If $\tan^{-1} (x^2 + 3|x|-4) = \tan^{-1} (4\pi + \sin^{-1}(\sin 14))$,then the value of $\cos^{-1}(\cos 3|x|)$ is equal to

Let $Z$ denote the set of integers. Then match the items in List-$I$ with those of the items in List-$II$.

List-$I$	List-$II$
$A$. $\sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)+\sin ^{-1} \frac{1}{3}$	$I$. $k \pi \pm(-1)^k \frac{\pi}{6}, k \in Z$
$B$. $\sin ^{-1}\left(\frac{(-1)^n}{2}\right), n \in Z$	$II$. $k \pi \pm 1, k \in Z$
$C$. $\tan ^{-1}\left(\sec \frac{\pi}{4}+\tan \frac{\pi}{4}\right)$	$III$. $\frac{3}{2}$
$D$. $\sin ^{-1}|\sin x|=\sqrt{\sin ^{-1}|\sin x|} \Rightarrow x \in$	$IV$. $\frac{3 \pi}{8}$
	$V$. $\frac{\pi}{2}$

The correct match is:

$\sin ^{-1}\left(\frac{12}{13}\right)+\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{63}{16}\right)=$

If $f(x) = \cot^{-1} \left( \frac{3x - x^3}{1 - 3x^2} \right)$ and $g(x) = \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)$,then for $0 < a < \frac{1}{\sqrt{3}}$,the value of $\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$ is

Identify the pair$(s)$ of functions which are identical.

Solve $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2},$ then $x$ is equal to