 2 for Class 9 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 9 About 2.html Polynomials Chapter 2 NCERT Solutions for Class 9 About 2.html Polynomials Chapter 2 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 9 About 2.html.

    Question 1
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    Is zero a rational number? Can you write it in the form straight p over straight q commawhere p and q are integers and q ≠ 0?

    Solution
    Yes! zero is a rational number. We can write zero in the form straight p over straight q comma where p and q are integers and q ≠ 0 as follows:  
    0 equals 0 over 1 equals 0 over 2 equals 0 over 3 space e t c..
    Note. Denominator q can also be taken as negative integer.

    Question 2
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    Find six rational numbers between 3 and 4.

    Solution
    There can be infinitely many rational numbers between 3 and 4.
    fraction numerator 3 plus 4 over denominator 2 end fraction equals 7 over 2 fraction numerator 3 plus begin display style 7 over 2 end style over denominator 2 end fraction equals 13 over 4 fraction numerator 3 plus begin display style 13 over 4 end style over denominator 2 end fraction equals 25 over 8 fraction numerator 3 plus begin display style 25 over 8 end style over denominator 2 end fraction equals 49 over 16 fraction numerator 3 plus begin display style 49 over 16 end style over denominator 2 end fraction equals 97 over 32 fraction numerator 3 plus begin display style 97 over 32 end style over denominator 2 end fraction equals 193 over 64
    Thus, six ratonal numbers between 3 and 4 are   7 over 2 comma space 13 over 4 comma space 25 over 8 comma space 49 over 16 comma space 97 over 32 space and space 193 over 64.
    Alter 
    open table attributes columnalign right end attributes row cell 3 equals 3 over 1 equals fraction numerator 3 cross times 7 over denominator 3 cross times 7 end fraction equals 21 over 7 end cell row cell 4 equals 4 over 1 equals fraction numerator 4 cross times 7 over denominator 4 cross times 7 end fraction equals 28 over 7 end cell end table close curly brackets space 6 plus 1 equals 7
    Thus, six rational numbers between 3 and 4 can be taken as  
    22 over 7 comma space 23 over 7 comma space 24 over 7 comma space 25 over 7 comma space fraction numerator 26 over denominator space 7 end fraction space and space 27 over 7.
    Note. This is known as the method of finding rational numbers in one step.
    Question 3
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    Find five rational numbers between  3 over 5 space and space space 4 over 5.

    Solution
    3 over 5 equals fraction numerator 3 cross times 10 over denominator 5 cross times 10 end fraction equals 30 over 50 4 over 5 equals fraction numerator 4 cross times 10 over denominator 5 cross times 10 end fraction equals 40 over 50
    Therefore, five rational numbers between 3 over 4 space a n d space 4 over 5 can be taken as  31 over 50 comma space 32 over 50 comma space 33 over 50 comma space 34 over 50 comma space 35 over 50.
    Question 4
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    State whether the following statements are true or false. Give reasons for your answers.

    (i)     Every natural number is a whole number.
    (ii)    Every integer is a whole number.
    (iii)   Every rational number is a whole number.
    Solution

    (i) True, since the collection of whole numbers contains all natural numbers.
    (ii) False, for example, - 2 is an integer but not a whole number.
    (iii) False, for example, 1 half is a rational number but not a whole number.

    Question 5
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     Find four rational numbers be-tween  3 over 7 space a n d space 5 over 7
    Solution

    Since,
    3 over 7 equals 30 over 70 5 over 7 equals 50 over 70 space
    Four rational numbers between  3 over 7 space a n d space 5 over 7  can be taken as 
    31 over 70 comma space 32 over 70 comma space 33 over 70 space space a n d space 34 over 70 i. e. space 31 over 70 comma space 16 over 35 comma space 33 over 70 space a n d space 17 over 35

    Question 6
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    Find four rational numbers be-tween 1 third space a n d space 4 over 5.

    Solution
    <pre>uncaught exception: <b>file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/3c/da/d4a3938b4d71e2799d196755596e.ini): failed to open stream: Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php at line #12file_put_contents(/home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/web/../../../../../../formulas/3c/da/d4a3938b4d71e2799d196755596e.ini): failed to open stream: Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php line 12<br />#0 [internal function]: _hx_error_handler(2, 'file_put_conten...', '/home/config_ad...', 12, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/sys/io/File.class.php(12): file_put_contents('/home/config_ad...', 'mml=<math xmlns...') #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(48): sys_io_File::saveContent('/home/config_ad...', 'mml=<math xmlns...') #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(112): com_wiris_util_sys_Store->write('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #6 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #7 {main}</pre>
    Hence, four rational numbers can be taken as  1 half comma space 7 over 12 comma space 2 over 3 space and space 3 over 4.
    Question 7
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    Find three rational numbers between 0 and 0.1. Also, find twenty rational numbers between 0 and 0.1.

    Solution
    0 equals 0 over 1 equals fraction numerator 0 cross times 100 over denominator 1 cross times 100 end fraction equals 0 over 100 0.1 equals 1 over 10 equals fraction numerator 1 cross times 10 over denominator 10 cross times 10 end fraction equals 10 over 100
    Thus, three rational numbers between 0 and 0.1 can be taken as
    1 over 100 comma space 2 over 100 space space and space 3 over 100. space straight i. straight e. space 0.01 comma space 0.02 space space and space 0.03
    Again,  
    0 equals 0 over 1 equals fraction numerator 0 cross times 1000 over denominator 1 cross times 1000 end fraction equals 0 over 1000 equals 0.1 equals 1 over 10 equals fraction numerator 1 cross times 100 over denominator 10 cross times 100 end fraction equals 100 over 1000
    Thus, twenty rational numbers between 0 and 0.1 can be taken as 
    1 over 1000 space space 2 over 1000 comma space 3 over 1000 comma space 4 over 1000 comma space 5 over 1000 6 over 1000 comma space 7 over 1000 comma space 8 over 1000 comma 9 over 1000 comma 10 over 1000 11 over 1000 comma space 12 over 1000 comma 13 over 1000 comma 14 over 1000 comma space 15 over 1000 comma 16 over 1000 comma 17 over 1000 comma 18 over 1000 comma 19 over 1000 space a n d space 20 over 1000 comma
    i.e., 0.001,0.002,0.003,0.004,0.005,0.006,0.007, 0.008,0.009,0.01,0.011,0.012,0.013,0.014,0.015, 0.016,0.017, 0.018, 0.019 and 0.02
    Question 8
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    Insert three rational numbers between negative 2 over 5 space and space minus 1 fifth
    Solution
    fraction numerator open parentheses negative begin display style 2 over 5 end style close parentheses plus open parentheses negative begin display style 1 fifth end style close parentheses over denominator 2 end fraction equals negative 3 over 10 fraction numerator open parentheses negative begin display style 2 over 5 end style close parentheses plus open parentheses negative begin display style 3 over 10 end style close parentheses over denominator 2 end fraction equals fraction numerator open parentheses negative begin display style 7 over 10 end style close parentheses over denominator 2 end fraction equals negative 7 over 20 fraction numerator open parentheses negative begin display style 3 over 10 end style close parentheses plus open parentheses negative begin display style 1 fifth end style close parentheses over denominator 2 end fraction equals fraction numerator open parentheses negative begin display style 3 over 10 end style close parentheses plus open parentheses negative begin display style 2 over 10 end style close parentheses over denominator 2 end fraction equals space fraction numerator open parentheses negative begin display style 5 over 10 end style close parentheses over denominator 2 end fraction equals negative 5 over 20 equals negative 1 fourth

    Thus, three rational numbers which can be inserted between negative 2 over 5 space and space minus 1 fifth space are space minus 7 over 20 comma space minus 3 over 10 space and space minus 1 fourth. 
    Question 9
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    Find three rational numbers 2 over 5 space and space space 7 over 8

    Solution
    3 over 5 equals 24 over 40 7 over 8 equals 35 over 40
    Hence, three rational numbers can be taken as
    space space space space space space space space space space space space space space 25 over 40 comma space 26 over 40 comma space 27 over 40 or comma space space space space space space space space space space space space 5 over 8 comma space 13 over 20 comma space 27 over 40
    Question 10
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    Are the following statements true or false? Give reasons for your answers.
    (i) Every whole number is a natural number.
    (ii) Every integer is a rational number.
    (iii) Every rational number is an integer.
    Solution

    (i) False, because zero is a whole number but not a natural number.

    (ii) True, because every integer m can be expressed in the form straight m over 1 comma and so it is a rational number.

    (iii) False, because 3 over 5 is a rational number but not an integer.

    Question 11
    CBSEENMA9001444
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    Find five rational numbers between 1 and 2.

    Solution

    Solution not provided.
    Ans.  7 over 6 comma space 4 over 3 comma space 3 over 2 comma space 5 over 3 space and space 11 over 6

    Question 12
    CBSEENMA9001445
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    Find two rational numbers between 3 over 4 and 5 over 9.

    Solution

    Solution not provided.
    Ans.  101 over 144 comma space 47 over 72

    Question 13
    CBSEENMA9001446
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    Find two rational numbers between 0.1 and 0.2.

    Solution

    Solution not provided.
    Ans.  0.11,  0.15

    Question 14
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    Find three rational numbers between negative 5 over 6 space and space 3 over 8.

    Solution

    Solution not provided.
    Ans.  negative 11 over 48 comma space 7 over 96 comma space 43 over 192

    Question 15
    CBSEENMA9001448
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    Give three rational numbers between - 3 and - 2.

    Solution

    Solution not provided.
    Ans.  negative 5 over 2 comma space minus 11 over 4 comma space minus 9 over 4

    Question 16
    CBSEENMA9001449
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    Write three rational numbers between negative 2 over 5 space a n d space minus 1 fifth.

    Solution

    Solution not provided.
    Ans.  negative 3 over 10 comma space minus 7 over 20 comma space minus 1 fourth

    Question 17
    CBSEENMA9001450
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    State whether the following statements are true or false. Justify your answers.

    (i)    Every irrational number is a real number.
    (ii)   Every point on the number line is of the form square root of straight m comma where m is a natural number.
    (iii)  Every real number is an irrational number.
    Solution

    (i) True, since collection of real numbers is made up of rational and irrational numbers.
    (ii) False, because no negative number can be the square root of any natural number.
    (iii) False, for example, 2 is real but not irrational.

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    Question 19
    CBSEENMA9001452
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    Show how square root of 5 can be represented on the number line.

    Solution
    Representation of square root of 5 on the number line
    Consider a unit square OABC and transfer it onto the number line making sure that the vertex O coincides with zero.
    Then OB = square root of 1 squared plus 1 squared end root equals square root of 2
    Construct BD of unit length perpendicular to OB.
    Then OD = square root of open parentheses square root of 2 close parentheses squared plus 1 squared end root space equals space square root of 3
    Construct BD of unit length perpendicular to OB.
    Then OE = square root of left parenthesis square root of 3 right parenthesis squared plus 1 squared end root space equals space square root of 4 space equals 2
    Construct BD of unit length perpendicular to OB.
    Then OF = square root of 2 squared plus 1 squared end root equals square root of 5
    Using a compass, with centre O and radius OF, draw an arc which intersects the number line in the point R. Then R corresponds to  square root of 5.

    Representation of square root of 5.
    Question 21
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    Locate square root of 8 on the number line.
    Solution

    Solution not provided.

    Question 22
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    Show how square root of 6 spacecan be represented on the number line.

    Solution

    Solution not provided.

    Question 23
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    Every point on the number line corresponds to a _____ number which may be either ___ or______

    Solution

    Real

    ,

    Rational Number

    ,

    Irrational Number

    Question 24
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     0 is a/an ______ number.

    Solution

    rational

    Question 25
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    straight pi is a/an......number.

    Solution

    irrational

    Question 26
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    square root of 4 π is a/an __________ number.

    Solution
    rational
    Question 27
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    square root of 8 is a/an __________ number.

