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

$\operatorname{sech}^{-1}\left(\frac{3}{5}\right)-\tanh ^{-1}\left(\frac{3}{5}\right)=$

The domain of the real valued function $f(x)=\sin ^{-1}\left(\log _{2}\left(\frac{x^{2}}{2}\right)\right)$ is

The trigonometric equation $\sin ^{-1} x=2 \sin ^{-1} a$, has a solution

If the real valued function $f(x)=\sin ^{-1}\left(x^2-1\right)-3 \log _3\left(3^x-2\right)$ is not defined for all $x \in(-\infty, a) \cup(b, \infty)$, then $3^a+b^2=$

If $\sin ^{-1}(4 x)-\cos ^{-1}(3 x)=\frac{\pi}{6}$, then $x=$

If $\sin h^{-1}(-\sqrt{3})+\cos ^{-1}(2)=K$, then $\cosh K=$

If $y=\cos ^{-1}\left(\frac{6 x-2 x^2-4}{2 x^2-6 x+5}\right)$, then $\frac{d y}{d x}=$

If $2 \tan ^{-1} x=3 \sin ^{-1} x$ and $x \neq 0$, then $8 x^2+1=$

Match the functions given in List I with their relevant characteristics from List II.

List I	List II
(A) sinh x	(I) Domains is (-1,1), even function
(B) sec hx	(II) Domain is [1,∞), neither even nor odd function
(C) tan hx	(III) Even function
(D) cosec h⁻¹x	(IV) Range is R, odd function
	(V) Range is (-1,1), odd function

The correct answer is

$\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\tan ^{-1} \frac{16}{63}=$

If $\cosh ^{-1}\left(\frac{5}{3}\right)+\sinh ^{-1}\left(\frac{3}{4}\right)=k$, then $e^k=$