Chapter 3 : Rationalisation - Rd Sharma Solutions for Class 9 Maths CBSE

If you love numbers & have a deep interest in statistics, then Mathematics is probably one of the scoring subject ever. A Class 9th CBSE Mathematics is one of the challenging subject one can undergo as, it involves a lot of crunching of complex numbers, geometrical formulae, diagrams & much more. Hence, to simplify the mathematical complexity, TopperLearning has framed a customise solution that involves Test preparation notes, Textbook solutions, Videos, & several other study material that help the student to memorise the concepts quickly. We have bifurcated our CBSE Class 9 Study Material (Solution) into 3 Different parts namely: 

- CBSE Class 9 Practice Test includes Revision notes, Question bank & Sample Paper 
- TopperLearning Classroom that includes Videos, Ask the Experts & Preparation Tips
- Text Book Solutions, etc

TopperLearning packages involves all the ingredients of CBSE Class 9 Study Material that includes Solved Question Papers, High Animated Videos, Solutions by SME (Subject Matter Expert), CBSE class 9 Preparation Tips, Update to CBSE 9th Mathematics syllabus, Practice books, Reference Materials, etc that help you to score great marks with glorious marks.

Getting a good score is now relatively easy if you prefer TopperLearning solutions for CBSE Class 9th Mathematics subject. By purchasing our package, you will be accessed to guaranteed success in your examination!

Read  more
Page / Exercise

Chapter 3 - Rationalisation Excercise Ex. 3.1

Question 1
Solution 1
Question 2

Solution 2
Question 3

Solution 3

Question 4

Solution 4

Chapter 3 - Rationalisation Excercise Ex. 3.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8

Solution 8
Question 9

Solution 9
Question 10

Solution 10
Question 11

Solution 11
Question 12

Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32
Solution 32
Question 33
Solution 33

Question 34
Solution 34

Question 35
Solution 35

Question 36
Solution 36

Question 37
Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Simplify:

Solution 44

  

Question 45

Simplify:

Solution 45

  

Chapter 3 - Rationalisation Excercise 3.16

Question 1

begin mathsize 11px style square root of 10 space cross times space square root of 15 space is space equal space to
left parenthesis straight a right parenthesis space 5 square root of 6
left parenthesis straight b right parenthesis space 6 square root of 5
left parenthesis straight c right parenthesis space square root of 30
left parenthesis straight d right parenthesis space square root of 25 end style

Solution 1

begin mathsize 11px style 10 space equals space 5 space straight x space 2
15 space equals space 5 space straight x space 3
therefore space square root of 10 space cross times space square root of 15 space equals space square root of 5 cross times 2 end root space cross times space square root of 5 cross times 3 end root space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space square root of 5 cross times square root of 2 cross times square root of 5 cross times 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 space equals space open parentheses square root of 5 cross times square root of 5 close parentheses space cross times space square root of 2 cross times 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 space space equals space 5 square root of 6
Hence comma space straight c orrect space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 11px style fifth root of 6 space cross times fifth root of 6 space space is space equal space to
left parenthesis straight a right parenthesis space fifth root of 36
left parenthesis straight b right parenthesis space fifth root of 6 cross times 0 end root space
left parenthesis straight c right parenthesis space fifth root of 6 space
left parenthesis straight d right parenthesis space fifth root of 12 end style

Solution 2

begin mathsize 11px style fifth root of 6 space space equals space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent
so space fifth root of 6 space cross times fifth root of 6 space space equals space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent space cross times space open parentheses 6 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 6 cross times 6 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 36 close parentheses to the power of 1 divided by 5 end exponent
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fifth root of 36
Hence comma space correct space option space is space open parentheses straight a close parentheses. end style

Chapter 3 - Rationalisation Excercise 3.17

Question 1

begin mathsize 11px style The space rationalisation space space factor space of space square root of 3 space space is space
left parenthesis straight a right parenthesis space minus square root of 3
left parenthesis straight b right parenthesis space fraction numerator 1 over denominator square root of 3 end fraction
left parenthesis straight c right parenthesis space 2 square root of 3
left parenthesis straight d right parenthesis space minus 2 square root of 3 end style

