The number of real solutions of $\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\frac{\pi}{2}$ is

Let $S=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0$ in the set S is equal to

The number of four letter words that can be formed using letters of the word BARRACK

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circular wave is 12 cm . how fast is the enclosed area increasing?