Mathematics

Multiple Choice Questions (MCQ)
MCQ | Class 12 | Chapter 2
INVERSE TRIGONOMETRIC FUNCTIONS

MCQ Based on the Domain and range of the Inverse Trigonometric Functions
MCQ Based on the Properties of Inverse Trigonometric Functions.
MCQ Based on the Principal Value of the Inverse Trigonometric Functions.
MCQ Based on the general problems of Inverse Trigonometric Functions.

Features

In this pdf given below you find the important MCQ which are strictly according to the CBSE syllabus and are very useful for the CBSE Examinations.
Solution Hints are also given to some difficult problems.
Each MCQ contains four options from which one option is correct.
On the right hand side column of the pdf Answer option is given.

Action Plan

First of all students should Learn and write all basic points and Formulas related to the Inverse Trigonometric Functions.
Start solving the NCERT Problems with examples.
Solve the important assignments on the Inverse Trigonometric Functions.
Then start solving the following MCQ.

MCQ | CHAPTER 2 | CLASS 12

INVERSE TRIGONOMETRIC FUNCTIONS

Q 1) The value of $tan^{-1}\left [ sin\left ( \frac{-\pi }{2{\color{Orchid} }} \right ) \right ]$ is =

a) -1

b) 1

$c)\: \: \pi ^{2}$

$d)\: \: \frac{-\pi }{4}$

Ans: d
Q 2) The value of $cos^{-1}\left ( \frac{1}{2} \right )+2sin^{-1}\left ( \frac{1}{2} \right )$ is equal to

$a)\: \: \frac{\pi }{4}$

$b)\: \: \frac{\pi }{6}$

$c)\: \: \frac{\pi }{3}$

$d)\: \: \frac{2\pi }{3}$

Ans: d

Q 3) The value of sin^-1{cos (4095^o) is equal to

$a)\: \: \frac{-\pi }{3}$

$b)\: \: \frac{\pi }{6}$

$c)\: \: \frac{-\pi }{4}$

$d)\: \: \frac{\pi }{4}$

Ans: c

Q 4) The value of $cos(tan^{-1}x)\: =$

$a)\: \: \sqrt{1+x^{2}}$

$b)\: \: \frac{1}{\sqrt{1+x^{2}}}$

$c)\: \: 1+x^{2}$

d) None of these

Ans: b

Q 5) The value of $tan\: (sex^{-1}\sqrt{1+x^{2}})\: \: is$

$a)\: \: \frac{1}{x}$

b) x
$c)\: \: \: \frac{1}{\sqrt{1+x^{2}}}$

$d)\: \: \: \frac{x}{\sqrt{1+x^{2}}}$
Ans: b

Q 6) The value of $sec^{-1}[sec(-30^{o})]\: \: is=$
a) - 60
b) -30
c) 30
d) 150
Ans: c
Q 7) The value of $tan^{-1}\left [ \frac{cosx}{1+sinx} \right ]=$
$a)\: \: \: \frac{\pi }{4}-\frac{x}{2}$

$b)\: \: \: \frac{\pi }{4}+\frac{x}{2}$

$c)\: \: \: \frac{x}{2}$

$d)\: \: \: \frac{\pi }{4}-x$
Ans: a
Q 8) The value of $tan^{-1}\left ( \frac{1}{\sqrt{x^{2}-1}} \right )\: \: is=$

$a)\: \: \: \frac{\pi }{2}+cosec^{-1}x$

$b)\: \: \: \frac{\pi }{2}+sec^{-1}x$

$c)\: \: \: cosec^{-1}x$

$d)\: \: \: sec^{-1}x$
Ans: c
Q 9) The value of $cos^{-1}\left ( cos\frac{7\pi }{5} \right )$ is =

$a)\: \: \: \frac{3\pi }{5}$

$b)\: \: \: \frac{2\pi }{5}$

$c)\: \: \: \frac{-7\pi }{5}$

$d)\: \: \: \frac{7\pi }{5}$
Ans: a
Q 10) The principal value of $sin^{-1}\left ( \frac{-1}{2} \right )\: \: is=$

$a)\: \: \: \frac{\pi }{3}$

$b)\: \: \: \frac{\pi }{6}$

$c)\: \: \: \frac{-\pi }{3}$

$d)\: \: \: \frac{-\pi }{6}$

Ans: d

Q 11) Sec²(tan^-12) + cosec²(cot^-13) =

a) 5
b) 13
c) 15
d) 16
Ans: c

Q 12) $sin^{-1}\left [x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^{2}} \right ] =$

$a)\: \: sin^{-1}x+sin^{-1}\sqrt{x}$

$b)\: \: sin^{-1}x-sin^{-1}\sqrt{x}$

$c)\: \: sin^{-1}\sqrt{x}-sin^{-1}x$

d) None of these

Ans: b

Q 13) $sin\left [ \frac{1}{2}cot^{-1}\left ( \frac{-3}{4} \right ) \right ]\: \: is=$
$a)\: \: \frac{1}{\sqrt{5}}$

$b)\: \: \frac{-2}{\sqrt{5}}$

$c)\: \: \frac{2}{\sqrt{5}}$

$d)\: \: \frac{-1}{\sqrt{5}}$

Ans: c

Q 14) If $tan^{-1}\left ( \frac{1-x}{1+x} \right )=\frac{1}{2}tan^{-1}x$ , then x =

a) 1

$b)\:\: \sqrt{3}$

$c)\:\: \frac{1}{\sqrt{3}}$

d) None of these

Ans: c

Q 15) The value of $cos^{-1}\left ( cos\frac{7\pi }{6} \right )$

$a)\:\:\frac{7\pi }{6}$

$b)\:\:\frac{5\pi }{6}$

$c)\:\:\frac{\pi }{6}$

d) None of these

Ans: b

Q 16) If $cos^{-1}\left ( \frac{5}{13} \right )-sin^{-1}\left ( \frac{12}{13} \right )=cos^{-1}x$ , then x is equal to

a) 1

$b)\:\:\frac{1}{\sqrt{2}}$

c) 0

$d)\:\:\frac{\sqrt{3}}{2}$

Ans: a

Q 17) Principal value of $sin^{-1}\left ( \frac{-1}{2} \right )$ is

a) $a)\: \: \frac{\pi }{3}$

$b)\: \: \frac{-\pi }{3}$

c) $c)\: \: \frac{5\pi }{6}$

d) $d)\: \: \frac{-\pi }{2}$

Ans: d

Explanation:

$\theta =sin^{-1}\left ( \frac{-1}{2} \right )$

$\Rightarrow \: \: sin\theta =\frac{-1}{2}=sin\left ( \frac{-\pi }{6} \right )$

$\Rightarrow \: \: \theta =\frac{-\pi }{6}$

Q 18) If $sin^{-1}x+sin^{-1}y=\frac{2\pi }{3}$ , then find the value of $cos^{-1}x+cos^{-1}y$
$a)\: \: \frac{2\pi }{3}$

$b)\: \: \frac{-2\pi }{3}$

$c)\: \: \frac{\pi }{3}$

$d)\: \: \pi$

Ans: c

Explanation

$sin^{-1}x+sin^{-1}y=\frac{2\pi }{3}$

$\left ( \frac{\pi }{2}-cos^{-1} \right )+\left ( \frac{\pi }{2}-cos^{-1}y \right )=\frac{2\pi }{3}$

$\pi -cos^{-1}x-cos^{-1}y=\frac{2\pi }{3}$

$cos^{-1}x+cos^{-1}y=\pi -\frac{2\pi }{3}$

$cos^{-1}x+cos^{-1}y=\frac{\pi }{3}$

Q 19) $tan^{-1}\left [ sin\left ( \frac{-\pi }{2} \right ) \right ]$ is equal to

a) -1
b) 1
$c)\: \: \frac{\pi }{2}$

$d)\: \: \frac{-\pi }{4}$
Ans: d

Q 20) $sin\left [ cot^{-1}\left\{ cos\left ( tan^{-1}x \right )\right\} \right ] =$

$a)\: \: \sqrt{\frac{1+x^{2}}{2+x^{2}}}$

$b)\: \: \sqrt{\frac{1-x^{2}}{2+x^{2}}}$

$c)\: \: \sqrt{\frac{1+x^{2}}{2-x^{2}}}$

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

Ans: a

Q21) The value of $sec\left\{tan^{-1}\left ( \frac{-y}{3} \right ) \right\}$ is equal to

a) $\frac{\sqrt{9+y^{2}}}{9}$

b) $\frac{\sqrt{9+y^{2}}}{3}$

c) $\frac{3}{\sqrt{9+y^{2}}}$

d) $\frac{9}{\sqrt{9+y^{2}}}$

Ans b

Explanation

$sec\left ( tan^{-1}\frac{y}{3} \right )=\sqrt{sec^{2}\left ( tan^{-1}\frac{y}{3} \right )}$

