Inverse Trigonometric Functions Set-1

Question 1

$	ext{cosec}^{-1} (-\frac{2}{\sqrt{3}})$

Accepted Answer

$-\pi/3$

Answer

$\pi/3$

Answer

$\pi/2$

Answer

$-\pi/2$

Question 2

$	ext{sec}^{-1} (2)$

Accepted Answer

$\pi/3$

Answer

$\pi/6$

Answer

$2\pi/3$

Answer

$5\pi/6$

Question 3

Which of the following is the principal value branch of $	ext{cosec}^{-1} x$?

Accepted Answer

$[−\pi/2, \pi/2] – \{0\}$

Answer

$(-\pi/2, \pi/2)$

Answer

$(0, \pi) – \{\pi/2\}$

Answer

$[−\pi/2, \pi/2]$

Question 4

The value of $\sin [	ext{cos}^{-1} (\frac{7}{25})]$ is

Accepted Answer

$24/25$

Answer

$25/24$

Answer

$25/7$

Answer

$7/24$

Question 5

If $x + (1/x) = 2$ then the principal value of $	ext{sin}^{-1} x$ is

Accepted Answer

$\pi/2$

Answer

$\pi/4$

Answer

$\pi$

Answer

$3\pi/2$

Question 6

The value of $	ext{cos}^{-1} [	ext{cos}(\frac{33\pi}{5})]$ is

Accepted Answer

$3\pi/5$

Answer

$-3\pi/5$

Answer

$\pi/10$

Answer

$-\pi/10$

Question 7

$	ext{sin}^{-1} (-1/2)$

Accepted Answer

$-\pi/6$

Answer

$\pi/3$

Answer

$-\pi/3$

Answer

$\pi/6$

Question 8

$	ext{cot}^{-1} (1)$

Accepted Answer

$\pi/4$

Answer

$\pi/3$

Answer

$\pi/2$

Answer

0

Question 9

$	ext{cos}^{-1} (-1/2) + 2	ext{sin}^{-1} (-1/2)$

Accepted Answer

$\pi/3$

Answer

$2\pi/3$

Answer

$3\pi/4$

Answer

$5\pi/8$

Question 10

$	ext{cos}^{-1} (\sqrt{3}/2)$

Accepted Answer

$5\pi/6$

Answer

$\pi/6$

Answer

$4\pi/9$

Answer

$2\pi/3$

Question 11

If $	ext{sin}^{-1}(\frac{2a}{1+a^2}) + 	ext{cos}^{-1}(\frac{1−a^2}{1+a^2}) = 	ext{tan}^{-1}(\frac{2x}{1−x^2})$ where $a, x \in |0, 1|$ then the value of $x$ is

Accepted Answer

$\frac{2a}{1−a^2}$

Answer

0

Answer

$a^2$

Answer

$a$

Question 12

Which of the following is the principal value branch of $	ext{cos}^{-1} x$?

Accepted Answer

$[0, \pi]$

Answer

$[−\pi/2, \pi/2]$

Answer

$(0, \pi)$

Answer

$(0, \pi) – \{\pi/2\}$

Question 13

$	ext{cosec}^{-1} (2)$

Accepted Answer

$\pi/6$

Answer

$2\pi/3$

Answer

$5\pi/6$

Answer

0

Question 14

$	ext{sec}^{-1} (−2/\sqrt{3})$

Accepted Answer

$5\pi/6$

Answer

$\pi/6$

Answer

$\pi/3$

Answer

$-2\pi/3$

Question 15

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

Accepted Answer

$\pi/3$

Answer

$\pi/6$

Answer

$2\pi/3$

Answer

$5\pi/6$

Question 16

If $\sin \{\sin^{-1} (1/2) + \cos^{-1} x\} = 1$, then the value of $x$ is

Accepted Answer

$1/2$

Answer

0

Answer

1

Answer

None of these

Question 17

Solve for $x : \{x\cos(\cot^{-1} x) + \sin(\cot^{-1} x)\}^2 = \frac{51}{50}$

Accepted Answer

$\frac{1}{5 \sqrt{2}}$

Answer

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

Answer

$2\sqrt{2}$

Answer

$5\sqrt{2}$

Question 18

If $	ext{sin}^{-1}(x^2 – 7x + 12) = n\pi, \forall n \in I$, then $x =$

Accepted Answer

4

Answer

-2

Answer

-3

Answer

5

Question 19

If $	ext{cos}^{-1} x + 	ext{sin}^{-1} x = \pi$, then the value of $x$ is

Accepted Answer

$\frac{\sqrt{3}}{2}$

Answer

$\frac{3}{2}$

Answer

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

Answer

$\frac{2}{\sqrt{3}}$

Question 20

If $	ext{sin}^{-1} x – 	ext{cos}^{-1} x = \frac{\pi}{6}$, then $x =$

Accepted Answer

$\frac{\sqrt{3}}{2}$

Answer

$\frac{1}{2}$

Answer

$-\frac{1}{2}$

Answer

$-\frac{\sqrt{3}}{2}$

Question 21

If $	ext{tan}^{-1} (\cot 	heta) = 2	heta$, then $	heta$ is equal to

Accepted Answer

$\frac{\pi}{6}$

Answer

$\frac{\pi}{3}$

Answer

$\frac{\pi}{4}$

Answer

None of these

Question 22

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

Accepted Answer

7

Answer

6

Answer

5

Answer

None of these

Question 23

If $	ext{tan}^{-1} 3 + 	ext{tan}^{-1} x = 	ext{tan}^{-1} 8$, then $x =$

Accepted Answer

$\frac{1}{5}$

Answer

5

Answer

$\frac{5}{14}$

Answer

$\frac{14}{5}$

Question 24

If $	ext{tan}^{-1} (x – 1) + 	ext{tan}^{-1} x + 	ext{tan}^{-1} (x + 1) = 	ext{tan}^{-1} 3x$, then the values of $x$ are