Solution 1

begin mathsize 11px style Rationalisation space factor space of space any space number space like space space square root of straight a space is space fraction numerator 1 over denominator square root of straight a end fraction space or space fraction numerator 1 over denominator square root of straight a end fraction space is space square root of straight a. space space
So comma space Rationalisation space factor space of space square root of 3 space is space fraction numerator 1 over denominator square root of 3 end fraction.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 11px style The space rationalisation space factor space of space 2 space plus space square root of 3 space is
left parenthesis straight a right parenthesis space 2 minus square root of 3
left parenthesis straight b right parenthesis space square root of 2 plus 3
left parenthesis straight c right parenthesis space square root of 2 minus 3
left parenthesis straight d right parenthesis space square root of 3 minus 2 end style

Solution 2

begin mathsize 11px style Rationalisation space factor space of space any space number space straight a plus-or-minus square root of straight b space is space straight a space minus-or-plus square root of straight b.
So comma space Rationalisation space factor space of space 2 plus square root of 3 space will space be space 2 space minus space square root of 3
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

begin mathsize 11px style If space space straight x space equals space square root of 5 plus 2 comma text   end text then text   x -  end text 1 over straight x text   end text equals
left parenthesis straight a right parenthesis space 2 square root of 5
left parenthesis straight b right parenthesis text  4    end text
left parenthesis straight c right parenthesis space text 2    end text
left parenthesis straight d right parenthesis square root of 5 end style

Solution 3

begin mathsize 11px style straight x equals square root of 5 plus 2
rightwards double arrow 1 over straight x equals fraction numerator 1 over denominator square root of 5 plus 2 end fraction equals fraction numerator 1 over denominator square root of 5 plus 2 end fraction straight space fraction numerator square root of 5 minus 2 over denominator square root of 5 space minus space 2 end fraction equals straight space fraction numerator square root of 5 minus 2 over denominator 5 minus 4 end fraction space equals space fraction numerator square root of 5 minus 2 over denominator 1 end fraction equals square root of 5 minus 2
Now comma space space straight x minus 1 over straight x straight space equals square root of 5 plus 2 minus open parentheses square root of 5 minus 2 close parentheses equals square root of 5 plus 2 minus square root of 5 plus 2 equals 4
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 4

begin mathsize 11px style table attributes columnalign left end attributes row cell If space space fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction equals straight space straight a minus straight b square root of 3 comma straight space then end cell row blank end table
left parenthesis straight a right parenthesis straight space straight a equals 2 comma space straight b equals 1
left parenthesis straight b right parenthesis space straight a equals 2 comma space straight b equals negative 1
left parenthesis straight c right parenthesis space straight a equals negative 2 comma space straight b equals 1
left parenthesis straight d right parenthesis straight a equals straight b equals 1
end style

Solution 4

fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction
Multiplying space and space dividing space by space the space rationalisation space factor
of space denominator comma space we space get
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
equals fraction numerator open parentheses square root of 3 minus 1 close parentheses squared over denominator open parentheses square root of 3 close parentheses squared minus 1 squared end fraction
equals fraction numerator 3 minus 2 square root of 3 plus 1 over denominator 3 minus 1 end fraction
equals fraction numerator 4 minus 2 square root of 3 over denominator 2 end fraction
equals fraction numerator 2 open parentheses 2 minus square root of 3 close parentheses over denominator 2 end fraction
equals 2 minus square root of 3
Comparing space with space straight a minus straight b square root of 3 comma space we space get space straight a equals 2 space and space straight b equals 1.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.