    Solution
    irrational.
    Question 28
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    Write the following in decimal form and say what kind of decimal expansion each has: 

    36 over 100

    Solution
    36 over 100
    36 over 100 equals 0.36
    The decimal expansion is terminating.
    Question 29
    CBSEENMA9001462
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    Write the following in decimal form and say what kind of decimal expansion each has: 

    1 over 11

    Solution
    1 over 11
    11 right parenthesis space 1.000000 space left parenthesis 0.090909.......... space space space space space space bottom enclose thin space thin space thin space 99 thin space thin space thin space thin space thin space end enclose space space space space space space space space space space 100 space space space space space space space space space bottom enclose thin space thin space space thin space 99 thin space thin space space space end enclose space space space space space space space space space space space space space 100 space space space space space space space space space space space space space bottom enclose thin space thin space thin space 99 thin space thin space thin space end enclose space space space space space space space space space space space space space space space space space 1 therefore space space 1 over 11 equals 0.090909 space...... equals space 0. top enclose 09
    The decimal expansion is non-terminating repeating.
    Question 30
    CBSEENMA9001463
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    Write the following in decimal form and say what kind of decimal expansion each has: 

    4 1 over 8

    Solution
    4 1 over 8
    4 1 over 8 equals fraction numerator 4 cross times 8 plus 1 over denominator 8 end fraction equals fraction numerator 32 plus 1 over denominator 8 end fraction equals 33 over 8
    8 space right parenthesis space 33.000 space left parenthesis 4.125 space space space space bottom enclose space 32 space end enclose space space space space space 10 space space space space space bottom enclose thin space space 8 thin space space end enclose space space space space space space 20 space space space space space space space bottom enclose thin space thin space thin space 16 thin space thin space thin space end enclose space space space space space space space space 40 space space space space space space thin space bottom enclose thin space space 40 thin space thin space thin space end enclose space space space space space space space space space space space x therefore space space 4 1 over 8 equals 4.125
    The decimal expansion is terminating.
    Question 31
    CBSEENMA9001464
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    Write the following in decimal form and say what kind of decimal expansion each has: 

    3 over 13

    Solution
    3 over 13
    13 space right parenthesis space 3.00000000000 space left parenthesis 0.23076923079...... space space space space bottom enclose 26 space end enclose space space space space space 40 space space space space space bottom enclose space 39 thin space space space end enclose space space space space space space 100 space space space space space space space bottom enclose thin space thin space space 91 thin space space space end enclose space space space space space space space space 90 space space space space space space thin space bottom enclose thin space space 78 space space space thin space thin space end enclose space space space space space space space space space space space 120 space space space space space space space space space bottom enclose space space space 117 space space space end enclose space space space space space space space space space space space space space space 30 space space space space space space space space space space space space bottom enclose space space 26 space space space space end enclose space space space space space space space space space space space space space space space space space space 40 space space space space space space space space space space space space space space space bottom enclose space space 39 space space space space end enclose space space space space space space space space space space space space space space space space space space space 100 space space space space space space space space space space space space space space space space space bottom enclose space space space space 91 space space space end enclose space space space space space space space space space space space space space space space space space space space space space 90 space space space space space space space space space space space space space space space space space space bottom enclose space space space 78 space space space end enclose space space space space space space space space space space space space space space space space space space space space 120 space space space space space space space space space space space space space space space space space space space bottom enclose space 117 space space end enclose space space space space space space space space space space space space space space space space space space space space space space space 3 therefore space space 3 over 13 equals space 0.230769230769 space...... space equals space 0. top enclose 230769

    The decimal expansion is non-terminating repeating.
    Question 32
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    Write the following in decimal form and say what kind of decimal expansion each has:

    2 over 11

    Solution
    2 over 11
    11 right parenthesis space 2.0000 space left parenthesis space 0.1818 space..... space space space space bottom enclose space 11 space space end enclose space space space space space space space 90 space space space space space bottom enclose space space space 88 space space end enclose space space space space space space space space 20 space space space space space space bottom enclose space space 11 space space end enclose space space space space space space space space space 90 space space space space space space space bottom enclose space space 88 space space end enclose space space space space space space space space space space 2 therefore space space 2 over 11 space equals space 0.1818 space...... space equals space 0. top enclose 18.

    The decimal expansion is non-terminating repeating.
    Question 33
    CBSEENMA9001466
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    Write the following in decimal form and say what kind of decimal expansion each has:

    329 over 400

    Solution
    329 over 400
    400 space right parenthesis space 329.0000 space left parenthesis space 0.8225 space space space space space space bottom enclose space space 3200 space space space end enclose space space space space space space space space space space space space 900 space space space space space space space space space space bottom enclose space space space 800 space space end enclose space space space space space space space space space space space space 1000 space space space space space space space space space space space bottom enclose space space 800 space space end enclose space space space space space space space space space space space space 1000 space space space space space space space space space space space bottom enclose space space 800 space space end enclose space space space space space space space space space space space space space 2000 space space space space space space space space space space space bottom enclose space space 2000 space space end enclose space space space space space space space space space space space space space space space space X space space space space space space space space space space space space therefore space space 329 over 400 space equals space 0.8225

    The decimal expansion is terminating. 
    Question 34
    CBSEENMA9001467
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     You know that 1 over 7 equals 0. top enclose 142857  Can you predict what the decimal expansions of 

    2 over 7 comma space 3 over 7 comma space 4 over 7 comma space 5 over 7 comma space 6 over 7     are, without actually doing the long division? If so, how?



    [Hint: Study the remainders while finding the value of 1 over 7 carefully.]

    Solution

    Yes! We can predict the decimal expansions of 2 over 7 comma space 3 over 7 comma space 4 over 7 comma space 5 over 7 comma space 6 over 7 without actually doing the long division as follows:
                                     downwards arrow
    7 space right parenthesis space 1.0000000 space left parenthesis space 0.1428571..... space space space space bottom enclose space 7 space space end enclose space space space space space 30 space space space space space bottom enclose space 28 space space end enclose space space space space space space space 20 space space leftwards arrow space space space space space space bottom enclose space 14 space space end enclose space space space space space space space space 60 space space space space space space space bottom enclose space 56 space space end enclose space space space space space space space space space 40 space space space space space space space space bottom enclose space 35 space space end enclose space space space space space space space space space space 50 space space space space space space space space space bottom enclose space 49 space space end enclose space space space space space space space space space space space space 10 space space space space space space space space space space space bottom enclose space space 7 space space end enclose space space space space space space space space space space space space 3 space
    To predict the decimal expansion of 2 over 7 locate when the remainder becomes 2 and respective quotient (here it is 2). Then write the new quotient beginning from there using the repeating digits 1, 4, 2, 8, 5, 7.
    therefore space space space space space 2 over 7 space equals space 0. top enclose 285714


    Similarly,
               space space space space space space space space space space 3 over 7 equals space 0. top enclose 428571 space space space space space space space space space space 4 over 7 equals space 0. top enclose 571428 space space space space space space space space space space 5 over 7 equals space 0. top enclose 714285 a n d space space space space space 6 over 7 equals space 0. top enclose 857142
     
    Question 35
    CBSEENMA9001468
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    Express the following in the form straight p over straight q where p and q are integers and q ≠ 0.
    left parenthesis straight i right parenthesis space 0. top enclose 6

    Solution

    Let   straight x space equals space 0. top enclose 6 equals 0.6666.....
    Multiplying both sides by 10 (since one digit is repeating), we get
               10x = 6.666 .......
    rightwards double arrow       10x = 6 + 0.6666 .....
    rightwards double arrow       10x = 6 + x
    rightwards double arrow       10x - x = 6
    rightwards double arrow         9x = 6
    rightwards double arrow space        straight x space equals space 6 over 9
    rightwards double arrow        straight x space equals space 2 over 3
    Thus,     0. top enclose 6 equals 2 over 3
    Here     p =  2
                q = 3 (not equal to 0)

    Question 36
    CBSEENMA9001469
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    Express the following in the form p over q where p and q are integers and q ≠ 0.

    0. top enclose 47

    Solution

    Let x = 0. top enclose 47 = 0.47777 ......
    Multiplying both sides by 10 (since one digit is repeating), we get
                  10x = 4.7777 ......
    rightwards double arrow          10x = 4.3 + 0.47777 .......
    rightwards double arrow          10x = 4.3 + x
    rightwards double arrow   10x - x = 4.3
    rightwards double arrow        9x = 4.3
    rightwards double arrow space space space space straight x space equals space fraction numerator 4.3 over denominator 9 end fraction equals 43 over 90 Thus comma space space space space 0. top enclose 47 equals 43 over 90
    Here,    p = 43
                q = 90 (not equal to 0)
     

    Question 37
    CBSEENMA9001470
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    Express the following in the form straight p over straight q where p and q are integers and q ≠ 0.
    0. top enclose 001

    Solution

    Let x = 0. top enclose 001 equals 0.001001001....
    Multiplying both sides by 1000 (since three digits are repeating), we get
                 1000x = 1.001001 .......
    rightwards double arrow        1000x = 1 + 0.001001001 ........
    rightwards double arrow        1000x = 1 + x
    rightwards double arrow space       1000x - x = 1
    rightwards double arrow         999x = 1
    rightwards double arrow         x space equals space 1 over 999 
    Thus.     0. top enclose 001 equals 1 over 999
    Here,         p = 1
                     q = 999 left parenthesis not equal to 0 right parenthesis

    Question 38
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    Express 0.99999...... in the form straight p over straight q comma Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

    Solution

    Let x = 0.99999......
    Multiplying both sides by 10 (since one digit is repeating), we get
                          10x = 9.9999 .........
    rightwards double arrow                 10x = 9 + 0.99999 ......
    rightwards double arrow                 10x = 9 + x
    rightwards double arrow                 10x - x = 9
    rightwards double arrow                   9x = 9
    rightwards double arrow space               x equals 9 over 9 equals 1
    Thus,          0.99999 ....... = 1 = 1 over 1
    Here,            p = 1
                        q = 1
    Since 0.99999......goes on for ever, so there is no gap between 1 and 0.99999......and hence they are equal.
                

    Question 39
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    What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1 over 17 ? Perform the division to check your answer.

    Solution
    he maximum number of digits in the repeating block of digits in the decimal expansion of 1 over 17 can be 16.



    Thus comma space space 1 over 17 equals 0. top enclose 0588235294117647
    By Long Division, the number of digits in the repeating block of digits in the decimal
    expansion of  1 over 17 equals 16
    ∴ The answer is verified.

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    Question 40
    CBSEENMA9001473
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    Look at several examples of rational numbers in the form straight p over straight q left parenthesis q not equal to 0 right parenthesis commawhere p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

    Solution
    Examples
    left parenthesis straight i right parenthesis space 1 half equals fraction numerator 1 cross times 5 over denominator 2 cross times 5 end fraction equals 5 over 10 equals 0.5 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose left parenthesis ii right parenthesis space 1 fourth equals fraction numerator 1 cross times 25 over denominator 4 cross times 25 end fraction equals 25 over 100 equals 0.25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose squared left parenthesis iii right parenthesis space 3 over 8 equals fraction numerator 1 cross times 125 over denominator 8 cross times 125 end fraction equals 375 over 1000 equals 0.375 space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose cubed left parenthesis iv right parenthesis space 31 over 25 equals fraction numerator 31 cross times 4 over denominator 25 cross times 4 end fraction equals 124 over 100 equals 1.24 space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 5 squared end enclose left parenthesis straight v right parenthesis space 7 over 125 equals fraction numerator 7 cross times 8 over denominator 125 cross times 8 end fraction equals 56 over 1000 equals 0.056 space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 5 cubed end enclose left parenthesis vi right parenthesis space 13 over 20 equals fraction numerator 13 cross times 5 over denominator 20 cross times 5 end fraction equals 65 over 100 equals 0.65 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose squared cross times 5 left parenthesis vii right parenthesis space 29 over 16 equals fraction numerator 29 cross times 625 over denominator 16 cross times 625 end fraction equals 18125 over 10000 equals 1.8125 space space space space space space space space space space space space space space space space space left enclose space straight q equals 2 end enclose to the power of 4
    The property that q must satisfy is that the prime factorisation of q must have only powers of 2 or powers of 5 or both.
    Question 41
    CBSEENMA9001474
    Wired Faculty App

    Write three numbers whose decimal expansions are non-terminating non-recurring.

    Solution

    0.01001 0001 00001.......,
    0.20 2002 20002 200002.......,
    0.003000300003......,

    Question 42
    CBSEENMA9001475
    Wired Faculty App

    Find three different irrational numbers between the rational numbers 5 over 7 and 

    Solution
    7 space right parenthesis space 5.000000 space left parenthesis space 0.714285......... space space space space bottom enclose space space 49 space space space space end enclose space space space space space space space space 10 space space space space space space bottom enclose space space space 7 space space space end enclose space space space space space space space space space 30 space space space space space space space space bottom enclose space space 28 space space end enclose space space space space space space space space space 20 space space space space space space space space bottom enclose space 14 space end enclose space space space space space space space space space space 60 space space space space space space space space bottom enclose space space 56 space space end enclose space space space space space space space space space space space 40 space space space space space space space space space space bottom enclose space space 35 space end enclose space space space space space space space space space space space space space 5 Thus comma space space 5 over 7 equals 0.714258.... equals space 0. top enclose 714285
             11 space right parenthesis space 9.0000 space left parenthesis space 0.8181..... space space space space space bottom enclose space space 88 space space end enclose space space space space space space space space 20 space space space space space space space bottom enclose space 11 space space end enclose space space space space space space space space space 90 space space space space space space space space bottom enclose space space 88 space space end enclose space space space space space space space space space space 20 space space space space space space space space bottom enclose space space 11 space space end enclose space space space space space space space space space space space 9 Thus comma space space 9 over 11 equals 0.8181...... equals 0. top enclose 81 space space space space space space

    Three different irrational numbers between the rational numbers 5 over 7 and 9 over 11 can be taken as

    0.75 075007500075000075......
    0.7670767000767.......
    0.808008000800008.......

