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CBSE 12th Maths Integrals Important Questions And Answers Study Material - QB365 Set A

Published on: 20/08/2026

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Std: 12th Standard CBSE - Maths

chapter: Integrals - Important Questions And Answers

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Questions:

Multiple Choice Question

25 x 1 = 25

\(\sin \left(\tan ^{-1} x\right)\), where |x| < 1 is equal to

(a)

\(\frac{x}{\sqrt{1-x^2}}\)

(b)

\(\frac{1}{\sqrt{1-x^2}}\)

(c)

\(\frac{1}{\sqrt{1+x^2}}\)

(d)

\(\frac{x}{\sqrt{1+x^2}}\)

The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at

(a)

(0.6, 1.6) only

(b)

(3, 0) only

(c)

(0.6, 1.6) and (3, 0) only

(d)

at every point of the line-segment joining the points (0.6, 1.6) and (3,0)

The function f : R→ R defined as f(x) = x³ is

(a)

one-one but not onto

(b)

not one-one but onto

(c)

neither one-one nor onto

(d)

both one-one and onto

\(\text { The value of } \cos ^{-1}\left(\frac{1}{2}\right)+3 \sin ^{-1}\left(\frac{1}{2}\right) \text { is equal to }\)

(a)

\(\frac{\pi}{4}\)

(b)

\(\frac{\pi}{6}\)

(c)

\(\frac{2\pi}{3}\)

(d)

\(\frac{5\pi}{6}\)

\(\text { The principal value of } \sin ^{-1}\left(\sin \frac{2 \pi}{3}\right) \text { is }\)

(a)

\(\frac{2 \pi}{3}\)

(b)

\(\frac{ \pi}{3}\)

(c)

\(-\frac{ \pi}{6}\)

(d)

\(\frac{ \pi}{6}\)

The maximum value of z= 4x + 3y, if the feasible region for an LPP is as shown below, is

(a)

112

(b)

100

(c)

(d)

110

The feasible region for an LPP is shown in the following figure. Then, the minimum value of Z = 11x + 7y is

(a)

(b)

(c)

(d)

\(\int\left(\frac{10 x^{9}+10^{x} \log _{e} 10}{x^{10}+10^{x}}\right) d x\) equal to

(a)

\(10^{x}-x^{10}+C\)

(b)

\(10^{x}+x^{10}+C\)

(c)

\(\left(10^{x}-x^{10}\right)^{-1}+C\)

(d)

\(\log \left|10^{x}+x^{10}\right|+C\)

The domain in which sine function will be one-one, is

(a)

\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

(b)

\(\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]\)

(c)

\([0, \pi]\)

(d)

Both 'a' and 'b'

10.

The value of \(\sin \left(2 \tan ^{-1} \frac{2}{3}\right)-\cos \left(2 \tan ^{-1} \sqrt{3}\right)\) is

(a)

\(\frac{26}{37}\)

(b)

\(\frac{37}{26}\)

(c)

\(\frac{12}{13}\)

(d)

\(\frac{13}{15}\)

11.

The value of \(\tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right]\) is

(a)

\(\frac{\pi}{3}\)

(b)

\(\frac{2 \pi}{3}\)

(c)

\(\frac{-\pi}{3}\)

(d)

\(\frac{\pi}{6}\)

12.

∫ f(x) dx = F(x) + c, then \(\frac { d }{ dx } (\int { f(x)dx+c) } \) =

(a)

F(x) + c

(b)

F(x)

(c)

f(x) + c

(d)

f(x)

13.

Domain of function \({ cos }^{ -1 }\left( \frac { 2x+1 }{ 3 } \right) \) is

(a)

(-2,0)

(b)

[-2,0]

(c)

[-2,1]

(d)

(-2,1)

14.

The principal value of tan^-1 1 is given by

(a)

π/2

(b)

π/3

(c)

π/6

(d)

π/4

15.

Identify the graph above

(a)

y = sin^-1x

(b)

y = cos^-1x

(c)

y = sin x

(d)

y = cos x

16.

What is the principal value of \({ sec }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \)

(a)

\(\frac { \pi }{ 3 } \)

(b)

not defined

(c)

\(-\frac { \pi }{ 3 } \)

(d)

\(\frac { 2\pi }{ 3 } \)

17.

The value of \(\int _{ 0 }^{ 1 }{ \left( \frac { 2x-1 }{ 1+x-{ x }^{ 2 } } \right) } dx\) is

(a)

(b)

(c)

-1

(d)

\(\frac{\pi}{4}\)

18.

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(a)

(b)

(c)

(d)

19.

Let f : R ⟶ R be defined as f(x) = x⁴. Choose the correct answer

(a)

f is one-one onto

(b)

f is many-one onto

(c)

f is one-one but not onto

(d)

f is neither one-one nor onto

20.

