Grade 12 : Mathematics | Inverse Trigonometric Functions

Question 1

What is the principal value of $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$?

Accepted Answer

B) $-\frac{\pi}{3}$

Answer

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

Answer

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

Answer

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

Question 2

Find the principal value of $\cos^{-1}\left(-\frac{1}{2}\right)$.

Accepted Answer

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

Answer

A) $-\frac{\pi}{3}$

Answer

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

Answer

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

Question 3

Determine the value of $ an^{-1}(-1) + \cos^{-1}\left(-\frac{\sqrt{3}}{2} ight)$.

Accepted Answer

A) $\frac{7\pi}{12}$

Answer

B) $\frac{5\pi}{12}$

Answer

C) $\frac{11\pi}{12}$

Answer

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

Question 4

What is the domain of the function $f(x) = \sin^{-1}(2x - 3)$?

Accepted Answer

B) $[1, 2]$

Answer

A) $[-1, 1]$

Answer

C) $[2, 3]$

Answer

D) $[0, 2]$

Question 5

Evaluate the expression: $\cos\left(\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}\left(\frac{3}{5}\right)\right)$.

Accepted Answer

C) 0

Answer

A) 1

Answer

B) $\frac{3}{5}$

Answer

D) -1

Question 6

What is the principal value of $\sec^{-1}(-2)$?

Accepted Answer

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

Answer

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

Answer

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

Answer

C) $-\frac{2\pi}{3}$

Question 7

Calculate the exact value of $\sin\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right)$.

Accepted Answer

B) 1

Answer

A) $\frac{1}{2}$

Answer

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

Answer

D) 0

Question 8

Find the value of $ an^{-1}\left( an\left(\frac{3\pi}{4} ight) ight)$.

Accepted Answer

A) $-\frac{\pi}{4}$

Answer

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

Answer

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

Answer

D) $-\frac{3\pi}{4}$

Question 9

Find the value of $\cos^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right)$.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 10

What is the domain of $g(x) = \cos^{-1}(x^2 - 4)$?

Accepted Answer

B) $[-\sqrt{5}, -\sqrt{3}] \cup [\sqrt{3}, \sqrt{5}]$

Answer

A) $[-\sqrt{3}, \sqrt{3}]$

Answer

C) $[-\sqrt{5}, \sqrt{5}]$

Answer

D) $[3, 5]$

Question 11

Assertion (A): The domain of $f(x) = \sec^{-1} x$ is $(-\infty, -1] \cup [1, \infty)$.
Reason (R): The range of $\sec x$ is $(-\infty, -1] \cup [1, \infty)$.

Choose the correct option:

Accepted Answer

A) Both (A) and (R) are true and (R) is the correct explanation of (A).

Answer

B) Both (A) and (R) are true but (R) is not the correct explanation of (A).

Answer

C) (A) is true, but (R) is false.

Answer

D) (A) is false, but (R) is true.

Question 12

Assertion (A): $\sin^{-1}\left(\sin\left(\frac{2\pi}{3} ight) ight) = \frac{2\pi}{3}$.
Reason (R): $\sin^{-1}(\sin heta) = heta$ holds if and only if $ heta \in \left[-\frac{\pi}{2}, \frac{\pi}{2} ight]$.

Choose the correct option:

Accepted Answer

C) (A) is false, but (R) is true.

Answer

A) Both (A) and (R) are true and (R) is the correct explanation of (A).

Answer

B) Both (A) and (R) are true but (R) is not the correct explanation of (A).

Answer

D) (A) is true, but (R) is false.

Question 13

A civil engineer models the angle of inclination $ heta$ of an access ramp as $ heta = an^{-1}\left(\frac{h}{b} ight)$, where $h$ is the vertical elevation and $b$ is the horizontal run. For a specific design, $h = 4 m$ and $b = 4\sqrt{3} m$.

What is the design angle of inclination $ heta$?

Accepted Answer

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

Answer

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

Answer

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

Answer

D) $\frac{5\pi}{12}$

Question 14

An air traffic monitoring system calculates the angular deviation $\alpha$ of an incoming drone using the formula $\alpha = \sin^{-1}\left(\frac{v_x}{v}\right)$, where $v_x$ is the horizontal velocity and $v$ is the total velocity. If $v_x = -250 km/h$ and $v = 500 km/h$:

Find the angular deviation $\alpha$ in radians.

Accepted Answer

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

Answer

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

Answer

B) $-\frac{\pi}{3}$

Answer

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

Question 15

Find the value of $x$ satisfying the equation $ an^{-1}(x) + \cot^{-1}\left(\frac{1}{3} ight) = \frac{\pi}{2}$ for $x > 0$.

Accepted Answer

D) $\frac{1}{3}$

Answer

A) $\frac{1}{2}$

Answer

B) 1

Answer

C) 2

Question 16

If $\sin^{-1} x + \sin^{-1} y = \pi$, then find the value of $x^2 + y^2$.

Accepted Answer

B) 2

Answer

A) 1

Answer

C) 0

Answer

D) 4

Question 17

What is the value of $\cot\left( an^{-1}\sqrt{3} + \cot^{-1}\sqrt{3} ight)$?

Accepted Answer

C) 0

Answer

A) $\sqrt{3}$

Answer

B) $\frac{1}{\sqrt{3}}$

Answer

D) Undefined

Question 18

Evaluate $\sin\left(2\cos^{-1}\left(-\frac{3}{5}\right)\right)$.

Accepted Answer

B) $-\frac{24}{25}$

Answer

A) $\frac{24}{25}$

Answer

C) $-\frac{7}{25}$

Answer

D) $\frac{12}{25}$

Question 19

Find the simplified form of $ an^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x} ight)$ where $-\frac{\pi}{4} < x < \frac{\pi}{4}$.

Accepted Answer

A) $\frac{\pi}{4} - x$

Answer

B) $\frac{\pi}{4} + x$

Answer

C) $x - \frac{\pi}{4}$

Answer

D) $\frac{\pi}{2} - x$

Question 20

If $\cos^{-1} x + \cos^{-1} y + \cos^{-1} z = 3\pi$, what is the value of $xy + yz + zx$?

Accepted Answer

D) 3

Answer

A) -3

Answer

B) 0

Answer

C) 1

Set 3Grade 12 : Mathematics | Inverse Trigonometric Functions