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RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 21 - Areas of Bounded Regions.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions Page/Excercise 21.1

Solution 1



Solution 2



Solution 3



Solution 4



Solution 5



Solution 6



Solution 7



Thus, Required area =2 over 3 open parentheses 5 to the power of 3 over 2 end exponent minus 1 close parentheses square units

Solution 8



Solution 9



Solution 11

9x2 + 4y2 = 36

Area of Sector OABCO =

Area of the whole figure = 4 × Ar. D OABCO

= 6p sq. units

 

Solution 12



Solution 13



Solution 14



Solution 15



Solution 16



Solution 17

C o n s i d e r space t h e space s k e t c h space o f space t h e space g i v e n space g r a p h : y equals open vertical bar x minus 5 close vertical bar

T h e r e f o r e comma space
R e q u i r e d space a r e a equals integral subscript 0 superscript 1 y d x
equals integral subscript 0 superscript 1 open vertical bar x minus 5 close vertical bar d x
equals integral subscript 0 superscript 1 minus open parentheses x minus 5 close parentheses d x
equals open square brackets fraction numerator minus x squared over denominator 2 end fraction plus 5 x close square brackets subscript 0 superscript 1
equals open square brackets minus 1 half plus 5 close square brackets
equals 9 over 2 s q. space u n i t s
T h e r e f o r e comma space t h e space g i v e n space i n t e g r a l space r e p r e s e n t s space t h e space a r e a space b o u n d e d space b y space t h e space c u r v e s comma space
x equals 0 comma y equals 0 comma space x equals 1 space a n d space y equals minus open parentheses x minus 5 close parentheses.

Solution 18

Solution 19



Solution 20



Solution 21




Solution 22




Solution 23

Solution 24







Solution 25





Solution 26

Solution 10

  

 

 

 

  

Solution 27

  

 

 

 

 

  

Solution 28

  

 

  

Solution 29

  

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions Page/Excercise 21.2

Solution 1



Solution 2



Solution 3

Solution 4

Solution 5

  

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions Page/Excercise 21.3

Solution 2

Solution 3



Solution 4



Solution 5



Solution 6





Solution 7





Solution 8



Solution 9



Solution 10



Solution 11

Solution 12



Solution 13



Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 14


Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 15



Solution 16



Solution 17



Solution 19



Solution 20



Solution 21



Solution 22



Solution 23

Equation of side AB,

 

Equation of side BC,

 

Equation of side AC,

 

Area of required region

= Area of EABFE + Area of BFGCB - Area of AEGCA

Solution 25

C o n s i d e r space t h e space f o l l o w i n g space g r a p h.

 

W e space h a v e comma space y equals square root of 3 x
S u b s t i t u t i n g space t h i s space v a l u e space i n space x squared plus y squared equals 16 comma space
x squared plus open parentheses square root of 3 x close parentheses squared equals 16
rightwards double arrow x squared plus 3 x squared equals 16
rightwards double arrow 4 x squared equals 16
rightwards double arrow x squared equals 4
rightwards double arrow x equals plus-or-minus 2
S i n c e space t h e space s h a d e d space r e g i o n space i s space i n space t h e space f i r s t space q u a d r a n t comma space l e t space u s space t a k e space t h e space p o s i t i v e
v a l u e space o f space x.
T h e r e f o r e comma space x equals 2 space a n d space y equals 2 square root of 3 space a r e space t h e space c o o r d i n a t e s space
o f space t h e space i n t e r s e c t i o n space p o i n t space A.
T h u s comma space a r e a space o f space t h e space s h a d e d space r e g i o n space O A B equals A r e a space O A C plus A r e a space A C B
rightwards double arrow A r e a space O A B equals integral subscript 0 superscript 2 square root of 3 x d x plus integral subscript 2 superscript 4 square root of 16 minus x squared end root d x
rightwards double arrow A r e a space O A B equals open parentheses fraction numerator square root of 3 x squared over denominator 2 end fraction close parentheses subscript 0 superscript 2 plus 1 half open square brackets x square root of 16 minus x squared end root plus 16 sin to the power of minus 1 end exponent open parentheses x over 4 close parentheses close square brackets subscript 2 superscript 4
rightwards double arrow A r e a space O A B equals open parentheses fraction numerator square root of 3 cross times 4 over denominator 2 end fraction close parentheses plus 1 half open square brackets 16 sin to the power of minus 1 end exponent open parentheses 4 over 4 close parentheses close square brackets minus 1 half open square brackets 4 square root of 16 minus 12 end root plus 16 sin to the power of minus 1 end exponent open parentheses 2 over 4 close parentheses close square brackets
rightwards double arrow A r e a space O A B equals 2 square root of 3 plus 1 half open square brackets 16 cross times straight pi over 2 close square brackets minus 1 half open square brackets 4 square root of 3 plus 16 sin to the power of minus 1 end exponent open parentheses 1 half close parentheses close square brackets
rightwards double arrow A r e a space O A B equals 2 square root of 3 plus 4 straight pi minus 2 square root of 3 minus fraction numerator 4 straight pi over denominator 3 end fraction
rightwards double arrow A r e a space O A B equals 4 straight pi minus fraction numerator 4 straight pi over denominator 3 end fraction
rightwards double arrow A r e a space O A B equals fraction numerator 8 straight pi over denominator 3 end fraction s q. space u n i t s.

