Question

Let $f: [-1,1] \rightarrow [-\frac{\pi}{2},\frac{\pi}{2}]$ be defined by $f(x) = \sin^{-1}x$

$g: [-1, 1] \rightarrow [0, \pi]$ be defined by $g(x) = \cos^{-1}x$

$h:R\rightarrow(-\frac{\pi}{2},\frac{\pi}{2})$ be defined by $h(x) = \tan^{-1}x$

Match each entry in List-l to the correct entries in List-II.

List-I	List-II
P) $f(x) + g(x)$	1) 0
Q) $f(x) + f(-x)$	2) $\frac{\pi}{2}$
R) $g(x) + g(-x)$	3) $\frac{-\pi}{2}$
S) $h(x) + h(-x)$	4) $\pi$

• A. P $\rightarrow$ 1, Q $\rightarrow$ 2, R $\rightarrow$ 3, S $\rightarrow$ 4
• B. P $\rightarrow$ 2, Q $\rightarrow$ 1, R $\rightarrow$ 4, S $\rightarrow$ 1
• C. P $\rightarrow$ 3, Q $\rightarrow$ 4, R $\rightarrow$ 1, S $\rightarrow$ 2
• D. P $\rightarrow$ 4, Q $\rightarrow$ 3, R $\rightarrow$ 2, S $\rightarrow$ 1

Answer 1

Answer: (B)

P $\rightarrow$ 2, Q $\rightarrow$ 1, R $\rightarrow$ 4, S $\rightarrow$ 1

Answer 2

P) $f(x) + g(x)$:
Using the identity $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$.

Q) $f(x) + f(-x)$:
Using the property $\sin^{-1}(-x) = -\sin^{-1}x$.

R) $g(x) + g(-x)$:
Using the property $\cos^{-1}(-x) = \pi - \cos^{-1}x$.

S) $h(x) + h(-x)$:
Using the property $\tan^{-1}(-x) = -\tan^{-1}x$.

Therefore, the correct matching is: P $\rightarrow$ 2, Q $\rightarrow$ 1, R $\rightarrow$ 4, S $\rightarrow$ 1