For the equation $\cos^{-1} x + \cos^{-1} 2x + \pi = 0$,the number of real solutions is:

The value of $\sec ^2(\tan ^{-1} 2)+\operatorname{cosec}^2(\cot ^{-1} 3)$ is

Assume that $3.13 \leq \pi \leq 3.15$. The integer closest to the value of $\sin ^{-1}(\sin 1 \cos 4+\cos 1 \sin 4)$,where $1$ and $4$ appearing in $\sin$ and $\cos$ are given in radians,is

If $\sin ^{-1} x + \sin ^{-1}(1-x) = \cos ^{-1} x$, then $x \in$

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

If $x, y, z$ are in $A.P.$ and $\tan^{-1} x, \tan^{-1} y, \tan^{-1} z$ are also in another $A.P.$,then: