Inverse Trigonometric Functions — Class 12 Mathematics PYQs

The graph of a trigonometric function is as shown. Which of the following will represent graph of its inverse ?

(A) Graph A
(B) Graph B
(C) Graph C
(D) Graph D

(This question references a diagram.)

If $y={\sin }^{-1}x$, $-1\le x\le 0$, then the range of $y$ is

(A) $\left[-\frac{\pi }{2},0\right]$
(B) $\left[\frac{\pi }{2},0\right]$
(C) $\left[-\frac{\pi }{2},0\right)$
(D) $\left[-\frac{\pi }{2},0\right]$

The principal value of ${\sin }^{-1}\left(\cos \frac{43\pi }{5}\right)$ is

(A) $-\frac{7\pi }{5}$
(B) $-\frac{\pi }{10}$
(C) $\frac{\pi }{10}$
(D) $\frac{3\pi }{5}$

The value of $\cos \left(\frac{\pi }{6}+{\cot }^{-1}(-\sqrt{3})\right)$ is :

(A) $-1$
(B) $-\frac{\sqrt{3}}{2}$
(C) $0$
(D) $1$

Evaluate : ${\tan }^{-1}\left[2\sin \left(2{\cos }^{-1}\frac{\sqrt{3}}{2}\right)\right]$

Evaluate : ${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right)$

Solve for $x$ :

$$2{\tan }^{-1}x+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)=4\sqrt{3}$$

If $y={\cos }^{-1}(e^x)$, then $\frac{dy}{dx}$ is :

(a) Find the value of ${\tan }^{-1}\left(1-\frac{1}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}\left[\sin \left(-\frac{\pi }{2}\right)\right]$.

OR

(b) Find the domain of the function $f(x)={\sin }^{-1}(x^2-4)$. Also, find its range.

(a) Find the value of ${\sin }^{-1}\left(\frac{1}{2}\right)+{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)+{\cot }^{-1}\left(\tan \frac{4\pi }{3}\right)$.

OR

(b) Find the domain of $f(x)={\cos }^{-1}(1-x^2)$. Also, find its range.