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 $\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$ for $0 < a < \frac{1}{2}$ is

Let $f : R \to R$,$f(x) = \max\{|\tan^{-1}x|, \cot^{-1}x\}$. Consider the following statements:
$I.$ The function is continuous and derivable $\forall x \in R$.
$II.$ The range of the function is $\left[ \frac{\pi}{4}, \pi \right]$.
$III.$ $f(x)$ is many-one into.
Identify the correct option.

Considering the principal values of inverse trigonometric functions, the value of the expression $\tan\left(2 \sin^{-1}\left(\frac{2}{\sqrt{13}}\right)-2 \cos^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to:

Match List $I$ with List $II$ and select the correct answer using the code given below the lists:

List $I$	List $II$
$P$. $\left(\frac{1}{y^2}\left(\frac{\cos (\tan ^{-1} y)+y \sin (\tan ^{-1} y)}{\cot (\sin ^{-1} y)+\tan (\sin ^{-1} y)}\right)^2+y^4\right)^{1 / 2}$ takes value	$1$. $\frac{1}{2} \sqrt{\frac{5}{3}}$
$Q$. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is	$2$. $\sqrt{2}$
$R$. If $\cos (\frac{\pi}{4}-x) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+\cos (\frac{\pi}{4}+x) \cos 2 x$ then possible value of $\sec x$ is	$3$. $\frac{1}{2}$
$S$. If $\cot (\sin ^{-1} \sqrt{1-x^2})=\sin (\tan ^{-1}(x \sqrt{6})), x \neq 0$,then possible value of $x$ is	$4$. $1$

Codes: $P \quad Q \quad R \quad S$

If $y = \tan^{-1} \left( \frac{\log(e/x^3)}{\log(ex^3)} \right) + \tan^{-1} \left( \frac{\log(e^4x^3)}{\log(e/x^{12})} \right)$, for $x \in (e^{-1/3}, e^{1/12})$, then $\frac{dy}{dx}$ is equal to...

The number of solutions of $\tan^{-1} 2x + \tan^{-1} 3x = \frac{\pi}{4}$ is