Question 5

begin mathsize 10px style table attributes columnalign left end attributes row cell The text   end text simplest text   end text rationalising text   end text factor text   end text of text    end text cube root of 500 text  is end text end cell row blank end table
left parenthesis straight a right parenthesis text   end text cube root of 2
left parenthesis straight b right parenthesis text   end text cube root of 5
left parenthesis straight c right parenthesis text   end text square root of 3
left parenthesis straight d right parenthesis text  none of these end text end style

Solution 5

begin mathsize 11px style table attributes columnalign left end attributes row cell cube root of 500 equals left parenthesis 500 right parenthesis to the power of bevelled 1 third end exponent equals open parentheses fraction numerator 500 space straight x space 2 over denominator 2 end fraction close parentheses to the power of bevelled 1 third end exponent space equals space open parentheses 1000 over 2 close parentheses to the power of bevelled 1 third end exponent equals space left parenthesis 10 to the power of up diagonal strike 3 end exponent right parenthesis to the power of bevelled fraction numerator 1 over denominator up diagonal strike 3 end fraction end exponent space. space 1 over 2 to the power of bevelled 1 third end exponent rightwards double arrow space 10 space. space 2 to the power of bevelled fraction numerator negative 1 over denominator 3 end fraction end exponent end cell row cell table attributes columnalign left end attributes row cell The space simplest space Rationalisation space factor space of space cube root of 500 straight space end cell row cell after space simplify space it space to space left parenthesis 10.2 to the power of negative bevelled 1 third end exponent right parenthesis space space is space space 2 to the power of bevelled 1 third end exponent space or space cube root of 2 space. end cell end table
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. space space space space space space space end cell end table
end style


Question 6

begin mathsize 11px style The text   end text simplest text   end text Rationalising text   end text Factor text   end text of text    end text square root of 3 plus square root of 5 text  is end text
left parenthesis straight a right parenthesis text   end text square root of 3 minus 5
left parenthesis straight b right parenthesis text  3- end text square root of 5
left parenthesis straight c right parenthesis text   end text square root of 3 text  -  end text square root of 5
left parenthesis straight d right parenthesis text   end text square root of 3 plus square root of 5 end style

Solution 6

begin mathsize 11px style Rationalising straight space factor straight space of straight space any straight space number straight space of straight space kind straight space square root of straight a plus-or-minus square root of straight b space space space is space space square root of straight a minus-or-plus square root of straight b
So comma space for space given space number space square root of 3 plus square root of 5 comma space Rationalising space factor space would space be space square root of 3 minus square root of 5.
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

begin mathsize 11px style The straight space simplest straight space rationalising straight space factor straight space of straight space 2 square root of 5 minus straight space square root of 3 space is
left parenthesis straight a right parenthesis space 2 square root of 5 plus 3
left parenthesis straight b right parenthesis straight space 2 square root of 5 plus square root of 3
left parenthesis straight c right parenthesis straight space square root of 5 space plus space square root of 3
left parenthesis straight d right parenthesis straight space square root of 5 space minus space square root of 3 end style

Solution 7

Error converting from MathML to accessible text.

Question 8

begin mathsize 11px style If text    x= end text fraction numerator 2 over denominator 3 plus square root of 7 end fraction comma text   end text then left parenthesis straight x minus 3 right parenthesis squared equals
left parenthesis straight a right parenthesis text  1 end text
left parenthesis straight b right parenthesis text  3 end text
left parenthesis straight c right parenthesis text  6 end text
left parenthesis straight d right parenthesis text 7 end text end style

Solution 8

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals fraction numerator 2 over denominator 3 plus square root of 7 end fraction equals space fraction numerator 2 over denominator 3 plus square root of 7 end fraction cross times space fraction numerator 3 minus square root of 7 over denominator 3 minus square root of 7 end fraction equals fraction numerator 2 left parenthesis 3 minus square root of 7 right parenthesis over denominator left parenthesis 3 right parenthesis squared minus left parenthesis square root of 7 right parenthesis squared end fraction equals fraction numerator 6 minus 2 square root of 7 over denominator 9 minus 7 end fraction equals fraction numerator 6 minus 2 square root of 7 over denominator 2 end fraction equals 3 minus 2 square root of 7 end cell row blank row cell Now space space space left parenthesis straight x minus 3 right parenthesis squared equals left parenthesis up diagonal strike 3 minus square root of 7 minus up diagonal strike 3 right parenthesis squared equals open parentheses negative square root of 7 close parentheses squared equals 7 end cell end table
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 9

begin mathsize 11px style If text  x = 7+ 4 end text square root of 3 text   and xy = 1, then  end text 1 over straight x squared plus 1 over straight y squared equals
left parenthesis straight a right parenthesis text   end text 64
left parenthesis straight b right parenthesis text  134 end text
left parenthesis straight c right parenthesis text  194 end text
left parenthesis straight d right parenthesis text   end text 1 over 49 end style

Solution 9

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals 7 plus 4 square root of 3 comma space space space space xy equals 1 rightwards double arrow straight y equals 1 over straight x end cell row cell therefore space straight y equals straight space fraction numerator 1 over denominator 7 plus 4 square root of 3 end fraction end cell row cell therefore space straight y space equals space fraction numerator 1 over denominator 7 plus 4 square root of 3 end fraction cross times fraction numerator 7 minus 4 square root of 3 over denominator 7 minus 4 square root of 3 end fraction equals fraction numerator 7 minus 4 square root of 3 over denominator left parenthesis 7 right parenthesis squared minus left parenthesis 4 square root of 3 right parenthesis squared end fraction equals fraction numerator 7 minus 4 square root of 3 over denominator 49 minus 48 end fraction equals 7 minus 4 square root of 3 end cell row blank row cell Now comma space space 1 over straight x squared plus 1 over straight y squared equals fraction numerator straight y squared plus straight x squared over denominator straight x squared straight y squared end fraction equals fraction numerator straight x squared plus straight y squared over denominator left parenthesis xy right parenthesis squared end fraction end cell row cell straight x to the power of straight 2 equals left parenthesis 7 plus 4 square root of 3 right parenthesis squared equals 49 plus 48 plus 56 square root of 3 equals 97 plus 56 square root of 3 end cell row cell straight y to the power of straight 2 equals straight space left parenthesis 7 minus 4 square root of 3 right parenthesis squared equals space 49 plus 48 minus 56 square root of 3 equals 97 minus 56 square root of 3 end cell row cell therefore space straight x to the power of straight 2 plus straight y squared equals 97 plus 56 square root of 3 plus 97 minus 56 square root of 3 equals 194 end cell row cell xy equals 1 end cell end table

therefore space fraction numerator straight x squared plus straight y squared over denominator left parenthesis xy right parenthesis squared end fraction equals space 194 over left parenthesis 1 right parenthesis squared equals 194
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 10

begin mathsize 11px style If text  x + end text square root of 15 equals 4 comma space text  then  x +  end text 1 over straight x equals
left parenthesis straight a right parenthesis text  2 end text
left parenthesis straight b right parenthesis text  4 end text
left parenthesis straight c right parenthesis text  8 end text
left parenthesis straight d right parenthesis text  1 end text end style

Solution 10

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x plus square root of 15 equals 4 end cell row cell rightwards double arrow straight x equals space 4 space minus space square root of 15 rightwards double arrow 1 over straight x equals fraction numerator 1 over denominator 4 minus square root of 15 end fraction end cell row cell 1 over straight x equals straight space fraction numerator 1 over denominator 4 minus square root of 15 end fraction cross times straight space fraction numerator 4 plus square root of 15 over denominator 4 plus square root of 15 end fraction equals fraction numerator 4 plus square root of 15 over denominator left parenthesis 4 right parenthesis squared minus left parenthesis square root of 15 right parenthesis squared end fraction equals fraction numerator 4 plus square root of 15 over denominator 16 minus 15 end fraction equals 4 plus square root of 15 end cell row blank row cell Now comma space straight x space plus space 1 over straight x equals 4 minus square root of 15 plus 4 plus square root of 15 equals 8 end cell row cell Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end cell end table end style

Question 11

begin mathsize 11px style if space straight x space equals space fraction numerator square root of 5 plus square root of 3 over denominator square root of 5 minus square root of 3 end fraction space space and space space straight y equals space fraction numerator square root of 5 minus square root of 3 over denominator square root of 5 plus square root of 3 end fraction space space then space straight x space plus space straight y space plus space xy equals
left parenthesis straight a right parenthesis space 9
left parenthesis straight b right parenthesis space 5
left parenthesis straight c right parenthesis space 17
left parenthesis straight d right parenthesis space 7 end style

Solution 11

Error converting from MathML to accessible text.