$=\sqrt{1+tan^{2}\left ( tan^{-1}\frac{y}{3} \right )}$

$=\sqrt{1+\frac{y^{2}}{9}}=\frac{\sqrt{9+y^{2}}}{3}$

Q22) If $sin^{-1}x+sin^{-1}y+sin^{-1}z = \frac{3\pi }{2}$ then the value of x + y² + z³is

a) 1

b) 3

c) 2

d) 5

Ans: b

Explanation

$sin^{-1}x+sin^{-1}y+sin^{-1}z = \frac{3\pi }{2}$

$\Rightarrow sin^{-1}x=\frac{\pi}{2}$ $\Rightarrow x=1$

$\Rightarrow sin^{-1}y=\frac{\pi}{2}$ $\Rightarrow y=1$

$\Rightarrow sin^{-1}z=\frac{\pi}{2}$ $\Rightarrow z=1$

$x+y^{2}+z^{3}=1+1+1=3$

Q23 Principal value of the expression $cos^{-1}\left [ cos\left ( -680^{o} \right ) \right ]$ is

a) $\frac{2\pi }{9}$

b) $-\frac{2\pi }{9}$

c) $\frac{34\pi }{9}$

d) $\frac{\pi }{9}$

Ans : a

Q24) The value of tan²(sec^-12) + cot²(cosec^-13) is

a) 5

b) 11

c) 13

d) 15

Ans: b

Q25) The value of $tan^{-1}\left ( \frac{1}{2} \right )+tan^{-1}\left ( \frac{1}{3} \right )+tan^{-1}\left ( \frac{7}{8} \right )$ is

a) $tan^{-1}\left ( \frac{7}{8} \right )$

b) $cot^{-1}(15)$

c) $tan^{-1}(15)$

d) $tan^{-1}\left ( \frac{25}{24} \right )$

Ans : c

Q26) Find the value of $tan^{-1}\left ( \sqrt{3} \right )-cot^{-1}\left ( -\sqrt{3} \right )$

a) π

b) $-\frac{\pi }{2}$

c) - π

d) $\frac{\pi }{2}$

Answer: b

Explanation

$tan^{-1}\sqrt{3}-cot^{-1}(-\sqrt{3})$

$=tan^{-1}\sqrt{3}-\left [\pi - cot^{-1}(\sqrt{3}) \right ]$

$=\left [ tan^{-1}\sqrt{3}+cot^{-1}(\sqrt{3}) \right ]-\pi$

$=\frac{\pi }{2}-\pi =-\frac{\pi }{2}$

Q27) If sec^-1x + sec^-1y = $\frac{\pi }{2}$ , then the value of cosec^-1x + cosec^-1y is

a) π

b) π/2

c) 3π/2

d) -π

Answer b

Explanation

sec^-1x + sec^-1y = $\frac{\pi }{2}$

⇒ $\frac{\pi }{2}$ - cosec^-1x + $\frac{\pi }{2}$ - cosec^-1y = $\frac{\pi }{2}$

⇒ $\frac{\pi }{2}$ - cosec^-1x - cosec^-1y = 0

⇒ cosec^-1x + cosec^-1y = $\frac{\pi }{2}$

Q28) $cot\left ( \frac{\pi }{4}-2cot^{-1}3 \right )=$

a) 5

b) 9

c) 12

d) 7

Answer 7

Explanation

$2cot^{-1}(3)=2tan^{-1}\left ( \frac{1}{3} \right )$

$tan^{-1}\left ( \frac{\frac{2}{3}}{1-\frac{1}{9}} \right )=tan^{-1}\left ( \frac{2/3}{8/9} \right )$

$=tan^{-1}\left ( \frac{2\times 9}{3\times 8} \right )=tan^{-1}\left ( \frac{3}{4} \right )$

Now given equations becomes

$cot\left ( \frac{\pi }{4}-2cot^{-1}3 \right )=$

$cot\left [ \frac{\pi }{4}-tan^{-1}\frac{3}{4} \right ]=cot\left [ \frac{\pi }{4}-cot^{-1}\frac{4}{3} \right ]$

$=\frac{cot\left ( \frac{\pi }{4} \right )cot^{-1}\left ( \frac{4}{3} \right )+1}{cot\left ( \frac{\pi }{4} \right )cot^{-1}\left ( \frac{4}{3} \right )-1}$

$=\frac{\frac{4}{3}+1}{\frac{4}{3}-1}=\frac{4+3}{4-3}=7$

Q29) If 3sin^-1x + cos^-1x = π, then x is equal to

a) 0

b) $\frac{1}{\sqrt{2}}$

c) -1

d) 1/2

Answer b

Explanation

3sin^-1x + cos^-1x = π

2sin^-1x + (sin^-1x + cos^-1x) = π

2sin^-1x + π / 2 = π

2sin^-1x = π - π / 2 = π / 2

sin^-1x = π / 4

x = sin(π / 4) = $\frac{1}{\sqrt{2}}$

Q30) If $sin^{-1}\left ( \frac{2\alpha }{1+\alpha ^{2}} \right )+cos^{-1}\left ( \frac{1-\beta ^{2}}{1+\beta ^{2}} \right )=tan^{-1}\left ( \frac{2x}{1-x^{2}} \right )$ , then the value of x is