Solution 26



Solution 27



Solution 29



Solution 31



Solution 32



Solution 33

 

                                                                                                                                                                                                       C l e a r l y comma space A r e a space o f space capital delta A B C equals A r e a space A D B plus A r e a space B D C
A r e a thin space A D B : space T o space f i n d space t h e space a r e a space A D B comma space w e space s l i c e space i t space i n t o space v e r t i c a l space s t r i p s.
W e space o b s e r v e space t h a t space e a c h space v e r t i c a l space s t r i p space h a s space i t s space l o w e r space e n d space o n space s i d e space A C space a n d space t h e
u p p e r space e n d space o n space A B. space S o space t h e space a p p r o x i m a t i n g space r e c tan g l e space h a s space
L e n g t h equals y subscript 2 minus y subscript 1
W i d t h equals capital delta x space
A r e a equals open parentheses y subscript 2 minus y subscript 1 close parentheses capital delta x
S i n c e space t h e space a p p r o x i m a t i n g space r e c tan g l e space c a n space m o v e space f r o m space x equals 4 space t o space 6 comma space
t h e space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses y subscript 2 minus y subscript 1 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open square brackets open parentheses fraction numerator 5 x over denominator 2 end fraction minus 9 close parentheses minus open parentheses 3 over 4 x minus 2 close parentheses close square brackets d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses fraction numerator 5 x over denominator 2 end fraction minus 9 minus 3 over 4 x plus 2 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals integral subscript 4 superscript 6 open parentheses fraction numerator 7 x over denominator 4 end fraction minus 7 close parentheses d x
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses fraction numerator 7 x squared over denominator 4 cross times 2 end fraction minus 7 x close parentheses subscript 4 superscript 6
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses fraction numerator 7 cross times 36 over denominator 8 end fraction minus 7 cross times 6 close parentheses minus open parentheses fraction numerator 7 cross times 16 over denominator 8 end fraction minus 7 cross times 4 close parentheses
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses 63 over 2 minus 42 minus 14 plus 28 close parentheses
rightwards double arrow space a r e a space o f space t h e space t r i a n g l e space A D B equals open parentheses 63 over 2 minus 28 close parentheses
S i m i l a r l y comma space A r e a space B D C equals integral subscript 6 superscript 8 open parentheses y subscript 4 minus y subscript 3 close parentheses d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open parentheses y subscript 4 minus y subscript 3 close parentheses d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open square brackets open parentheses minus x plus 12 close parentheses minus open parentheses 3 over 4 x minus 2 close parentheses close square brackets d x
rightwards double arrow A r e a space B D C equals integral subscript 6 superscript 8 open square brackets fraction numerator minus 7 x over denominator 4 end fraction plus 14 close square brackets d x
rightwards double arrow A r e a space B D C equals open square brackets minus fraction numerator 7 x squared over denominator 8 end fraction plus 14 x close square brackets subscript 6 superscript 8
rightwards double arrow A r e a space B D C equals open square brackets minus fraction numerator 7 cross times 64 over denominator 8 end fraction plus 14 cross times 8 close square brackets minus open square brackets minus fraction numerator 7 cross times 36 over denominator 8 end fraction plus 14 cross times 6 close square brackets
rightwards double arrow A r e a space B D C equals open square brackets minus 56 plus 112 plus 63 over 2 minus 84 close square brackets
rightwards double arrow A r e a space B D C equals open parentheses 63 over 2 minus 28 close parentheses
T h u s comma space A r e a space A B C equals A r e a space A D B plus A r e a space B D C
rightwards double arrow A r e a space A B C equals open parentheses 63 over 2 minus 28 close parentheses plus open parentheses 63 over 2 minus 28 close parentheses
rightwards double arrow A r e a space A B C equals 63 minus 56
rightwards double arrow A r e a space A B C equals 7 space s q. space u n i t s

 



 

 

                                                                                                                                                                                                      

 



Solution 34



Solution 35






Solution 36



Solution 37

  Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.  

Solution 39



Solution 40



Solution 41




Solution 42



Solution 43

T h e space e q u a t i o n space o f space t h e space g i v e n space c u r v e s space a r e
x squared plus y squared equals 4.... left parenthesis 1 right parenthesis
x plus y equals 2....... left parenthesis 2 right parenthesis
C l e a r l y space x squared plus y squared equals 4 space r e p r e s e n t s space a space c i r c l e space a n d space x plus y equals 2 space i s space t h e space e q u a t i o n space o f space a
s t r a i g h t space l i n e space c u t t i n g space x space a n d space y space a x e s space a t space left parenthesis 0 comma 2 right parenthesis space a n d space left parenthesis 2 comma 0 right parenthesis space r e s p e c t i v e l y.
T h e space s m a l l e r space r e g i o n space b o u n d e d space b y space t h e s e space t w o space c u r v e s space i s space s h a d e d space i n space t h e space
f o l l o w i n g space f i g u r e.

L e n g t h space equals y subscript 2 minus y subscript 1
W i d t h equals capital delta x space a n d
A r e a equals open parentheses y subscript 2 minus y subscript 1 close parentheses capital delta x
S i n c e space t h e space a p p r o x i m a t i n g space r e c tan g l e space c a n space m o v e space f r o m space x equals 0 space t o space x equals 2 comma space t h e
r e q u i r e d space a r e a space i s space g i v e n space b y space
A equals integral subscript 0 superscript 2 open parentheses y subscript 2 minus y subscript 1 close parentheses d x
W e space h a v e space y subscript 1 equals 2 minus x space a n d space y subscript 2 equals square root of 4 minus x squared end root
T h u s comma
A equals integral subscript 0 superscript 2 open parentheses square root of 4 minus x squared end root minus 2 plus x close parentheses d x
rightwards double arrow A equals integral subscript 0 superscript 2 open parentheses square root of 4 minus x squared end root close parentheses d x minus 2 integral subscript 0 superscript 2 d x plus integral subscript 0 superscript 2 x d x
rightwards double arrow A equals open square brackets fraction numerator x square root of 4 minus x squared end root over denominator 2 end fraction plus a squared over 2 sin to the power of minus 1 end exponent open parentheses x over 2 close parentheses close square brackets subscript 0 superscript 2 minus 2 open parentheses x close parentheses subscript 0 superscript 2 plus open parentheses x squared over 2 close parentheses subscript 0 superscript 2
rightwards double arrow A equals 4 over 2 sin to the power of minus 1 end exponent open parentheses 2 over 2 close parentheses minus 4 plus 2
rightwards double arrow A equals 2 sin to the power of minus 1 end exponent open parentheses 1 close parentheses minus 2
rightwards double arrow A equals 2 cross times straight pi over 2 minus 2
rightwards double arrow A equals straight pi minus 2 space sq. units

Solution 44



Solution 45



Solution 46



Solution 47



Solution 48



Solution 49



Solution 1

  

 

 

  

 

Solution 18

 

  

 

  

Solution 24

  

 

  

 

Solution 28

  

 

 

  

Solution 30

  

 

  

Solution 38

  

 

 

  

Solution 50

  

 

 

  

Solution 51

  

 

 

 

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions Page/Excercise 21.4

Solution 1

  

 

  

 

Solution 2

  

 

 

  

Solution 3

  

 

 

  

Solution 4

  

 

 

 

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 21 - Areas of Bounded Regions Page/Excercise MCQ

Solution 1

Correct option: (b)

  

 

Solution 2

Correct option: (c)

  

 

Solution 3

Correct option: (b)

  

 

Solution 4

Correct option: (b)

  

Solution 5

Correct option: (b)

  

Solution 6

Correct option: (a)

  

 

Solution 7

Correct option: (c)

 

Solution 8

Correct option: (a)

   

Solution 9

Correct option: (d)

 

NOTE: Answer not matching with back answer. 

Solution 10

Correct option: (a)

  

 

Solution 11

Correct option: (b)

  

 

Solution 12

Correct option: (b)

  

Solution 13

Correct option: (b)

  

 

Solution 14

Correct option: (c)

 

NOTE: Answer not matching with back answer. 

 

Solution 15

Correct option: (c)

  

 

Solution 16

Correct option: (d)

  

 

Solution 17

Correct options: (d)

 

NOTE: Answer not matching with back answer. 

Solution 18

Correct option: (d)

  

Solution 19

Correct option: (a)

 

NOTE: Options are modified. 

Solution 20

Correct option: (c)

  

Solution 21

Correct option: (b)

  

Solution 22

Correct option: (b)

  

Solution 23

Correct option: (c)

 

Solution 24

 

NOTE: Answer is not matching with back answer. 

Solution 25

Correct option: (a)

 

NOTE: Answer is not matching with back answer. 

Solution 26

Correct option: (d)

  

Solution 27

Correct option: (c)

  

Solution 28

Correct option:(b)

  

 

Solution 29

Correct option: (c)

  

 

Solution 30

Correct option: (b)

 

Solution 31

Correct option: (b)

  

Solution 32

Correct option: (a)

  

Solution 33

Correct option: (b)

  

 

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 21 - Areas of Bounded Regions for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 21 - Areas of Bounded Regions.

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CBSE XII Science - Mathematics

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