\(\int { { e }^{ x }{ \left( \frac { 1-x }{ 1+{ x }^{ 2 } } \right) }^{ 2 } } dx\) is equal to

(a)

\(\frac { { e }^{ x } }{ 1+{ x }^{ 2 } } +C\)

(b)

\(-\frac { { e }^{ x } }{ 1+{ x }^{ 2 } } +C\)

(c)

\(\frac { { e }^{ x } }{ (1+{ x }^{ 2 })^{ 2 } } +C\)

(d)

\(\frac { { e }^{ x } }{ (1+{ x }^{ 2 })^{ 2 } } +C\)

21.

∫cot²x dx equals to

(a)

cot x – x + C

(b)

cot x + x + C

(c)

-cot x + x + C

(d)

-cot x – x + C

22.

Given ∫ 2^x dx = f(x) + C, then f(x) is

(a)

2^x

(b)

2^x log_e2

(c)

\(\frac { { 2 }^{ x } }{ { log }_{ e }2 } \)

(d)

\(\frac { { 2 }^{ x } }{ { log }_{ e }2 } \)

23.

The domain of the function y = sin^-1(x²) is

(a)

[0, 1]

(b)

(0, 1)

(c)

[-1, 1]

(d)

24.

tan^-1{sin (-\(\frac{\pi}{2}\))} is equal to

(a)

-1

(b)

(c)

\(\frac{\pi}{2}\)

(d)

\(-\frac{\pi}{4}\)

25.

Principal value of sin^-1\(-\frac{1}{2}\)is

(a)

\(\frac{\pi}{3}\)

(b)

-\(\frac{\pi}{3}\)

(c)

\(\frac{5\pi}{3}\)

(d)

\(-\frac{\pi}{6}\)

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Answers Key:

Multiple Choice Question

25 x 1 = 25

(d)
\(\frac{x}{\sqrt{1+x^2}}\)

(d)
at every point of the line-segment joining the points (0.6, 1.6) and (3,0)

(d)
both one-one and onto

(d)
\(\frac{5\pi}{6}\)

(b)
\(\frac{ \pi}{3}\)

(a)
112

(a)
21

(d)
\(\log \left|10^{x}+x^{10}\right|+C\)

(d)
Both 'a' and 'b'

10.

\( 2 \tan ^{-1} x=\sin ^{-1} \frac{2 x}{1+x^{2}}\)
\(\therefore \ 2 \tan ^{-1} \frac{2}{3}=\sin ^{-1} \frac{2\left(\frac{2}{3}\right)}{1+\left(\frac{2}{3}\right)^{2}}=\sin ^{-1} \frac{12}{13}\)
\(\cos \left(2 \tan ^{-1} x\right)=\cos \left(\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right) \)

11.

\(\begin{array}{l} \text { Given, } \tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right] \\ =\tan ^{-1}\left[2 \sin \left(2 \times \frac{\pi}{6}\right)\right]=\tan ^{-1}\left(2 \sin \frac{\pi}{3}\right) \\ =\tan ^{-1}\left(2 \times \frac{\sqrt{3}}{2}\right)=\tan ^{-1} \sqrt{3}=\frac{\pi}{3} \end{array}\)

12.

(d)
f(x)

13.

14.

(d)
π/4

15.

(a)
y = sin^-1x

16.

(b)
not defined

17.

(b)
0

18.

(b)
2

19.

(d)
f is neither one-one nor onto

20.

As \(\int { { e }^{ x }\left( \frac { 1+{ x }^{ 2 }-2x }{ { { (1+x }^{ 2 }) }^{ 2 } } \right) } dx\)
= \(\int { { e }^{ x }\left\{ \frac { 1 }{ 1+{ x }^{ 2 } } -\frac { 2x }{ { { (1+x }^{ 2 }) }^{ 2 } } \right\} } dx\)
\(f(x)=\frac { 1 }{ 1+{ x }^{ 2 } } ;\)
\(f'(x)=\frac { -2x }{ (1+{ x }^{ 2 })^{ 2 } } \)
Using \(\int { { e }^{ x }\{ f(x)+f'(x)\} } dx\)
\(={ e }^{ x }.f(x)+C\)
\(={ e }^{ x }.\frac { 1 }{ 1+{ x }^{ 2 } } +C\)

21.

∫ (cosec²x -1)dx = -cot x – x + C

22.

As \(\frac { d }{ dx } \left( \frac { { 2 }^{ x } }{ { log }_{ e }2 } \right) \)
\(=\frac { 1 }{ { log }_{ e }2 } .{ 2 }^{ x }.{ log }_{ e }2 ={ 2 }^{ x }\)

23.

As -1 ≤ -x² < 1
⇒ 1 ≥ x² ≥ -1
⇒ 0 ≤ x² ≤ 1
⇒ |x| ≤ 1
⇒ -1 ≤ x ≤ 1.

24.

As sin (\(-\frac{\pi}{2}\))= -1, and tan<sup>-1</sup>(-1) = \(-\frac{\pi}{4}\).

25.

Let θ = sin^-1 \(-(\frac12)\)
⇒ sin^-1 = \(-\frac12\) = sin \((-\frac{\pi}{6})\) = θ = \(-\frac{\pi}{6}\)