Question

If equation $\sin ^{-1} \sqrt{ x }+\cos ^{-1} \sqrt{ x ^2-1}+\tan ^{-1} \tan y = k$ has atleast one solution, then $k \in$ $\left(\frac{p \pi}{2}, \frac{q \pi}{2}\right)$, where $p, q \in I$, then value of $(p+q)$ is greater than or equal to

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (A), (B), (C)

$ \sin ^{-1} \sqrt{ x }+\cos ^{-1} \sqrt{ x ^2-1}+\tan ^{-1} \tan y = k$
$0 \leq x \leq 1$ and $0 \leq x ^2-1 \leq 1$
Hence, $x=1$
$\frac{\pi}{2}+\frac{\pi}{2}+\tan ^{-1} \tan y = k $
$\frac{-\pi}{2}<\tan ^{-1} \tan y <\frac{\pi}{2}$
$\therefore k \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
Hence, $(p+q)=4$.