Question 12

begin mathsize 10px style text If  x =  end text fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction text   and y =  end text 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 text  then x end text to the power of text 2 end text end exponent plus xy plus straight y squared equals
left parenthesis straight a right parenthesis text  101 end text
left parenthesis straight b right parenthesis text  99 end text
left parenthesis straight c right parenthesis text  98 end text
left parenthesis straight d right parenthesis text  102 end text end style

Solution 12

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x space equals space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction end cell row cell therefore space straight x space equals space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 plus square root of 2 end fraction straight space straight x space fraction numerator square root of 3 minus square root of 2 over denominator square root of 3 minus square root of 2 end fraction space equals space fraction numerator left parenthesis square root of 3 minus square root of 2 right parenthesis squared over denominator 3 minus 2 end fraction equals space 5 space minus space 2 square root of 6 end cell row cell straight y space equals space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 minus square root of 2 end fraction space end cell row cell therefore space straight y equals space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 minus square root of 2 end fraction straight space straight x space fraction numerator square root of 3 plus square root of 2 over denominator square root of 3 plus square root of 2 end fraction space equals space fraction numerator left parenthesis square root of 3 plus square root of 2 right parenthesis squared over denominator 3 minus 2 end fraction equals space 5 space plus space 2 square root of 6 end cell end table

Now comma space straight x to the power of straight 2 space plus space xy space plus space straight y to the power of straight 2
equals left parenthesis 5 space minus space 2 square root of 6 right parenthesis squared plus left parenthesis 5 space minus space 2 square root of 6 right parenthesis left parenthesis 5 space plus space 2 square root of 6 right parenthesis plus left parenthesis 5 space plus space 2 square root of 6 right parenthesis squared
equals open parentheses 25 plus 24 minus 20 square root of 6 close parentheses plus open parentheses 25 minus 24 close parentheses plus open parentheses 25 plus 24 plus 20 square root of 6 close parentheses
equals 49 minus 20 square root of 6 plus 1 plus 49 plus 20 square root of 6
equals 9
Hence comma space correct space option space is space left parenthesis straight b right parenthesis.

end style

Question 13

begin mathsize 11px style fraction numerator 1 over denominator square root of 9 minus square root of 8 end fraction text  is equal to end text
left parenthesis straight a right parenthesis text  3+2 end text square root of 2
left parenthesis straight b right parenthesis text   end text fraction numerator 1 over denominator 3 plus 2 square root of 2 end fraction
left parenthesis straight c right parenthesis text  3-2 end text square root of 2
left parenthesis straight d right parenthesis text   end text 3 over 2 minus square root of 2 end style

Solution 13

begin mathsize 11px style fraction numerator 1 over denominator square root of 9 minus square root of 8 end fraction
equals fraction numerator 1 over denominator square root of 9 minus square root of 8 end fraction straight space straight x straight space fraction numerator square root of 9 space plus space square root of 8 straight space over denominator square root of 9 space plus space square root of 8 end fraction
fraction numerator square root of 9  +  square root of 8 over denominator open parentheses square root of 9 close parentheses squared minus open parentheses square root of 8 close parentheses squared end fraction
equals fraction numerator square root of 9 plus square root of 8 over denominator 9 minus 8 end fraction
equals square root of 9 straight space plus straight space square root of 8
equals 3 plus 2 square root of 2
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 14

begin mathsize 11px style The text   end text value text   end text of text   end text fraction numerator square root of 48 plus square root of 32 over denominator square root of 27 plus square root of 18 end fraction text   is end text
left parenthesis straight a right parenthesis text   end text 4 over 3
left parenthesis straight b right parenthesis text  4 end text
left parenthesis straight c right parenthesis text  3 end text
left parenthesis straight d right parenthesis text   end text 3 over 4 end style