    Question 43
    CBSEENMA9001476
    Wired Faculty App

    Classify the following numbers as rational or irrational:


    square root of 23

    Solution
    square root of 23

    Thus,  square root of 23 = 4.795831523......
    ∵ The decimal expansion is non-terminating non-recurring.
    therefore space square root of 23 is an irrational number.
    Question 44
    CBSEENMA9001477
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    square root of 225
    Solution
    square root of 225


    because space square root of 225 space equals space 15 space equals space 15 over 1 open parentheses p over q space f o r m close parentheses
    therefore space square root of 225 is a rational number
    Here,     p = 15
                 q = 1 open parentheses not equal to 0 close parentheses
    Question 45
    CBSEENMA9001478
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    0.3796
    Solution

     0.3796
    ∵ The decimal expansion is terminating.
    ∴ 0.3796 is a rational number.

    Question 46
    CBSEENMA9001479
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    7.478478......
    Solution
    7.478478...... = 7. top enclose 478

    ∵    The decimal expansion is non-terminating recurring.
    ∴ 7.478478......is a rational number.

    Question 47
    CBSEENMA9001480
    Wired Faculty App

    Classify the following numbers as rational or irrational:
    1.101001000100001......

    Solution

    1.101001000100001......
    ∵    The decimal expansion is non-terminating non-recurring.
    ∴ 1.101001000100001...... is an irrational number.

    Question 48
    CBSEENMA9001481
    Wired Faculty App

    Find two irrational numbers between 0.1 and 0.12.

    Solution

    The two irrational numbers between 0.1 and 0.12 can be taken as
    0.1010010001......
    and    0.11010010001.....

    Question 49
    CBSEENMA9001482
    Wired Faculty App

    Find two irrational numbers between 2 and 2.5.

    Solution

    First Method
    The two irrational numbers between 2 and 2.5 can be taken as
    2.101001000100001......
    and 2.201001000100001.......
    Second Method
    Note: If a and b are two distinct positive rational numbers such that ab is not a perfect square of a rational number, then square root of ab is an irrational number lying between a and b.
    ∴ An irrational number between 2 and 2.5
    equals square root of 2 cross times left parenthesis 2.5 right parenthesis end root equals square root of 5
    Similarly, an irrational number between 2 and square root of 5
    equals square root of 2 square root of 5 end root
    ∴ Two irrational numbers between 2 and 2.5 are square root of 5 and square root of 2 square root of 5. end root

    Question 50
    CBSEENMA9001483
    Wired Faculty App

    Give two rational numbers lying between
    0.232 332 333 2 3333 2......    and 0.212 112 111 2 1111......
    Solution

    Two rational numbers lying between
    0.232 332 333 2 3333 2......
    and 0.212 112 111 2 1111......
    can be taken as 2.221 and 2.222.

    Question 51
    CBSEENMA9001484
    Wired Faculty App

    Find one irrational number between the numbers a and b given below:

    a = 0.1111 ........ = 0. top enclose 1
    b = 0.1101.

    Solution
    An irrational number between 0.1111 ....... = 0. top enclose 1 and 0.1101 can be taken as 0.111010010001.
    Question 52
    CBSEENMA9001485
    Wired Faculty App

    Find two rational numbers in the form straight p over straight q between 0.34 344 3444 34444 3 ...... and 0.36 366 3666 36666 3......

    Solution
    Two rational numbers can be taken as 0.35 and 0.355.
    Now comma space 0.35 space equals space 35 over 100 equals 7 over 20 space space space space space space 0.355 space equals space 355 over 1000 equals 71 over 200
    Question 53
    CBSEENMA9001486
    Wired Faculty App

    Express 5. top enclose 347in the form straight p over straight q where p and q are integers and q ≠ 0.

    Solution

    Let space straight x space equals space 5. top enclose 347

    ⇒    x = 5.3474747......

    ⇒ 10x = 53.474747...... ...(1)

    ⇒ 1000x = 5347.474747...... ...(2)

    Subtracting (1) from (2), we get
    990x = 5294
    rightwards double arrow space space straight x space 5294 over 990 rightwards double arrow space space space straight x space equals space 2647 over 495 rightwards double arrow space space space straight p space equals space 2647 comma space space space space straight q space equals space 495 space space left parenthesis not equal to 0 right parenthesis

    Question 54
    CBSEENMA9001487
    Wired Faculty App

    Express 0. top enclose 001 in the form of straight p over straight q where p and q are integers and q ≠ 0.

     
    Solution

    Let straight x equals 0. top enclose 001

    Then, x = 0.001 001 001 001...... ...(1)
    ⇒ 1000x = 1.001 001 001 001...... ...(2)
    Subtracting (1) from (2), we get
    99x = 1
    rightwards double arrow space space space space space space space space straight x equals 1 over 99
    rightwards double arrow space space space space space space space space straight p equals 1 comma space space space straight q space equals space 999 space left parenthesis not equal to 0 right parenthesis

    Question 55
    CBSEENMA9001488
    Wired Faculty App

    Express 5. top enclose 2 in the form of straight p over straight q comma where p and q are integers and q ≠ 0.
    Solution
    Let straight x equals 5. top enclose 2
    ⇒ x = 5.2222...... ...(1)
    ⇒ 10x = 52.2222...... ...(2)
    Subtracting (1) from (2), we get
    9x = 47
    straight x equals 47 over 9 rightwards double arrow space space space space straight p space equals space 47 comma space space straight q space equals space 9 space left parenthesis not equal to space 0 right parenthesis
    Question 56
    CBSEENMA9001489
    Wired Faculty App

    How many irrational numbers lie between square root of 2 and square root of 3 ? Find any three irrational numbers lying between square root of 2 and square root of 3.

    Solution

    Infinitely many irrational numbers lie between square root of 2 and square root of 3.

    One irrational number between square root of 2 and square root of 3 is
    square root of square root of 2 space square root of 3 end root space equals square root of square root of 6 end root space equals space 1 over 6 to the power of 4
    Another irrational number between square root of 2 and square root of 3 is square root of square root of 2 space 1 over 6 to the power of 4 end root
    equals 1 over 2 to the power of 4.1 over 6 to the power of 8 equals fraction numerator 1 over denominator 2 to the power of 4. end fraction fraction numerator 1 over denominator 2 to the power of 8. end fraction 1 over 3 to the power of 8 equals space 1 over 2 to the power of 4 plus 1 over 8.1 over 3 to the power of 8 equals 2 to the power of 3 over 8 end exponent.3 to the power of 1 over 8 end exponent
    Third irrational number between square root of 2 and square root of 3 is 
    square root of square root of 2.2 to the power of 3 over 8 end exponent.3 to the power of 1 over 8 end exponent end root equals 2 to the power of 1 fourth end exponent.2 to the power of 3 over 16 end exponent.3 to the power of 1 over 16 end exponent equals 2 to the power of 7 over 16 end exponent.3 to the power of 1 over 16 end exponent

    Question 57
    CBSEENMA9001490
    Wired Faculty App

    Find two irrational numbers between 1 over 7 and 2 over 7.

    Solution

    One irrational number is square root of 1 over 7 cross times 2 over 7 end root equals fraction numerator square root of 2 over denominator 7 end fraction

    Another irrational number is square root of 1 over 7 cross times fraction numerator square root of 2 over denominator 7 end fraction end root equals 2 to the power of begin display style 1 fourth end style end exponent over 7.

    Question 58
    CBSEENMA9001491
    Wired Faculty App

     Find the decimal expansions of 10 over 3 comma space 7 over 8 space a n d space 1 over 7 and 

    Solution

    Solution not provided.
    Ans. 3.3333...... = 3. top enclose 3 space colon0.875; 0.142857  142857 .......

    Question 60
    CBSEENMA9001493
    Wired Faculty App

     Show that 0.3333..... = 0. top enclose 3can be expressed in the form straight p over straight qwhere p and q are integers and q ≠ 0.

    Solution

    Solution not provided.
    Ans.  1 third

    Question 61
    CBSEENMA9001494
    Wired Faculty App

    Show that 0.2353535........ equals 0.2 top enclose 35 can be expressed in the form straight p over straight qwhere p and q are integers and q ≠ 0.

    Solution

    Solution not provided.
    Ans.   233 over 990

    Question 62
    CBSEENMA9001495
    Wired Faculty App

    Find an irrational number betweenspace 1 over 7 and 2 over 7. 

    Solution

    Solution not provided.
    Ans.   0.150150015000150000

    Question 63
    CBSEENMA9001496
    Wired Faculty App

    Express 15.7 top enclose 12 in the form  where straight p over straight q and q are integers and q ≠ 0.

    Solution

    Solution not provided.
    Ans.   1037 over 66

    Question 64
    CBSEENMA9001497
    Wired Faculty App

    Show that 1.3 top enclose 23can be expressed in the form straight p over straight qwhere p and q are integers and q ≠ 0.

    Solution

    Solution not provided.
    Ans.   131 over 99

    Question 65
    CBSEENMA9001498
    Wired Faculty App

     Express 0. top enclose 001 as a rational number in the form  where p and q are integers and q ≠ 0.

    Solution

    Solution not provided.
    Ans.  1 over 999

    Question 66
    CBSEENMA9001499
    Wired Faculty App

    Express 18. top enclose 48 in the form of straight p over straight q where p and q are integers, q ≠ 0.

    Solution

    Solution not provided.
    Ans.  610 over 33

    Question 69
    CBSEENMA9001502
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    2 minus square root of 5

    Solution
    2 minus square root of 5
    because 2 is a rational number and square root of 5 is an irrational number.
    therefore The difference of a rational number and irrantional number is irrational.
    Question 70
    CBSEENMA9001503
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    left parenthesis 3 plus square root of 23 right parenthesis minus square root of 23

    Solution
    left parenthesis 3 plus square root of 23 right parenthesis minus square root of 23
    left parenthesis 3 plus square root of 23 right parenthesis minus square root of 23 equals 3 plus square root of 23 minus square root of 23 equals 3
    which is a rational number.
    Question 71
    CBSEENMA9001504
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    fraction numerator 2 square root of 7 over denominator 7 square root of 7 end fraction

    Solution
    fraction numerator 2 square root of 7 over denominator 7 square root of 7 end fraction equals 2 over 7
    which is a rational number.
    Question 72
    CBSEENMA9001505
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    fraction numerator 1 over denominator square root of 2 end fraction

    Solution
    fraction numerator 1 over denominator square root of 2 end fraction
    because space space 1 space left parenthesis not equal to 0 right parenthesis is a rational number and square root of 2 left parenthesis not equal to 0 right parenthesis is irrational number.
    therefore space space fraction numerator 1 over denominator square root of 2 end fraction is an irrational number.
    because The quotient of a non-zero rational number with an irrational number is irrational.
    Question 73
    CBSEENMA9001506
    Wired Faculty App

    Classify the following numbers as rational or irrational:

    2 straight pi

    Solution
    2 straight pi
    because 2 space left parenthesis not equal to 0 right parenthesis is a rational number and straight pi is an irrational number.
    therefore 2 straight pi is an irrational number.
    because The product of a non-zero rational number with an irrational numbers is irrational.
    Question 74
    CBSEENMA9001507
    Wired Faculty App

    Simplify each of the following expressions:

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

    Solution
    left parenthesis 3 plus square root of 3 right parenthesis space left parenthesis 2 plus square root of 2 right parenthesis
    left parenthesis 3 plus square root of 3 right parenthesis space left parenthesis 2 plus square root of 2 right parenthesis
    left enclose table row cell table row cell Left space Distributive space law space of end cell row cell multiplication space over space addition end cell end table end cell end table end enclose
    equals 3 left parenthesis 2 plus square root of 3 right parenthesis space plus square root of 3 left parenthesis 2 plus square root of 2 right parenthesis equals 6 space plus space 3 square root of 2 plus 2 square root of 3 plus square root of left parenthesis 3 right parenthesis left parenthesis 2 right parenthesis end root 1 space because space space square root of straight a space square root of straight b space equals space square root of ab equals space 6 plus 3 square root of 2 plus 2 square root of 3 plus square root of 6.
    Question 75
    CBSEENMA9001508
    Wired Faculty App

    Simplify each of the following expressions:

    left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 3 minus square root of 3 right parenthesis

    Solution
    left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 3 minus square root of 3 right parenthesis space equals space left parenthesis 3 right parenthesis squared minus left parenthesis square root of 3 right parenthesis squared because space space left parenthesis straight a plus square root of straight b right parenthesis space left parenthesis straight a minus square root of straight b right parenthesis end root equals straight a squared minus straight b space space space space space equals space 9 minus 3 space equals space 6
    Question 76
    CBSEENMA9001509
    Wired Faculty App

    Simplify each of the following expressions:

    left parenthesis square root of 5 plus square root of 2 right parenthesis squared

    Solution
    left parenthesis square root of 5 plus square root of 2 right parenthesis squared
    left parenthesis square root of 5 plus square root of 2 right parenthesis squared space equals open parentheses square root of 5 close parentheses squared plus 2 square root of 5 space square root of 2 plus open parentheses square root of 2 close parentheses squared equals space 5 plus 2 square root of 10 plus 2 because space space space square root of a space square root of b space equals space square root of a b end root equals 7 plus 2 square root of 10.
    Question 77
    CBSEENMA9001510
    Wired Faculty App

    Simplify each of the following expressions:

    left parenthesis space square root of 5 minus square root of 2 right parenthesis space left parenthesis square root of 5 plus square root of 2 right parenthesis

    Solution
    left parenthesis space square root of 5 minus square root of 2 right parenthesis space left parenthesis square root of 5 plus square root of 2 right parenthesis space equals left parenthesis square root of 5 right parenthesis squared minus left parenthesis square root of 2 right parenthesis squared
    because   left parenthesis straight a minus square root of straight b right parenthesis end root left parenthesis straight a plus square root of straight b right parenthesis end root equals straight a squared minus straight b equals space 5 minus 2 space equals space 3
    Question 79
    CBSEENMA9001512
    Wired Faculty App

    Represent square root of 9.3 end root on the number line.