Accepted Answer

$0, \pm \frac{1}{2}$

Answer

$\pm \frac{1}{2}$

Answer

$0, \frac{1}{2}$

Answer

$0, -\frac{1}{2}$

Question 25

If $6	ext{sin}^{-1} (x^2 – 6x + 8.5) = \pi$, then $x$ is equal to

Accepted Answer

2

Answer

1

Answer

3

Answer

8

Question 26

$	ext{sin}^{-1} (1 – x) – 2	ext{sin}^{-1} x = \frac{\pi}{2}$

Accepted Answer

0

Answer

$1/2$

Answer

$0, 1/2$

Answer

$-1/2$

Question 27

$2	ext{tan}^{-1}(\cos x) = 	ext{tan}^{-1}(2	ext{cosec } x)$

Accepted Answer

$\pi/4$

Answer

0

Answer

$\pi/3$

Answer

$\pi/2$

Question 28

The domain of the function defined by $f(x) = 	ext{sin}^{-1} \sqrt{x-1}$ is

Accepted Answer

A

Answer

Answer

[-1, 1]

Answer

Answer

none of these

Question 29

The value of $\sin (2	ext{tan}^{-1} (0.75))$ is equal to

Accepted Answer

0.96

Answer

0.75

Answer

1.5

Answer

$\sin 1.5$

Question 30

The value of expression $2 	ext{sec}^{-1} 2 + 	ext{sin}^{-1}(\frac{1}{2})$

Accepted Answer

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

Answer

$\frac{\pi}{6}$

Answer

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

Answer

1

Question 31

The value of the expression $	an (\frac{1}{2} \cos^{-1} \frac{2}{\sqrt{3}})$

Accepted Answer

$\sqrt{5} – 2$

Answer

$2 + \sqrt{5}$

Answer

$\frac{\sqrt{5}+2}{2}$

Answer

$5 + \sqrt{2}$

Question 32

$	ext{cos}^{-1}[\cos(2\cot^{-1}(\sqrt{2} – 1))] = \_\_\_\_\_\_$

Accepted Answer

$\frac{3 \pi}{4}$

Answer

$\sqrt{2} – 1$

Answer

$1 + \sqrt{2}$

Answer

$\frac{\pi}{4}$

Question 33

The range of $	ext{sin}^{-1} x + 	ext{cos}^{-1} x + 	ext{tan}^{-1} x$ is

Accepted Answer

$\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]$

Answer

$[0, \pi]$

Answer

$(0, \pi)$

Answer

$\left[0, \frac{\pi}{2}\right]$

Question 34

Find the value of $	ext{sec}^2 (	ext{tan}^{-1} 2) + 	ext{cosec}^2 (	ext{cot}^{-1} 3)$

Accepted Answer

15

Answer

12

Answer

5

Answer

9

Question 35

The equation $	ext{sin}^{-1} x – 	ext{cos}^{-1} x = 	ext{cos}^{-1}(\frac{\sqrt{3}}{2})$ has

Accepted Answer

unique solution

Answer

no solution

Answer

infinitely many solution

Answer

none of these

Question 36

The equation $2	ext{cos}^{-1} x + 	ext{sin}^{-1} x = \frac{11 \pi}{6}$ has

Accepted Answer

no solution

Answer

only one solution

Answer

two solutions

Answer

three solutions

Question 37

If $	ext{tan}^{-1} 2x + 	ext{tan}^{-1} 3x = \frac{\pi}{4}$, then $x$ is

Accepted Answer

$\frac{1}{6}$

Answer

1

Answer

$(\frac{1}{6}, -1)$

Answer

none of these

Question 38

If $	ext{tan}^{-1} x – 	ext{tan}^{-1} y = 	ext{tan}^{-1} A$, then $A$ is equal to

Accepted Answer

$\frac{x-y}{1+x y}$

Answer

$x – y$

Answer

$x + y$

Answer

$\frac{x+y}{1-x y}$

Question 39

The value of $	ext{cot}^{-1} 9 + 	ext{cosec}^{-1}(\frac{\sqrt{41}}{4})$ is given by

Accepted Answer

$\frac{\pi}{4}$

Answer

0

Answer

$	ext{tan}^{-1} 2$

Answer

$\frac{\pi}{2}$

Question 40

Which of the following is the principal value branch of $	ext{cosec}^{-1} x$?

Accepted Answer

$[-\pi/2, \pi/2] – \{0\}$

Answer

$(-\pi/2, \pi/2)$

Answer

$[0, \pi] – \{\pi/2\}$

Answer

$[-\pi/2, \pi/2]$

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