Solution 14

begin mathsize 11px style table attributes columnalign left end attributes row cell square root of 48 equals text   end text square root of 16 text  x 3 end text end root text  = 4 end text square root of 3 end cell row cell square root of 32 equals text   end text square root of 16 text  x 2 end text end root text  = 4 end text square root of 2 end cell row cell square root of 27 equals text   end text square root of text 9 x 3 end text end root text  = 3 end text square root of 3 end cell row cell square root of 18 equals text   end text square root of text 9 x 2 end text end root text  = 3 end text square root of 2 end cell row cell Now comma text    end text fraction numerator square root of 48 text  +  end text square root of 32 over denominator square root of 27 text  +  end text square root of 18 end fraction text  =  end text fraction numerator 4 square root of 3 text  + 4 end text square root of 2 over denominator 3 square root of 3 text  + 3 end text square root of 2 end fraction end cell row cell text        =  end text fraction numerator 4 text    end text left parenthesis square root of up diagonal strike 3 end root text + end text square root of up diagonal strike 2 end root right parenthesis over denominator 3 text    end text left parenthesis square root of up diagonal strike 3 end root text + end text square root of up diagonal strike 2 end root right parenthesis end fraction end cell row cell text        =   end text 4 over 3 end cell end table
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Chapter 3 - Rationalisation Excercise 3.18

Question 1

begin mathsize 11px style If text   end text fraction numerator 5 minus square root of 3 over denominator 2 plus square root of 3 end fraction equals straight x plus straight y square root of 3 space then
left parenthesis straight a right parenthesis space space straight x equals 13 comma space straight y equals negative 7
left parenthesis straight b right parenthesis space straight x equals negative 13 comma space straight y equals 7
left parenthesis straight c right parenthesis space straight x equals negative 13 comma space space straight y equals negative 7
left parenthesis straight d right parenthesis space straight x equals 13 comma space straight y equals 7
end style

Solution 1

begin mathsize 11px style fraction numerator 5 minus square root of 3 over denominator 2 plus square root of 3 end fraction
equals fraction numerator 5 minus square root of 3 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 open parentheses 5 minus square root of 3 close parentheses open parentheses 2 minus square root of 3 close parentheses over denominator open parentheses 2 close parentheses squared minus open parentheses square root of 3 close parentheses squared end fraction
equals fraction numerator 10 minus 5 square root of 3 minus 2 square root of 3 plus 3 over denominator 4 minus 3 end fraction
equals fraction numerator 13 minus 7 square root of 3 over denominator 1 end fraction
equals 13 minus 7 square root of 3
rightwards double arrow straight x equals 13 space and space straight y equals negative 7
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 2

begin mathsize 11px style If text   x= end text cube root of 2 plus square root of 3 end root text   then x end text to the power of text 3 end text end exponent plus 1 over straight x cubed text   is end text
left parenthesis straight a right parenthesis text  2  end text
left parenthesis straight b right parenthesis text  4 end text
left parenthesis straight c right parenthesis text  8 end text
left parenthesis straight d right parenthesis text  9 end text end style

Solution 2

begin mathsize 11px style table attributes columnalign left end attributes row cell straight x equals straight space cube root of 2 plus square root of 3 end root  +  left parenthesis 2 plus square root of 3 right parenthesis to the power of bevelled 1 third end exponent end cell row cell straight x cubed straight space equals left curly bracket left parenthesis 2 plus square root of 3 right parenthesis to the power of bevelled fraction numerator 1 over denominator up diagonal strike 3 end fraction end exponent right curly bracket to the power of up diagonal strike 3 end exponent equals left parenthesis 2 plus square root of 3 right parenthesis end cell row cell rightwards double arrow 1 over straight x cubed 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 straight space 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 straight space equals 2 minus square root of 3 end cell row blank row cell Now comma space straight x to the power of straight 3 plus straight space 1 over straight x cubed equals straight space 2 plus square root of 3  + space 2 space minus space square root of 3 equals 4 end cell row cell space space space space space space space space space space space space space space space space space space space space space space end cell end table
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 3

begin mathsize 10px style The text  value of  end text square root of 3 minus 2 square root of 2 end root text  is end text
left parenthesis straight a right parenthesis text   end text square root of 2 minus 1
left parenthesis straight b right parenthesis text   end text square root of 2 plus 1
left parenthesis straight c right parenthesis text   end text square root of 3 minus square root of 2
left parenthesis straight d right parenthesis text   end text square root of 3 plus square root of 2 end style

Solution 3

begin mathsize 11px style square root of 3 minus 2 square root of 2 end root
equals square root of 2 plus 1 minus straight space 2 square root of 2 end root
equals square root of open parentheses square root of 2 close parentheses squared plus left parenthesis 1 right parenthesis squared minus 2 open parentheses square root of 2 close parentheses left parenthesis 1 right parenthesis end root
equals square root of open parentheses square root of 2 minus 1 close parentheses squared end root
equals square root of 2 minus 1
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.
end style

Question 4

begin mathsize 10px style The text  value of  end text square root of 5 plus 2 square root of 6 end root text  is end text
left parenthesis straight a right parenthesis text   end text square root of 3 minus square root of 2
left parenthesis straight b right parenthesis text   end text square root of 3 plus square root of 2
left parenthesis straight c right parenthesis text   end text square root of 5 plus square root of 6
left parenthesis straight d right parenthesis text   end text none space of space these end style

Solution 4

begin mathsize 11px style square root of 5 plus 2 square root of 6 end root
equals square root of 3 plus 2 space plus space 2 open parentheses square root of 3 close parentheses open parentheses square root of 2 close parentheses end root
equals square root of open parentheses square root of 3 close parentheses squared plus open parentheses square root of 2 close parentheses squared plus 2 open parentheses square root of 3 close parentheses open parentheses square root of 2 close parentheses end root
equals square root of open parentheses square root of 3 plus square root of 2 close parentheses squared end root
equals square root of 3 plus square root of 2
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

begin mathsize 11px style If text    end text square root of 2 equals text  1.4142,  then  end text square root of fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction end root text   is equal to end text
left parenthesis straight a right parenthesis text  0.1718 end text
left parenthesis straight b right parenthesis text  5.8282 end text
left parenthesis straight c right parenthesis text  0.4142 end text
left parenthesis straight d right parenthesis text  2.4142 end text end style

Solution 5

begin mathsize 11px style text By Rationalising   end text fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction text , end text space we space get
table attributes columnalign left end attributes row cell fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction space straight x space fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction  =  fraction numerator left parenthesis square root of 2 minus 1 right parenthesis squared over denominator left parenthesis square root of 2 right parenthesis squared minus 1 squared end fraction equals straight space left parenthesis square root of 2 minus 1 right parenthesis squared over 1 end cell row cell so straight space square root of fraction numerator square root of 2 minus 1 over denominator square root of 2 plus 1 end fraction end root space equals space square root of left parenthesis square root of 2 minus 1 right parenthesis squared over 1 end root  =  left parenthesis square root of 2 minus 1 right parenthesis equals 1.4142 minus 1 equals 0.4142 end cell end table
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

begin mathsize 11px style If text   end text square root of 2 text  = 1.414, then value of  end text square root of 6 minus square root of 3 text   upto three places of decimal is end text
text (a) 0.235 end text
text (b) 0.717 end text
text (c) 1.414 end text
text (d) 0.471 end text end style

Solution 6

begin mathsize 11px style square root of 6 minus square root of 3
equals straight space square root of 3 left parenthesis square root of 2 minus 1 right parenthesis
table attributes columnalign left end attributes row cell Now comma space square root of 3 equals   1.732 space end cell row cell      space space space space space square root of 2 equals   1.414 end cell row cell therefore space square root of 6 minus square root of 3 space equals 1.732 left parenthesis 1.414 minus 1 right parenthesis equals 1.732   left parenthesis 0.414 right parenthesis equals space 0.717 space left parenthesis upto space 3 space decimal space places right parenthesis end cell end table

Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 7

begin mathsize 11px style The text  positive square root of  end text 7 plus square root of 48
left parenthesis straight a right parenthesis text  7+2 end text square root of 3
text (b) 7+ end text square root of 3
text (c) 2+ end text square root of 3
left parenthesis straight d right parenthesis text  3+ end text square root of 2 end style

Solution 7

begin mathsize 11px style square root of 7 plus square root of 48 end root
equals square root of 7 plus 2 square root of 12 end root
equals square root of 4 space plus 3 space plus space 2 square root of 4 space straight x space square root of 3 end root
equals square root of open parentheses square root of 4 close parentheses squared plus open parentheses square root of 3 close parentheses squared plus space 2 space straight x space square root of 4 space straight x space square root of 3 end root
equals square root of open parentheses square root of 4 plus square root of 3 close parentheses squared end root
equals plus-or-minus straight space open parentheses square root of 4 plus square root of 3 close parentheses
Positve space value space is space square root of 4 plus straight space square root of 3 equals 2 plus square root of 3
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 8

begin mathsize 11px style If text  x =  end text square root of 6 text  +  end text square root of 5 text ,  then x end text to the power of text 2 end text end exponent text  + end text 1 over straight x squared text  - 2 = end text
left parenthesis straight a right parenthesis text  2 end text square root of 6
left parenthesis straight b right parenthesis text  2 end text square root of 5
left parenthesis straight c right parenthesis text  24 end text
left parenthesis straight d right parenthesis text  20 end text
end style

Solution 8

begin mathsize 11px style straight x squared plus 1 over straight x squared minus 2 equals text   end text open parentheses straight x minus 1 over straight x close parentheses squared
straight x equals text   end text square root of 6 plus square root of 5
rightwards double arrow 1 over straight x text  =  end text fraction numerator 1 over denominator square root of 6 plus square root of 5 end fraction equals text   end text fraction numerator 1 over denominator square root of 6 plus square root of 5 end fraction text   end text straight x text   end text fraction numerator square root of 6 minus square root of 5 over denominator square root of 6 minus square root of 5 end fraction text = end text fraction numerator square root of 6 minus square root of 5 over denominator 1 end fraction equals square root of 6 minus square root of 5
Now comma
open parentheses straight x minus 1 over straight x close parentheses squared equals open square brackets square root of 6 plus square root of 5 minus open parentheses square root of 6 minus square root of 5 close parentheses close square brackets squared equals open parentheses 2 square root of 5 close parentheses squared equals 4 cross times 5 equals 20
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 9

begin mathsize 11px style If text   end text square root of 13 minus straight a square root of 10 end root equals square root of 8 plus square root of 5 comma text  then a = end text

left parenthesis straight a right parenthesis thin space minus 5
left parenthesis straight b right parenthesis space minus 6
left parenthesis straight c right parenthesis space minus 4
left parenthesis straight d right parenthesis space minus 2 space space space space space space space end style

Solution 9

begin mathsize 11px style square root of 13 minus straight a square root of 10 end root equals text   end text square root of 8 plus square root of 5
Squaring space both space sides comma space we space get
table attributes columnalign left end attributes row cell 13 minus straight a square root of 10 equals text   end text 8 plus 5 plus 2 square root of 40 end cell row cell rightwards double arrow 1 up diagonal strike 3 minus straight a square root of 10 text  -  end text 1 up diagonal strike 3 text  = 2 x 2 end text square root of 10 end cell end table
rightwards double arrow negative straight a square root of 10 equals 4 square root of 10
rightwards double arrow straight a equals negative 4
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.
end style

CBSE Class 9 Maths Homework Help

Clear all your doubts instantly at our “Ask Doubt” section. Get expert help and guidance at your comfort. To know the syllabus in detail, click here.

 

NCERT Textbooks are the rich stimulant for those students who want to score better in the CBSE examinations. By solving our papers the students have achieved a better and higher result. One of the primary objectives of creating ncert solution for class 9  is that the student gets access to an easy solution; which acts as a strong catalyst in improving the marks. We usually focus that the students don’t find any difficulty in understanding the concepts and can memorize them easily.