    Solution

    Mark the distance 9.3 from a fixed point A on a given line to obtain a point B such that AB = 9.3 units. From B mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then BD equals square root of 9.3 end root.

    Sponsor Area

    Question 80
    CBSEENMA9001513
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 1 over denominator square root of 7 end fraction

    Solution
    fraction numerator 1 over denominator square root of 7 end fraction fraction numerator 1 over denominator square root of 7 end fraction equals fraction numerator 1 over denominator square root of 7 end fraction cross times fraction numerator square root of 7 over denominator square root of 7 end fraction
    Multiplying and dividing by square root of 7 = fraction numerator square root of 7 over denominator 7 end fraction.
     
    Question 81
    CBSEENMA9001514
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction

    Solution
    fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction equals fraction numerator 1 over denominator square root of 7 minus square root of 6 end fraction cross times fraction numerator square root of 7 plus square root of 6 over denominator square root of 7 plus square root of 6 end fraction
    Multipying and dividing by square root of 7 plus square root of 6

    equals fraction numerator square root of 7 plus square root of 6 over denominator 7 minus 6 end fraction equals square root of 7 plus square root of 6.
    Question 82
    CBSEENMA9001515
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 1 over denominator square root of 5 plus square root of 2 end fraction

    Solution
    fraction numerator 1 over denominator square root of 5 plus square root of 2 end fraction
    fraction numerator 1 over denominator square root of 5 plus square root of 2 end fraction equals fraction numerator 1 over denominator square root of 5 plus square root of 2 end fraction cross times fraction numerator square root of 5 minus square root of 2 over denominator square root of 5 minus square root of 2 end fraction
    Multiplying and dividing by square root of 5 minus square root of 2

    equals fraction numerator square root of 5 minus square root of 2 over denominator 5 minus 2 end fraction equals fraction numerator square root of 5 minus square root of 2 over denominator 3 end fraction
    Question 83
    CBSEENMA9001516
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 1 over denominator square root of 7 minus 2 end fraction equals fraction numerator 1 over denominator square root of 7 minus 2 end fraction cross times fraction numerator square root of 7 plus 2 over denominator square root of 7 plus 2 end fraction

     

    Solution
    fraction numerator 1 over denominator square root of 7 minus 2 end fraction fraction numerator 1 over denominator square root of 7 minus 2 end fraction equals fraction numerator 1 over denominator square root of 7 minus 2 end fraction cross times fraction numerator square root of 7 plus 2 over denominator square root of 7 plus 2 end fraction
    Multiplying and dividing by  square root of 7 plus 2

    equals fraction numerator square root of 7 plus 2 over denominator 7 minus 4 end fraction equals fraction numerator square root of 7 plus 2 over denominator 3 end fraction.
    Question 84
    CBSEENMA9001517
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    square root of 4

    Solution
    square root of 4 = 2, which is a rational number.
    Its decimal represenation is 2.0
    Question 85
    CBSEENMA9001518
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    3 square root of 18

    Solution
    3 square root of 18 equals 3 square root of 3 cross times 3 cross times 2 end root equals 3 cross times 3 square root of 2 equals 9 square root of 2
    because space 9 left parenthesis not equal to 0 right parenthesis is a rational number and square root of 2 is an irrational number.
    therefore space 9 square root of 2 is a irrational number.
    therefore the product of a non-zero rational number with an irrational number is irrational .
    therefore space 3 square root of 18 is an irrational number.
    Question 86
    CBSEENMA9001519
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    square root of 1.44 end root

    Solution
    square root of 1.44 end root
    square root of 1.44 end root space equals space square root of 1.2 space cross times 1.2 end root equals 1.2 equals 12 over 10 equals 6 over 5 which is a rational number.
    Here      p = 6
                 q = 5 left parenthesis not equal to 0 right parenthesis
    Its decimal representation is 1.2
    5 space right parenthesis space 6.0 space left parenthesis space 1.2 space space space bottom enclose space space 5 space space space end enclose space space space space space 10 space space space bottom enclose space space 10 space end enclose space space space space space space straight X
    Question 87
    CBSEENMA9001520
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    square root of 9 over 27 end root

    Solution
    square root of 9 over 27 end root square root of bold 9 over bold 27 end root space equals square root of 1 third end root equals square root of 1 third end root equals fraction numerator 1 over denominator square root of 3 end fraction
    because space 1 space left parenthesis not equal to space 0 right parenthesis is a rational and square root of 3 is an irrational number.
    therefore space space fraction numerator 1 over denominator square root of 3 end fraction is an  irrational number.
    because the quotient of a non-zero rational number with irrational number is irrational.
    Question 88
    CBSEENMA9001521
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    negative square root of 0.64 end root

    Solution
    negative square root of 06.4 end root space equals space minus space square root of 0.8 cross times 0.8 end root equals space minus 0.8 space equals space minus 8 over 10 equals negative 4 over 5
    which is a rational number
    Here      p -4
                 q = 5 left parenthesis not equal to 0 right parenthesis
    Its decimal representation is - 0.8
    5 space right parenthesis space 4. space 0 space left parenthesis space 0. space 8 space space space space bottom enclose 4.0 end enclose space space space space space straight x
    Question 89
    CBSEENMA9001522
    Wired Faculty App

    Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers.

    square root of 100

    Solution
    square root of 100 square root of 100 equals square root of 10 cross times 10 end root equals 10
    which is a rational number.
    Its decimal representation is 10.0.
    Question 91
    CBSEENMA9001524
    Wired Faculty App

    y2 = 9

    y2 = 9 rightwards double arrowy = square root of 9 equals square root of 3 cross times 3 end root equals 3
    which is a rational number.
    Solution
    Question 92
    CBSEENMA9001525
    Wired Faculty App

    straight z squared equals 0.04
    Solution
    straight z squared equals 0.04 space rightwards double arrow space straight z space equals space square root of 0.04 end root equals square root of 0.2 cross times 0.2 end root equals 0.2 equals 2 over 10 equals 1 fifth

    which is a rational number.
    Here    p = 1
    q = 5 (≠ 0)

    Question 93
    CBSEENMA9001526
    Wired Faculty App

    In the following equations, find which variables x, y, z etc., represent rational numbers and which represent irrational numbers:

    straight u squared equals 17 over 4

    Solution
    straight u squared equals 17 over 4
    straight u squared equals 17 over 4
    rightwards double arrow straight u space equals space square root of 17 over 4 end root rightwards double arrow straight u space equals space square root of 17 over 4 end root
    rightwards double arrow space straight u space equals fraction numerator square root of 17 over denominator 2 end fraction rightwards double arrow straight u space equals 1 half cross times square root of 17
    because space space 1 half space left parenthesis not equal to 0 right parenthesis is a rational number and square root of 17 is an irrational number.
    therefore   1 half space left parenthesis not equal to 0 right parenthesis  is a rational number and square root of 17 is an number irrational number.
    therefore space space space space 1 half space cross times square root of 17  is an irrational number.
    because   The product of a non-zero rational number with an irrational number is irrational.
    therefore     u is an  irratinal number.

     
     
    Question 94
    CBSEENMA9001527
    Wired Faculty App

    straight v squared equals 3

    Solution
    straight v squared equals 3
    straight v squared equals 3 space rightwards double arrow space straight v space equals square root of 3
    which is an irrational number.
    Question 96
    CBSEENMA9001529
    Wired Faculty App

    In the following equations, find which variables x, y, z etc., represent rational numbers and which represent irrational numbers:

    straight t squared equals 0.4

    Solution
    straight t squared equals 0.4 rightwards double arrow space space space straight t space equals space square root of 0.4 end root equals 0.63245553203.......
    which is an irrational number since the decimal expansion is non-terminating non-recurring.

    Question 97
    CBSEENMA9001530
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction

    Solution
    fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction equals fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction cross times fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction
    Multiplying the numerator and denominator by square root of 3 minus 1

    equals fraction numerator left parenthesis square root of 3 minus 1 right parenthesis squared over denominator open parentheses square root of 3 close parentheses squared minus left parenthesis 1 right parenthesis squared end fraction equals fraction numerator left parenthesis square root of 3 right parenthesis squared minus 2 left parenthesis square root of 3 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis squared over denominator 3 minus 1 end fraction equals space fraction numerator 3 minus 2 square root of 3 plus 1 over denominator 2 end fraction equals fraction numerator 4 minus 2 square root of 3 over denominator 2 end fraction equals 2 minus square root of 3.
    Question 98
    CBSEENMA9001531
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 3 plus square root of 2 over denominator 3 minus square root of 2 end fraction

    Solution
    fraction numerator 3 plus square root of 2 over denominator 3 minus square root of 2 end fraction fraction numerator 3 plus square root of 2 over denominator 3 minus square root of 2 end fraction equals fraction numerator 3 plus square root of 2 over denominator 3 minus square root of 2 end fraction cross times fraction numerator 3 plus square root of 2 over denominator 3 plus square root of 2 end fraction
    Mul;tiplying the numerator and denominator by 7 minus 4 square root of 3

    equals space fraction numerator left parenthesis 5 plus 2 square root of 3 left parenthesis left parenthesis 7 minus 4 square root of 3 right parenthesis over denominator left parenthesis 7 right parenthesis squared minus left parenthesis 4 square root of 3 right parenthesis squared end fraction equals space fraction numerator 5 left parenthesis 7 minus 4 square root of 3 right parenthesis end root plus 2 square root of 3 left parenthesis 7 minus 4 square root of 3 right parenthesis over denominator 49 minus 48 end fraction equals fraction numerator 35 minus 20 square root of 3 plus 14 square root of 3 minus 24 over denominator 1 end fraction equals 11 minus 6 square root of 3
    Question 99
    CBSEENMA9001532
    Wired Faculty App

    Rationalise the denominators of the following:

    fraction numerator 3 plus square root of 7 over denominator 3 minus 4 square root of 7 end fraction

    Solution
    fraction numerator 3 plus square root of 7 over denominator 3 minus 4 square root of 7 end fraction fraction numerator 3 plus square root of 7 over denominator 3 minus 4 square root of 7 end fraction equals fraction numerator 3 plus square root of 7 over denominator 3 minus 4 square root of 7 end fraction cross times fraction numerator 3 plus square root of 7 over denominator 3 plus 4 square root of 7 end fraction
    Multliplying the numerator and denominator by 3 plus 4 square root of 7

    equals fraction numerator left parenthesis 3 plus square root of 7 right parenthesis left parenthesis 3 plus 4 square root of 7 right parenthesis over denominator left parenthesis 3 right parenthesis squared minus left parenthesis 4 square root of 7 right parenthesis squared end fraction equals fraction numerator 3 left parenthesis 3 plus 4 square root of 7 right parenthesis plus square root of 7 left parenthesis 3 plus 4 square root of 7 right parenthesis over denominator 9 minus 112 end fraction equals space fraction numerator 9 plus 12 square root of 7 plus 3 square root of 7 plus 28 over denominator negative 103 end fraction equals fraction numerator 37 plus 15 square root of 7 over denominator negative 103 end fraction equals negative fraction numerator 37 plus 15 square root of 7 over denominator 103 end fraction
    Question 100
    CBSEENMA9001533
    Wired Faculty App

    Represent square root of 5on the number line.

    Solution
    Mark the distance 5 units from a fixed point A on a given line to obtain a point B such that AB = 5 units. From B, mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D.

