The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ w.r.t. $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^2}}{1-2 x^2}\right)$ at $x=0$ is

The normal to the curve $x=9(1+\cos \theta)$, $y=9 \sin \theta$ at $\theta$ always passes through the fixed point

In a triangle ABC with usual notations, if $3 \mathrm{a}=\mathrm{b}+\mathrm{c}$, then $\cot \frac{\mathrm{B}}{2} \cdot \cot \frac{\mathrm{C}}{2}=$

If $p$ : switch $S_1$ is closed, $q$ : switch $S_2$ is closed, $r$ : switch $S_3$ closed, then the symbolic form of the following switching circuit is equivalent to

Switching Circuit: