If $y=\tan ^{-1}\left(\frac{x}{1+2 x^2}\right)+\tan ^{-1}\left(\frac{x}{1+6 x^2}\right)$, then $\frac{d y}{d x}=$

The range of the real valued function $f(x)=\cos ^{-1}(-x)+\sin ^{-1}(-x)+\operatorname{cosec}^{-1}(x)$ is

The horizontal distance between a tower and a building is $10 \sqrt{3}$ units. If the angle of depression of the foot of the building from the top of the tower is $60^{\circ}$ and the angle of elevation of the top of the building from the foot of the tower is $30^{\circ}$, then the sum of the heights of the tower and the building is

If $x$ is a real number, then the number of solutions of $\tan ^{-1}(\sqrt{x(x+1)})+\sin ^{-1}\left(\sqrt{x^2+x+1}\right)=\frac{\pi}{2}$ is