    Then, BD equals square root of 5.

    Now, let us treat the line BC as the number line, with B as zero, C as 1, and so on. Draw an arc with centre B and radius BD, which intersects the number line in E.
    Then, E represents square root of 5.

    Question 101
    CBSEENMA9001534
    Wired Faculty App

    Represent square root of 4.7 end root on the number line.

    Solution



    Mark the distance 4.7 units from a fixed point A on a given line to obtain a point B such that AB = 4.7 units. From B, mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then,  BD space equals space square root of 4.7 end root.

    Now, let us treat the line BC as the number line, with B as zero, C as 1, and so on. Draw an arc with centre B and radius BD, which intersects the number line in E.

    Then, E represents square root of 4.7 end root.

    Question 102
    CBSEENMA9001535
    Wired Faculty App

    Check whether 7 square root of 5 comma fraction numerator 7 over denominator square root of 5 comma end fraction comma space square root of 2 plus 21 comma space space straight pi minus 5 are irrational numbers or not.

    Solution

    Sollution not provided.
    All are irrational numbers.

    Question 103
    CBSEENMA9001536
    Wired Faculty App

     Add 2 square root of 2 plus 5 square root of 3 space and space square root of 2 minus 3 square root of 3.

    Solution

    Sollution not provided.
    Ans.  3 square root of 2 plus 2 square root of 3

    Question 104
    CBSEENMA9001537
    Wired Faculty App

    Multiply 6 square root of 5 space by space 2 square root of 5.

    Solution

    Sollution not provided.
    Ans.  60

    Question 105
    CBSEENMA9001538
    Wired Faculty App

     Divide 8 square root of 15 space by space 2 square root of 5.

    Solution

    Sollution not provided.
    Ans.  4 square root of 5

    Question 106
    CBSEENMA9001539
    Wired Faculty App

    Rationalise the denominator of fraction numerator 1 over denominator square root of 2 end fraction.

    Solution

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

    Question 107
    CBSEENMA9001540
    Wired Faculty App

    Rationalise the denominator of fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction.

    Solution

    Sollution not provided.
    Ans.  open parentheses negative 5 over 2 close parentheses space open parentheses square root of 3 plus square root of 5 close parentheses

    Question 108
    CBSEENMA9001541
    Wired Faculty App

    If fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction equals space straight a plus straight b square root of 3 find the values of a and b.
    Solution

    Sollution not provided.
    Ans.  a = 2,   b = -1

    Question 109
    CBSEENMA9001542
    Wired Faculty App

    Simplify the following by rationalising the denominators: fraction numerator 3 over denominator 4 square root of 5 minus square root of 3 end fraction plus fraction numerator 2 over denominator 4 square root of 5 plus square root of 3 end fraction

    Solution

    Sollution not provided.
    Ans.  fraction numerator 20 square root of 5 plus square root of 3 over denominator 77 end fraction

    Question 110
    CBSEENMA9001543
    Wired Faculty App

    Simplify the following by rationalising the denominators: fraction numerator 3 square root of 3 minus 2 square root of 5 over denominator 3 square root of 3 plus 2 square root of 5 end fraction plus fraction numerator square root of 12 over denominator square root of 5 minus square root of 3 end fraction

    Solution

    Sollution not provided.
    Ans.  68 over 7 minus 12 over 7 square root of 5 plus square root of 15

    Question 111
    CBSEENMA9001544
    Wired Faculty App

    Simplify: 12 square root of 18 minus 6 square root of 20 minus 3 square root of 50 plus 8 square root of 45

    Solution

    Sollution not provided.
    Ans.  12 square root of 5 plus 21 square root of 2

    Question 112
    CBSEENMA9001545
    Wired Faculty App

    If straight x equals 3 plus 2 square root of 2 comma space find space square root of straight x plus fraction numerator 1 over denominator square root of straight x end fraction

    Solution

    Sollution not provided.
    Ans.  2 square root of 2

    Question 113
    CBSEENMA9001546
    Wired Faculty App

     Simplify:  square root of 2 left parenthesis square root of 6 minus square root of 8 right parenthesis plus square root of 3 left parenthesis square root of 27 minus square root of 6 right parenthesis

    Solution

    Sollution not provided.
    Ans.  5 plus 2 square root of 3 minus 3 square root of 2

    Question 114
    CBSEENMA9001547
    Wired Faculty App

     Simplify:  fraction numerator 1 plus square root of 2 over denominator square root of 5 plus square root of 3 end fraction plus fraction numerator 1 minus square root of 2 over denominator square root of 5 minus square root of 3 end fraction

    Solution

    Sollution not provided.
    Ans.  square root of 5 minus square root of 6

    Question 115
    CBSEENMA9001548
    Wired Faculty App

    Find the simplified value of fraction numerator 1 over denominator 5 minus 2 square root of 3 end fraction plus fraction numerator 1 over denominator 5 plus 2 square root of 3 end fraction

    Solution

    Sollution not provided.
    Ans.  10 over 13

    Question 116
    CBSEENMA9001549
    Wired Faculty App

    If straight x equals 3 minus 2 square root of 2 then find the value of space space open parentheses straight x squared minus 1 over straight x squared close parentheses.

    Solution

    Sollution not provided.
    Ans.  negative 4 square root of 2

    Question 117
    CBSEENMA9001550
    Wired Faculty App

    If straight x equals fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 minus square root of 2 end fraction comma space space straight y space equals fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction, then find the value of x2 + y2 - 10xy.
    Solution

    Sollution not provided.
    Ans.  88

    Question 118
    CBSEENMA9001551
    Wired Faculty App

    If square root of 5 equals 2.236 space a n d space square root of 10 equals 3.162 comma find the value of open parentheses fraction numerator square root of 10 minus square root of 5 over denominator square root of 2 end fraction close parentheses.
    Solution

    Sollution not provided.
    Ans.  1.31

    Question 119
    CBSEENMA9001552
    Wired Faculty App

    If straight a equals square root of 2 minus 1 comma space find the value of open parentheses a minus 1 over a close parentheses cubed.
    Solution

    Sollution not provided.
    Ans. - 8

    Sponsor Area

    Question 120
    CBSEENMA9001553
    Wired Faculty App

    Simplify the following by rationalising the denominators  fraction numerator 3 over denominator 4 minus square root of 3 end fraction plus fraction numerator 2 over denominator 4 plus square root of 3 end fraction.

    Solution

    Sollution not provided.
    Ans. fraction numerator 20 plus square root of 3 over denominator 9 end fraction

    Question 121
    CBSEENMA9001554
    Wired Faculty App

    Simplify the following by rationalising the denominators  fraction numerator 2 square root of 3 over denominator square root of 3 minus square root of 2 end fraction plus fraction numerator 3 square root of 2 over denominator square root of 3 plus square root of 2 end fraction.

    Solution

    Sollution not provided.
    Ans.   5 square root of 6

    Question 122
    CBSEENMA9001555
    Wired Faculty App

    Find 641/2 
    Solution

    641/2
    (64)1/2 = (82)1/2
    = 82 × 1/2 = 81 = 8.

    Question 123
    CBSEENMA9001556
    Wired Faculty App

    Find 321/5
    Solution

    321/5
    321/5 = (25)1/5
    = 25 × 1/3 = 21= 2.

    Question 124
    CBSEENMA9001557
    Wired Faculty App

    Find 1251/3
    Solution

    1251/3
    1251/3 = (53)1/3
    = 53 × 1/3 = 51 = 5.

    Question 125
    CBSEENMA9001558
    Wired Faculty App

    Find 93/2
    Solution

    93/2
    93/2 = (91/2)3 = {(32)1/2}3
    = (32×1/2)3 = (31)3 = 31×3
    = 33 = 27.

    Question 126
    CBSEENMA9001559
    Wired Faculty App

    Find 93/2
    Solution

    322/5
    322/5 = (25)2/5 = 25×2/5 = 22 = 4.

    Question 127
    CBSEENMA9001560
    Wired Faculty App

    Find 163/4
    Solution

    163/4
    163/4 = (24)3/4 = 24×3/4 = 23 = 8.



    Question 128
    CBSEENMA9001561
    Wired Faculty App

    Find 125-1/3
    Solution

    125-1/3
    125-1/3 = (53)-1/3
    equals 5 to the power of 3 straight X left parenthesis negative 1 divided by 3 right parenthesis end exponent equals 5 to the power of negative 1 end exponent equals 1 fifth

    Question 129
    CBSEENMA9001562
    Wired Faculty App

    Find the value of ‘p’ if 5p-3 × 32p - 8 = 225.

    Solution

    5p - 3 × 32p - 8 = 225
    ⇒ 5p - 3 × 32p - 8 = 52 × 32
    ⇒ p – 3 = 2
    2p - 8 = 2
    ⇒ p = 5

    Question 130
    CBSEENMA9001563
    Wired Faculty App

    Find a point corresponding to 3 plus square root of 2 on the number line.

    Solution
    Draw a number line l. On the number line OA = 3 units, OB = 4 units so that AB = 1 unit. At B, draw BC = 1 unit, perpendicular to l and join AC. By Pythagoras theorem, A C equals square root of 2 Using a compass with centre A and radius AC, draw an arc which intersects

    the number line at the point D. Then, point D represents 3 plus square root of 2.
    Question 131
    CBSEENMA9001564
    Wired Faculty App

    Prove that left parenthesis 3 minus square root of 7 right parenthesis squared is an irrational number.  

    Solution
    left parenthesis 3 minus square root of 7 right parenthesis squared equals space 9 plus 7 minus 2.3. square root of 7 equals space 16 minus 6 square root of 7
    6 square root of 7 is an irrational number as 6 is a non-zero rational number and square root of 7 is an irrational number and the product of a non-zero rational number with an irrational number is irrational.

    Again, 16 is a rational number and   is an irrational number. So, 
    Question 132
    CBSEENMA9001565
    Wired Faculty App

    Write three numbers whose decimal expansions are non-terminating non-recurring.

    Solution

    0.01001 0001 00001.......,
    0.20 2002 20002 200002.......,
    0.003000300003......,

    Question 133
    CBSEENMA9001566
    Wired Faculty App

    Simplify

     2 to the power of 2 over 3 end exponent.2 to the power of 1 fifth end exponent

    Solution
    2 to the power of 2 over 3 end exponent.2 to the power of 1 fifth end exponent 2 to the power of 2 over 3 end exponent.2 to the power of 1 fifth end exponent equals 2 to the power of 2 over 3 plus 1 fifth end exponent equals 2 to the power of fraction numerator 10 plus 3 over denominator 15 end fraction end exponent equals 2 to the power of 13 over 15 end exponent
    Question 134
    CBSEENMA9001567
    Wired Faculty App

    Simplify

     open parentheses 1 over 3 cubed close parentheses to the power of 7

    Solution
    open parentheses 1 over 3 cubed close parentheses to the power of 7
    open parentheses 1 over 3 cubed close parentheses to the power of 7 equals fraction numerator 1 to the power of 7 over denominator left parenthesis 3 cubed right parenthesis to the power of 7 end fraction equals 1 over 3 to the power of 21 equals 3 to the power of negative 21 end exponent
    Question 135
    CBSEENMA9001568
    Wired Faculty App

    Simplify

     11 to the power of 1 half end exponent over 11 to the power of 1 fourth end exponent

    Solution
    11 to the power of 1 half end exponent over 11 to the power of 1 fourth end exponent 11 to the power of 1 half end exponent over 11 to the power of 1 fourth end exponent equals 11 to the power of 1 half minus 1 fourth end exponent equals 11 to the power of 1 fourth end exponent
    Question 136
    CBSEENMA9001569
    Wired Faculty App

    Simplify

    7 to the power of 1 half end exponent.8 to the power of 1 half end exponent

     

    Solution
    7 to the power of 1 half end exponent.8 to the power of 1 half end exponent
    7 to the power of 1 half end exponent.8 to the power of 1 half end exponent space equals space left parenthesis 7.8 right parenthesis to the power of 1 half end exponent equals 56 to the power of 1 half end exponent
    Question 137
    CBSEENMA9001570
    Wired Faculty App

    Find the values of a and b if:


    fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b.

    Solution
    fraction numerator 7 plus 3 square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 7 minus 3 square root of 5 over denominator 3 minus square root of 5 end fraction equals straight a plus square root of 5 straight b. rightwards double arrow space space space fraction numerator left parenthesis 7 plus 3 square root of 5 right parenthesis left parenthesis 3 minus square root of 5 right parenthesis over denominator left parenthesis 3 plus square root of 5 right parenthesis left parenthesis 3 minus square root of 5 right parenthesis end fraction minus fraction numerator left parenthesis 7 minus 3 square root of 5 right parenthesis left parenthesis 3 plus square root of 5 right parenthesis over denominator left parenthesis 3 minus square root of 5 right parenthesis left parenthesis 3 plus square root of 5 right parenthesis end fraction equals straight a plus square root of 5 straight b. rightwards double arrow space space space space space space fraction numerator 21 minus 7 square root of 5 plus 9 square root of 5 minus 15 over denominator 9 minus 5 end fraction minus fraction numerator 21 plus 7 square root of 5 minus 9 square root of 5 minus 15 over denominator 9 minus 5 end fraction equals space 1 space plus space square root of 5 space b rightwards double arrow space fraction numerator 6 plus 2 square root of 5 over denominator 4 end fraction minus fraction numerator 6 minus 2 square root of 5 over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space fraction numerator left parenthesis 6 plus 2 square root of 5 right parenthesis minus left parenthesis 6 minus 2 square root of 5 right parenthesis over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space fraction numerator 4 square root of 5 over denominator 4 end fraction equals a plus square root of 5 b rightwards double arrow space space space square root of 5 space equals a plus square root of 5 b rightwards double arrow space a equals 0 comma space b space equals space 1
    Question 138
    CBSEENMA9001571
    Wired Faculty App

    If straight a equals 7 minus 4 square root of 3 comma find the value of square root of straight a plus fraction numerator 1 over denominator square root of a end fraction.
    Solution
    straight a equals 7 minus 4 square root of 3 equals 4 plus 3 minus 4 square root of 3 equals 2 squared plus left parenthesis square root of 3 right parenthesis squared minus 2 left parenthesis 2 right parenthesis left parenthesis square root of 3 right parenthesis equals space left parenthesis 2 minus square root of 3 right parenthesis squared rightwards double arrow space space space square root of straight a space equals 2 minus square root of 3 space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis space space space space space space fraction numerator 1 over denominator square root of straight a end fraction equals fraction numerator 1 over denominator 2 minus square root of 3 end fraction space space space space space equals space fraction numerator 1 over denominator 2 minus square root of 3 end fraction cross times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction space space space space space equals space fraction numerator 2 plus square root of 3 over denominator 4 minus 3 end fraction space space space space space equals space 2 plus square root of 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis
    From (1) and (2),
    square root of straight a plus fraction numerator 1 over denominator square root of a end fraction equals left parenthesis 2 minus square root of 3 right parenthesis plus left parenthesis 2 plus square root of 3 right parenthesis equals 4
    Question 139
    CBSEENMA9001572
    Wired Faculty App

    Simplify:  left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 2 plus square root of 2 right parenthesis squared.

    Solution
    left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 2 plus square root of 2 right parenthesis squared space space space space space space space space space space space equals left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 4 plus 2 plus 2.2. square root of 2 right parenthesis space space space space space space space space space space space equals space left parenthesis 3 plus square root of 3 right parenthesis left parenthesis 6 plus 4 square root of 2 right parenthesis space space space space space space space space space space space equals space 18 plus 12 square root of 2 plus 6 square root of 3 plus 4 square root of 6
    Question 141
    CBSEENMA9001574
    Wired Faculty App

    If  find the value of straight x squared plus 1 over straight x squared.
    Solution
    straight x squared plus 1 over straight x squared equals open parentheses straight x plus 1 over straight x close parentheses squared minus 2 space space space space space space space space space space space space.... left parenthesis 1 right parenthesis
    x = 2 plus square root of 3
    1 over straight x equals fraction numerator 1 over denominator 2 plus square root of 3 end fraction

    equals fraction numerator 1 over denominator 2 plus square root of 3 end fraction cross times fraction numerator 2 minus square root of 3 over denominator 2 minus square root of 3 end fraction
    equals fraction numerator 2 minus square root of 3 over denominator 4 minus 3 end fraction equals 2 minus square root of 3
    therefore space space space space straight x plus 1 over straight x equals 4
    From (1),
    straight x squared plus 1 over straight x equals left parenthesis 4 right parenthesis squared minus 2 equals 14
     

     
    Question 142
    CBSEENMA9001575
    Wired Faculty App

    Let ‘a’ he a rational number and ‘b’ be an irrational number. Is ‘ab’ necessarily an irrational? Justify your answer with an example.

    Solution

    a = 0 and b = square root of 2
    Then, ab = 0 x square root of 2 equals 0 which is rational.
    Hence, 'ab' is not necessarily an irrational.

    Question 143
    CBSEENMA9001576
    Wired Faculty App

    If straight a equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction space and space straight b space equals fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction, find a2 - b2.
    Solution

    a2 - b2 = (a + b) (a - b)               .......(1)

    straight a plus straight b equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction plus fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction equals fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction plus fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times fraction numerator 3 plus square root of 5 over denominator 3 plus square root of 5 end fraction equals space fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction plus fraction numerator left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction equals space fraction numerator left parenthesis 9 plus 5 minus 6 square root of 5 right parenthesis plus left parenthesis 9 plus 5 plus 6 square root of 5 right parenthesis over denominator 4 end fraction equals space 28 over 4 equals space 7 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis
    straight a minus straight b space equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction minus fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction equals space fraction numerator 3 minus square root of 5 over denominator 3 plus square root of 5 end fraction cross times space fraction numerator 3 minus square root of 5 over denominator 3 minus square root of 5 end fraction minus space fraction numerator 3 plus square root of 5 over denominator 3 minus square root of 5 end fraction cross times space fraction numerator 3 plus square root of 5 over denominator 3 plus square root of 5 end fraction equals fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction minus fraction numerator left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 9 minus 5 end fraction equals space fraction numerator left parenthesis 3 minus square root of 5 right parenthesis squared minus left parenthesis 3 plus square root of 5 right parenthesis squared over denominator 4 end fraction equals space minus fraction numerator 12 square root of 5 over denominator 4 end fraction equals negative 3 square root of 5 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
    From (1), (2) and (3),
    straight a squared minus straight b squared equals space left parenthesis 7 right parenthesis left parenthesis negative 3 square root of 5 right parenthesis equals negative 21 square root of 5
    Question 144
    CBSEENMA9001577
    Wired Faculty App

    If straight x equals 3 plus 2 square root of 2 comma check whether straight x plus 1 over straight x is rational or irrational.
    Solution
    x equals 3 plus 2 square root of 2 space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis
    1 over straight x equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction
    equals fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction cross times fraction numerator 3 minus 2 square root of 2 over denominator 3 minus 2 square root of 2 end fraction equals fraction numerator 3 minus 2 square root of 2 over denominator left parenthesis 3 right parenthesis squared minus left parenthesis 2 square root of 2 right parenthesis squared end fraction equals space fraction numerator 3 minus 2 square root of 2 over denominator 9 minus 8 end fraction equals fraction numerator 3 minus 2 square root of 2 over denominator 1 end fraction equals space 3 minus 2 square root of 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis

    Adding (1) and (2), we get
    straight x plus 1 over straight x equals left parenthesis 3 plus 2 square root of 2 right parenthesis plus left parenthesis 3 minus 2 square root of 2 right parenthesis equals 6
    Question 145
    CBSEENMA9001578
    Wired Faculty App

    If fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals straight a square root of 3 plus straight b square root of 5 comma find a and b.
    Solution
    fraction numerator 2 over denominator square root of 3 plus square root of 5 end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction equals a square root of 3 plus b square root of 5 rightwards double arrow space space space fraction numerator 2 left parenthesis square root of 3 minus square root of 5 right parenthesis over denominator left parenthesis square root of 3 plus square root of 5 right parenthesis left parenthesis square root of 3 minus square root of 5 right parenthesis end fraction plus fraction numerator 5 over denominator square root of 3 minus square root of 5 end fraction space fraction numerator square root of 3 plus square root of 5 over denominator square root of 3 plus square root of 5 end fraction space equals a square root of 3 plus b square root of 5 rightwards double arrow space space space space fraction numerator 2 left parenthesis square root of 3 minus square root of 5 right parenthesis over denominator 3 minus 5 end fraction plus fraction numerator 5 left parenthesis square root of 3 plus square root of 5 right parenthesis over denominator 3 minus 5 end fraction space equals space a square root of 3 plus b square root of 5 rightwards double arrow space space minus left parenthesis square root of 3 minus square root of 5 right parenthesis minus 5 over 2 left parenthesis square root of 3 plus square root of 5 right parenthesis space equals space a square root of 3 plus b square root of 5 equals negative 7 over 2 square root of 3 minus 3 over 2 square root of 5 equals a square root of 3 plus b square root of 5 rightwards double arrow space a equals negative 7 over 2. space space space space space space b equals negative 3 over 2 space
    Question 147
    CBSEENMA9001580
    Wired Faculty App

    Evaluate: fraction numerator square root of 5 plus square root of 2 over denominator square root of 5 minus square root of 2 end fraction comma given that square root of 10 equals 3.162.

    Solution
    fraction numerator square root of 5 plus square root of 2 over denominator square root of 5 minus square root of 2 end fraction space space space space space space space space space
             equals space fraction numerator square root of 5 plus square root of 2 over denominator square root of 5 minus square root of 2 end fraction. fraction numerator square root of 5 plus square root of 2 over denominator square root of 5 plus square root of 2 end fraction equals space fraction numerator left parenthesis square root of 5 plus square root of 2 right parenthesis squared over denominator 5 minus 2 end fraction equals space fraction numerator 5 plus 2 plus 2 square root of 5 square root of 2 over denominator 3 end fraction equals space fraction numerator 7 plus 2 square root of 10 over denominator 3 end fraction equals space fraction numerator 7 plus 2 cross times 3.162 over denominator 3 end fraction equals space fraction numerator 13.324 over denominator 3 end fraction equals space 4.441 top enclose 3
    Question 149
    CBSEENMA9001582
    Wired Faculty App

    Prove that left parenthesis 3 minus square root of 7 right parenthesis squared is an irrational number.

    Solution
    left parenthesis 3 minus square root of 7 right parenthesis squared space space space space space space equals space 9 plus 7 minus 2.3. square root of 7 space space space space space space space equals 16 minus 6 square root of 7 space space space space space space
    6 square root of 7 is an irrational number as 6 is a non-zero rational number and
    square root of 7 is an irrational number and the product of a non-zero rational number with an irrational number is irrational.

    Again, 16 is a rational number and 6 square root of 7 is an irrational number. So, 16 minus 6 square root of 7 is an irrational number since the difference of a rational number and an irrational number is irrational.
    Question 150
    CBSEENMA9001583
    Wired Faculty App

    Evaluate cube root of left parenthesis 343 right parenthesis to the power of negative 2 end exponent end root space space

    Solution
    cube root of left parenthesis 343 right parenthesis to the power of negative 2 end exponent end root space space
    equals space left parenthesis 343 right parenthesis to the power of negative 2 over 3 end exponent equals space left parenthesis 7 cross times 7 cross times 7 right parenthesis to the power of 2 over 3 end exponent equals space left parenthesis 7 cubed right parenthesis to the power of negative 2 over 3 end exponent equals space 7 to the power of 3 x open parentheses negative 2 over 3 close parentheses end exponent equals 7 to the power of negative 2 end exponent equals 1 over 7 to the power of negative 2 end exponent equals space 1 over 49
    Question 152
    CBSEENMA9001585
    Wired Faculty App

    Find the value of ‘p’ if 5p-3 × 32p - 8 = 225.

    Solution

    5p - 3 × 32p - 8 = 225
    ⇒ 5p - 3 × 32p - 8 = 52 × 32
    ⇒ p – 3 = 2
    2p - 8 = 2
    ⇒ p = 5

    Question 153
    CBSEENMA9001586
    Wired Faculty App

    Simplify:    open parentheses 81 over 16 close parentheses to the power of negative begin inline style 3 over 4 end style end exponent cross times open parentheses 25 over 9 close parentheses to the power of negative 3 over 2 end exponent

    Solution
    open parentheses 81 over 16 close parentheses to the power of negative begin inline style 3 over 4 end style end exponent cross times open parentheses 25 over 9 close parentheses to the power of negative 3 over 2 end exponent
    equals open curly brackets open parentheses 3 over 2 close parentheses to the power of 4 close curly brackets to the power of negative 3 over 4 end exponent cross times open curly brackets open parentheses 5 over 2 close parentheses squared close curly brackets to the power of negative 3 over 2 end exponent equals space open parentheses 3 over 2 close parentheses to the power of 4 straight x open parentheses negative 3 over 4 close parentheses end exponent cross times open parentheses 5 over 3 close parentheses to the power of 2 straight x open parentheses negative 1 third close parentheses end exponent equals space open parentheses 3 over 2 close parentheses to the power of negative 3 end exponent cross times open parentheses 5 over 3 close parentheses to the power of 2 straight x open parentheses negative 3 over 2 close parentheses end exponent equals space open parentheses 3 over 2 close parentheses to the power of negative 3 end exponent cross times open parentheses 5 over 3 close parentheses to the power of negative 3 end exponent equals space open parentheses 3 over 2 cross times 5 over 3 close parentheses to the power of negative 3 end exponent equals space open parentheses 5 over 2 close parentheses to the power of negative 3 end exponent equals open parentheses 2 over 5 close parentheses cubed equals space 8 over 125
    Question 154
    CBSEENMA9001587
    Wired Faculty App

    If a = 2, b = 3, then find the values of the following:

    (i) (ab + ba)-1    (ii) (aa + bb)-1
    Solution

    (i)  (ab+ba)-1
        = (23 + 32)-1
        = (8 + 9)-1
        = (17)-1 
       = 1 over 17
    (ii)      (aa + bb)
                     
             = (22 + 33)-1
             
             = (4 + 27)-1
        
             = (31)-1 
               
             equals space 1 over 31
         

    Question 155
    CBSEENMA9001588
    Wired Faculty App

    Prove that   fraction numerator 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 over denominator 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 end fraction equals 7 over 10
    Solution
    fraction numerator 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 over denominator 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 end fraction space space space space space space space space space space space equals space fraction numerator 2 to the power of 28 left parenthesis 2 squared plus 2 to the power of 1 plus 1 right parenthesis over denominator 2 to the power of 29 left parenthesis 2 squared plus 2 to the power of 1 minus 1 right parenthesis end fraction space space space space space space space space space space space equals space fraction numerator 2 squared plus 2 to the power of 1 plus 1 over denominator 2 to the power of 29 minus 28 end exponent left parenthesis 2 squared plus 2 to the power of 1 minus 1 right parenthesis end fraction space space space space space space space space space space space equals space space fraction numerator 4 plus 2 plus 1 over denominator 2 to the power of 1 left parenthesis 4 plus 2 plus negative 1 right parenthesis end fraction space space space space space space space space space space space space equals space fraction numerator 7 over denominator 2 left parenthesis 5 right parenthesis end fraction space space space space space space space space space space space space equals space 7 over 10  
    Question 156
    CBSEENMA9001589
    Wired Faculty App

    Show that


    x to the power of a left parenthesis b minus c right parenthesis end exponent over x to the power of b left parenthesis a minus c right parenthesis end exponent plus open parentheses x to the power of b over x to the power of a close parentheses to the power of c equals 1

    Solution
    straight x to the power of straight a left parenthesis straight b minus straight c right parenthesis end exponent over straight x to the power of straight b left parenthesis straight a minus straight c right parenthesis end exponent plus open parentheses straight x to the power of straight b over straight x to the power of straight a close parentheses to the power of straight c equals straight x to the power of straight a left parenthesis straight b minus straight c right parenthesis minus straight b left parenthesis straight a minus straight c right parenthesis end exponent plus left parenthesis straight x to the power of straight b minus straight a end exponent right parenthesis to the power of straight c equals straight x to the power of av minus ac minus ba plus bc end exponent plus straight x to the power of left parenthesis straight b minus straight a right parenthesis to the power of straight c end exponent equals straight x to the power of negative ac plus bc end exponent plus straight x to the power of bc minus ac end exponent equals space straight x to the power of left parenthesis negative ac plus bc right parenthesis minus left parenthesis bc minus ac right parenthesis end exponent equals space straight x to the power of 0 equals 1
    Question 157
    CBSEENMA9001590
    Wired Faculty App

    Simplify:

       open curly brackets 5 left parenthesis 8 to the power of 1 third end exponent plus 27 to the power of 1 third end exponent right parenthesis cubed close curly brackets to the power of 1 fourth end exponent space space space space space space

    Solution
    open curly brackets 5 left parenthesis 8 to the power of 1 third end exponent plus 27 to the power of 1 third end exponent right parenthesis cubed close curly brackets to the power of 1 fourth end exponent space space space space space space
    equals open square brackets 5 open curly brackets left parenthesis 2 cubed right parenthesis to the power of 1 third end exponent plus left parenthesis 3 cubed right parenthesis close curly brackets close square brackets to the power of 1 fourth end exponent
    equals left parenthesis 5 to the power of 1 5 cubed right parenthesis to the power of 1 fourth end exponent equals left parenthesis 5 to the power of 1 plus 3 end exponent right parenthesis to the power of 1 fourth end exponent equals left parenthesis 5 to the power of 4 right parenthesis to the power of 1 fourth end exponent equals 5 to the power of 4 straight x 1 fourth end exponent equals 5 to the power of 1 equals 5


    Question 158
    CBSEENMA9001591
    Wired Faculty App

    Simplify:

     fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space

    Solution
    fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space space space equals space fraction numerator left parenthesis 3 squared right parenthesis to the power of begin display style 1 third end style end exponent cross times left parenthesis 3 cubed right parenthesis to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 plus 1 third end style end exponent end fraction space space space space space equals space fraction numerator 3 to the power of 2 cross times begin display style 1 third end style end exponent straight x 3 to the power of 3 straight x begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent end fraction space space space space equals space 3 to the power of begin display style 2 over 3 plus 3 over 2 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space space 3 to the power of begin display style 13 over 6 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space 3 squared space space space space equals space 9
    Question 159
    CBSEENMA9001592
    Wired Faculty App

    Show that

    (xa - b)a + b. (xb - c)b + c. (xc - a)c + a = 1
    Solution

    (xa - b)a + b. (xb - c)b + c. (xc - a)c + a
    = x(a - b) (a + b). x(b - c) (b + c). x(c - a) (c + a)
    = xa2 - b2. xb2 - c2. xc2 - a2
    = xa2 - b2 b2 + b2 - c2 + c2 - a2
    = X0
    = 1

    Question 160
    CBSEENMA9001593
    Wired Faculty App

    Write the following in the ascend ing order of their magnitude fourth root of 3 comma space cube root of 2 comma space cube root of 4.

    Solution
    fourth root of 3 equals 3 to the power of 1 fourth end exponent equals 3 to the power of 3 over 12 end exponent equals left parenthesis 3 cubed right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 27 right parenthesis to the power of 1 over 12 end exponent cube root of 2 equals 2 to the power of 1 third end exponent equals 2 to the power of 4 over 12 end exponent equals left parenthesis 2 to the power of 4 right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 16 right parenthesis to the power of 1 over 12 end exponent cube root of 4 equals 4 to the power of 1 third end exponent equals 4 to the power of 4 over 12 end exponent equals left parenthesis 4 to the power of 4 right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 256 right parenthesis to the power of 1 over 12 end exponent because space space space 16 less than 27 less than 256 therefore space space left parenthesis 16 right parenthesis to the power of 1 over 12 end exponent less than left parenthesis 27 right parenthesis to the power of 1 over 12 end exponent less than left parenthesis 256 right parenthesis to the power of 1 over 12 end exponent rightwards double arrow space space space space cube root of 2 space less than space fourth root of 3 less than cube root of 4
    Hence, the ascending order is
    cube root of 2 comma space fourth root of 3 comma cube root of 4
    Question 161
    CBSEENMA9001594
    Wired Faculty App

    Simplify :

    172.175
    Solution

    Solution is not provided.
    Ans.    177

    Question 162
    CBSEENMA9001595
    Wired Faculty App

    Simplify :

    (52)7 
    Solution

    Solution is not provided.
    Ans. 514

    Question 163
    CBSEENMA9001596
    Wired Faculty App

    Simplify :

    23 to the power of 10 over 23 to the power of 7 space space
    Solution

    Solution is not provided.
    Ans. 233

    Question 164
    CBSEENMA9001597
    Wired Faculty App

    Simplify :

    23 to the power of 10 over 23 to the power of 7
    Solution

    Solution is not provided.
    Ans. 233

    Question 165
    CBSEENMA9001598
    Wired Faculty App

    Simplify :

    73.93
    Solution

    Solution is not provided.
    Ans. 633

    Question 166
    CBSEENMA9001599
    Wired Faculty App

    Simplify :

    172.17-5 
    Solution

    Solution is not provided.
    Ans. 1 over 17 cubed

    Question 167
    CBSEENMA9001600
    Wired Faculty App

    Simplify :

    (52)-7
    Solution

    Solution is not provided.
    Ans. 5-14

    Question 168
    CBSEENMA9001601
    Wired Faculty App

    Simplify :

    23 to the power of negative 10 end exponent over 23 to the power of 7
    Solution

    Solution is not provided.
    Ans. 23-17

    Question 169
    CBSEENMA9001602
    Wired Faculty App

    Simplify :

    (7)-3 (9)-3
    Solution

    Solution is not provided.
    Ans. 63-3

    Question 170
    CBSEENMA9001603
    Wired Faculty App

    Simplify :

    2 to the power of 2 over 3 end exponent.2 to the power of 1 third end exponent
    Solution

    Solution is not provided.
    Ans. 2

    Question 171
    CBSEENMA9001604
    Wired Faculty App

    Simplify :

    open parentheses 3 to the power of 1 third end exponent close parentheses to the power of 4
    Solution

    Solution is not provided.
    A n s. space space 3 to the power of 4 over 5 end exponent

    Question 172
    CBSEENMA9001605
    Wired Faculty App

    Simplify :

    7 to the power of begin display style 1 fifth end style end exponent over 7 to the power of begin display style 1 third end style end exponent
    Solution

    Solution is not provided.
     A n s. space space space space 7 to the power of fraction numerator negative 2 over denominator 15 end fraction end exponent

    Question 173
    CBSEENMA9001606
    Wired Faculty App

    Simplify :

    13 to the power of bevelled 1 fifth end exponent.17 to the power of bevelled 1 fifth end exponent
    Solution

    Solution is not provided.
     A n s. space space space space space 221 to the power of 1 fifth end exponent

    Question 174
    CBSEENMA9001607
    Wired Faculty App

    Simplify:  fraction numerator left parenthesis 25 right parenthesis to the power of begin display style 3 over 2 end style end exponent cross times left parenthesis 343 right parenthesis to the power of begin display style 1 fifth end style end exponent over denominator 16 to the power of begin display style 5 over 4 end style end exponent cross times 8 to the power of begin display style 4 over 3 end style end exponent cross times 7 to the power of begin display style 3 over 5 end style end exponent end fraction

    Solution

    Solution not provided.
    Ans.  125 over 512

    Question 175
    CBSEENMA9001608
    Wired Faculty App

    Find the value of x if 24 × 25 = (23)x 

    Solution

    Solution not provided.
    Ans.  3

    Question 176
    CBSEENMA9001609
    Wired Faculty App

    Simplify:

    open parentheses 64 over 125 close parentheses to the power of negative 2 over 3 end exponent
    Solution

    Solution not provided.
    Ans. space space space space space space space 25 over 16

    Question 177
    CBSEENMA9001610
    Wired Faculty App

    Find the value of:

    fraction numerator 4 over denominator left parenthesis 216 right parenthesis to the power of negative begin display style 2 over 3 end style end exponent end fraction minus fraction numerator 1 over denominator left parenthesis 256 right parenthesis to the power of negative begin display style 3 over 4 end style end exponent end fraction

    Solution

    Solution not provided.
    Ans.   80

    Question 178
    CBSEENMA9001611
    Wired Faculty App

    Simplify:

    open parentheses 15 to the power of begin display style 1 third end style end exponent over 9 to the power of begin display style 1 fourth end style end exponent close parentheses to the power of negative 6 end exponent

    Solution

    Solution not provided.
    Ans. space space space 27 over 225

    Question 179
    CBSEENMA9001612
    Wired Faculty App

    Simplify:

    open parentheses 15 to the power of begin display style 1 fourth end style end exponent over 3 to the power of begin display style 1 half end style end exponent close parentheses to the power of negative 2 end exponent

    Solution

    Solution not provided.
    Ans. space space space space square root of 3 over 5 end root

    Question 180
    CBSEENMA9001613
    Wired Faculty App

    Prove that:   

    open parentheses 64 over 125 close parentheses to the power of negative 2 over 3 end exponent plus 1 over open parentheses begin display style 256 over 625 end style close parentheses to the power of begin display style 1 fourth end style end exponent plus fraction numerator square root of 25 over denominator cube root of 64 end fraction

    Solution

    Solution not provided.
    Ans. space space space space 65 over 16

    Question 181
    CBSEENMA9001623
    Wired Faculty App

    If ‘x’ is a positive real number and exponents are rational numbers, simplify:

    open parentheses straight x to the power of straight b over straight x to the power of straight c close parentheses to the power of left parenthesis b plus c plus negative a right parenthesis end exponent cross times open parentheses x to the power of c over x to the power of a close parentheses to the power of left parenthesis c plus a minus b right parenthesis end exponent cross times open parentheses x to the power of a over x to the power of b close parentheses to the power of left parenthesis a plus b plus negative c right parenthesis end exponent

    Solution

    Solution not provided.
    Ans. 1

    Question 182
    CBSEENMA9001624
    Wired Faculty App

    Which of the following is a rational number?
    • square root of 5
    • straight pi
    • 0.101 001 0001 00001......
    • 0.85 385 3853 ......