a) $x=\frac{\alpha +\beta }{1-\alpha \beta }$

b) $x=\frac{\alpha -\beta }{1-\alpha \beta }$

c) $x=\frac{\alpha -\beta }{1+\alpha \beta }$

d) None of these

Ans: a

Explanation

$sin^{-1}\left ( \frac{2\alpha }{1+\alpha ^{2}} \right )=2tan^{-1}\alpha$

$cos^{-1}\left ( \frac{1-\beta ^{2}}{1+\beta ^{2}} \right )=2tan^{-1}\beta$

$tan^{-1}\left ( \frac{2x}{1-x^{2}} \right )=2tan^{-1}x$

Using these values in the given equation we get

$2tan^{-1}\alpha +2tan^{-1}\beta =2tan^{-1}x$

$tan^{-1}\alpha +tan^{-1}\beta =tan^{-1}x$

$\Rightarrow tan^{-1}\left ( \frac{\alpha +\beta }{1-\alpha \beta } \right )=tan^{-1}x$

$\Rightarrow x=\frac{\alpha +\beta }{1-\alpha \beta }$

Q31) The value of : $tan\left ( \frac{1}{2}cos^{-1}\frac{\sqrt{5}}{3} \right )$ is

a) $\frac{3-\sqrt{5}}{2}$

b) $\frac{3+\sqrt{5}}{2}$

c) $\frac{5+\sqrt{3}}{2}$

d) $\frac{\sqrt{5}}{3}$

Answer a

Q32) The value of : $sin^{-1}\left ( \frac{2\sqrt{2}}{3} \right )+sin^{-1}\left ( \frac{1}{3} \right )$

a) $\frac{\pi }{6}$

b) $\frac{\pi }{4}$

c) $\frac{\pi }{2}$

d) 0

Answer c

Q33) The value of expression $2sec^{-1}2 + sin^{-1}\left ( \frac{1}{2} \right )$ is

a) $\frac{\pi }{6}$

b) $\frac{5\pi }{6}$

c) $\frac{7\pi }{6}$

d) 1

Answer b

Q34) The value of : $sec^{2}(tan^{-1}3)+cosec^{2}(cot^{-1}2)$ is

a) 5

b) 13

c) 15

d) 25

Answer c

Q35) $sin\left [ \frac{\pi }{3}-sin^{-1}\left ( -\frac{1}{2} \right ) \right ]$ is equal to

a) 1/2

b) 1/3

c) 1/4

d) 1

Ans d

Q36) $cos\left [ 2cos^{-1}\left ( \frac{1}{5} \right ) +sin^{-1}\left ( \frac{1}{5} \right )\right ]$ is equal to

a) 1/5

b) $-\frac{2\sqrt{6}}{5}$

c) -1/5

d) $\frac{2\sqrt{6}}{5}$

Answer: d

Q37) The value of $sin^{-1}\left [ sin\left ( \frac{-17\pi }{8} \right ) \right ]$ is

a) $-\frac{\pi }{6}$

b) $\frac{\pi }{8}$

c) $\frac{-\pi }{8}$

d) $\frac{\pi }{12}$

Answer c

Q38) Evaluate: $sin^{-1}\left\{sin\left ( \frac{5\pi }{6} \right ) \right\}+tan^{-1}\left\{ tan\left ( \frac{\pi }{6} \right )\right\}$ is

a) $\frac{\pi }{6}$

b) $\frac{2\pi }{3}$

c) $\frac{\pi }{3}$

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

Answer: c

Q39) Find the value of : $sin\left [\frac{\pi }{3}-sin^{-1}\left ( -\frac{1}{2} \right ) \right ]$

Ans: 1

Q40) $sin^{-1}\left [ sin\left ( \frac{2\pi }{3} \right ) \right ]=\frac{2\pi }{3}$ , state true or false

Ans: False

Q41) What is the Principal branch of tan^-1x

Ans $\left ( -\frac{\pi }{2},\frac{\pi }{2} \right )$

Q42) If $tan^{-1}x=sin^{-1}\left ( \frac{1}{\sqrt{2}} \right )$ then find the value of x

Ans 1

Q43) $cos^{-1}\left ( cos\frac{7\pi }{6} \right )=\frac{7\pi }{6}$ state true or false

Ans : False

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INVERSE TRIGONOMETRIC FUNCTIONS

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Multiple Choice Questions (MCQ)MCQ | Class 12 | Chapter 2INVERSE TRIGONOMETRIC FUNCTIONS

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Action Plan

MCQ | CHAPTER 2 | CLASS 12

INVERSE TRIGONOMETRIC FUNCTIONS

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Case Study Based Questions Class 10 | Mathematics

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Multiple Choice Questions (MCQ)
MCQ | Class 12 | Chapter 2
INVERSE TRIGONOMETRIC FUNCTIONS