    Solution

    D.

    0.85 385 3853 ......
    Question 183
    CBSEENMA9001625
    Wired Faculty App

    2. π is:
    • a rational number
    • an integer
    •  an irrational number

    • a whole number  
    Solution

    C.

     an irrational number
    Question 184
    CBSEENMA9001626
    Wired Faculty App

    The decimal form of 56 over 1000 space i s
    • 0.56  
    • 0.056
    • 0.0056  
    • 5.6

    Solution

    B.

    0.056
    Question 185
    CBSEENMA9001627
    Wired Faculty App

    The value of fourth root of left parenthesis 64 right parenthesis to the power of negative 2 end exponent end root is:
    • 1 over 8
    • 1 half

    • 8

    • 1 over 64

    Solution

    A.

    1 over 8
    Question 187
    CBSEENMA9001629
    Wired Faculty App

    The value of fourth root of cube root of 2 squared end root end root is equal to:

    • 2 to the power of negative 1 over 6 end exponent
    • 2 to the power of negative 6 end exponent
    • 2 to the power of 1 over 6 end exponent

    • 26 
    Solution

    C.

    2 to the power of 1 over 6 end exponent
    Question 188
    CBSEENMA9001630
    Wired Faculty App

    Which one of the following is an irrational number?

    • 0.14

    • 0.1416 with rightwards harpoon with barb upwards on top
    • 0. top enclose 1416

    • 0.401 4001 4 00014......
    Solution

    D.

    0.401 4001 4 00014......

    Question 189
    CBSEENMA9001631
    Wired Faculty App

    left parenthesis 5 plus square root of 5 right parenthesis open parentheses 5 minus square root of 5 close parentheses on simplification gives:

    • 20

    • 2 square root of 5

    • 10

    • 25
    Solution

    A.

    20

    Question 190
    CBSEENMA9001632
    Wired Faculty App

    The simplest rationalisation factor of square root of 50 is:
    • 5 square root of 2
    • square root of 2

    • 50

    • square root of 50

    Solution

    B.

    square root of 2
    Question 191
    CBSEENMA9001633
    Wired Faculty App

    The value of cube root of 216 minus cube root of 125 space i s colon

    • 1

    • 0

    • 2

    • -1
    Solution

    A.

    1

    Question 192
    CBSEENMA9001634
    Wired Faculty App

    Which of the following is irrational number?

    • 0.15

    • 0.15 top enclose 16
    • 0. top enclose 1516

    • 0.501 5001 50001 ......
    Solution

    D.

    0.501 5001 50001 ......

    Question 193
    CBSEENMA9001635
    Wired Faculty App

    straight p over straight q form of the number 0. top enclose 3 is:
    • 3 over 10
    • 3 over 100
    • 1 third
    • 1 half

    Solution

    C.

    1 third
    Question 195
    CBSEENMA9001637
    Wired Faculty App

    Add 5 square root of 2 plus 3 square root of 3 space and space 2 square root of 2 minus 5 square root of 3

    • 7 square root of 2 minus 2 square root of 3
    • 6 square root of 2 minus 3 square root of 3
    • 6 square root of 2 minus 8 square root of 3
    • 6 square root of 2 plus 8 square root of 3

    Solution

    A.

    7 square root of 2 minus 2 square root of 3
    Question 196
    CBSEENMA9001638
    Wired Faculty App

    the value of fraction numerator 2 to the power of 0 cross times 7 to the power of 0 over denominator 5 to the power of 0 end fraction is:

    • 1

    • 0

    • 9 over 5
    • 1 fifth

    Solution

    A.

    1

    Question 197
    CBSEENMA9001639
    Wired Faculty App

    cube root of 54 over 250 end root equals:
    • 9 over 25
    • 3 over 5
    • 27 over 125
    • fraction numerator cube root of 2 over denominator 5 end fraction

    Solution

    B.

    3 over 5
    Question 198
    CBSEENMA9001640
    Wired Faculty App

    The decimal expansion of square root of 2 is:
    •  finite decimal
    • 1.4121
    • non-terminating recurring
    • non-terminating non-recurring

    Solution

    D.

    non-terminating non-recurring
    Question 200
    CBSEENMA9001642
    Wired Faculty App

    Two rational numbers between 2 over 3 space and space 5 over 3 space are colon
    • 1 over 6 space a n d space 2 over 6
    • 1 half space a n d space 2 over 7
    • 5 over 6 space a n d space 7 over 6
    • 2 over 3 space a n d space 4 over 3

    Solution

    C.

    5 over 6 space a n d space 7 over 6
    Question 202
    CBSEENMA9001644
    Wired Faculty App

    An irrational number between 5 over 7 space a n d space 7 over 9 is:

    • 0.75

    • square root of 6

    • 0.750 7500 75000 .......

    • 0.7512
    Solution

    C.

    0.750 7500 75000 .......

    Question 203
    CBSEENMA9001645
    Wired Faculty App

    Which of the following is a rational number?
    • 1 plus square root of 3
    • straight pi
    • 2 square root of 3

    • 0
    Solution

    D.

    0

    Question 204
    CBSEENMA9001646
    Wired Faculty App

    Simplified value of left parenthesis 16 right parenthesis to the power of 1 fourth end exponent cross times fourth root of 16 is:

    • 16

    • 4

    • 1

    • 0
    Solution

    C.

    1

    Question 205
    CBSEENMA9001647
    Wired Faculty App

    Simplified value of

    open parentheses 25 close parentheses to the power of 1 third end exponent cross times open parentheses 5 close parentheses to the power of 1 third end exponent is 

    • 25

    • 3

    • 1

    • 5
    Solution

    D.

    5

    Question 206
    CBSEENMA9001648
    Wired Faculty App

    When 15 square root of 15 is divided by 3 square root of 3,  the quotient is:

    • 5 square root of 3
    • 3 square root of 5
    • 5 square root of 5
    • 3 square root of 3            

    Solution

    C.

    5 square root of 5
    Question 207
    CBSEENMA9001649
    Wired Faculty App

    If square root of 3 equals 1.732 space and space square root of 2 equals 1.414 comma space the value of  fraction numerator 1 over denominator square root of 3 minus square root of 2 end fraction space is colon

    • 0.318

    • 3.146

    • fraction numerator 1 over denominator 3.146 end fraction
    • square root of 1.732 end root minus square root of 1.414 end root

    Solution

    B.

    3.146

    Question 208
    CBSEENMA9001650
    Wired Faculty App

    A rational number lying between square root of 2 space and space square root of 3 is.
    • fraction numerator square root of 2 plus square root of 3 over denominator 2 end fraction
    • square root of 6
    • 1.6
    • 1.9

    Solution

    C.

    1.6
    Question 209
    CBSEENMA9001651
    Wired Faculty App

    If square root of straight x is irrational number, then x is:
    • rational  
    • irrational
    • 0    
    • real

    Solution

    D.

    real
    Question 211
    CBSEENMA9001653
    Wired Faculty App

    The simplified form of  13 to the power of begin display style 1 fifth end style end exponent over 13 to the power of begin display style 1 third end style end exponent is :
    • 13 to the power of 2 over 15 end exponent
    • 13 to the power of 8 over 15 end exponent
    • 13 to the power of 1 third end exponent
    • 13 to the power of negative 2 over 15 end exponent

    Solution

    D.

    13 to the power of negative 2 over 15 end exponent
    Question 212
    CBSEENMA9001654
    Wired Faculty App

    Every rational number is:
    • a natural number
    • an integer
    • a real number
    • a whole number

    Solution

    C.

    a real number
    Question 213
    CBSEENMA9001655
    Wired Faculty App

    Value of fraction numerator 1 over denominator square root of 18 minus square root of 32 end fractionis equal to:
    • square root of 2
    • negative square root of 2
    • fraction numerator 1 over denominator square root of 2 end fraction
    • negative fraction numerator 1 over denominator square root of 2 end fraction

    Solution

    D.

    negative fraction numerator 1 over denominator square root of 2 end fraction
    Question 214
    CBSEENMA9001656
    Wired Faculty App

    Rationalisation of the denominator of fraction numerator 1 over denominator square root of 5 plus square root of 2 end fraction gives

    • fraction numerator 1 over denominator square root of 10 end fraction
    • square root of 5 plus square root of 2
    • square root of 5 minus square root of 2
    • fraction numerator open parentheses square root of 5 minus square root of 2 close parentheses over denominator 3 end fraction

    Solution

    D.

    fraction numerator open parentheses square root of 5 minus square root of 2 close parentheses over denominator 3 end fraction
    Question 215
    CBSEENMA9001657
    Wired Faculty App

    If straight x to the power of 1 over 12 end exponent equals 49 to the power of 1 over 24 end exponent then x is a equal to:

    • 49

    • 2

    • 12

    • 7
    Solution

    D.

    7

    Question 216
    CBSEENMA9001658
    Wired Faculty App

    Which of the following numbers is an irrational number?
    • square root of 16 minus 4 end root
    • left parenthesis 3 minus square root of 3 right parenthesis left parenthesis 3 plus square root of 3 right parenthesis
    • square root of 5 plus 3
    • negative square root of 25

    Solution

    C.

    square root of 5 plus 3
    Question 217
    CBSEENMA9001659
    Wired Faculty App

    A terminating decimal is:
    • natural number 
    • a whole number
    • a rational number
    • an integer

    Solution

    C.

    a rational number
    Question 218
    CBSEENMA9001660
    Wired Faculty App

    The rational number between negative 1 fifth space and space minus 2 over 5 space is space colon

    • 0

    • negative 1 fourth
    • negative 3 over 10
    • negative 7 over 25

    Solution

    C.

    negative 3 over 10
    Question 219
    CBSEENMA9001661
    Wired Faculty App

    The value of square root of left parenthesis 3 to the power of negative 2 end exponent end root right parenthesis space  is :
    • 1 over 9

    • 9

    • - 3

    • 1 third

    Solution

    D.

    1 third
    Question 220
    CBSEENMA9001662
    Wired Faculty App

    The value of left parenthesis 243 right parenthesis to the power of 1 fifth end exponent is equal to:

    • 5

    • 3

    • 6

    • 1
    Solution

    B.

    3

    Question 221
    CBSEENMA9001663
    Wired Faculty App

    The value of fraction numerator 2 to the power of 0 plus 7 to the power of 0 over denominator 5 to the power of 0 end fraction is

    • 2

    • 0

    • 9 over 5
    • 1 fifth

    Solution

    A.

    2

    Question 222
    CBSEENMA9001664
    Wired Faculty App

    Which of the following numbers is an irrational number?
    • square root of 23
    • square root of 225

    • 0.3796

    • 7. top enclose 478

    Solution

    A.

    square root of 23
    Question 223
    CBSEENMA9001665
    Wired Faculty App

    Which of the following is an irrational number? 
    • 3. top enclose 3

    • 3.763

    • 3. top enclose 763

    • 3.1011 0011 0001 .......
    Solution

    D.

    3.1011 0011 0001 .......

    Question 224
    CBSEENMA9001666
    Wired Faculty App

    fraction numerator square root of 12 over denominator square root of 3 end fraction is equal to 

    • 1

    • 2

    • 3

    • 12
    Solution

    B.

    2

    Question 225
    CBSEENMA9001667
    Wired Faculty App

    fraction numerator cube root of 135 over denominator cube root of 5 end fraction is equal to

    • 1

    • 2

    • 3

    • 4
    Solution

    C.

    3

    Question 226
    CBSEENMA9001668
    Wired Faculty App

    If 8 to the power of x equals 64 over 2 to the power of x then the value of x is 

    • 3

    • 1

    • 1 half
    • 3 over 2

    Solution

    D.

    3 over 2
    Question 227
    CBSEENMA9001669
    Wired Faculty App

    (0.001)1/3 is equal to
    • 0.1 
    • 0.001
    • 0.01 
    • 0.0001

    Solution

    A.

    0.1 
    Question 228
    CBSEENMA9001670
    Wired Faculty App

    If a = 2 and b = 3, then the value of ba is

    • 4

    • 9

    • 2

    • 3
    Solution

    B.

    9

    Question 229
    CBSEENMA9001671
    Wired Faculty App

    If xa/b = 1, then the value of a is

    • 0

    • 1

    • - 1

    • none of these.

    Solution

    A.

    0

    Question 230
    CBSEENMA9001672
    Wired Faculty App

     8-1/3 is equal to

    • 3

    • 1 third

    • 2

    • 1 half

    Solution

    D.

    1 half
    Question 231
    CBSEENMA9001673
    Wired Faculty App

    The decimal form of  1 over 500 is
    • 0.002   
    • 0.02

    • 0.2

    • 0.005

    Solution

    A.

    0.002   

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