Area of the region bounded by the curve $$y=\sqrt{49-x^2}$$ and $$\mathrm{X}$$-axis is

The solution of the equation $$\tan ^{-1}(1+x)+\tan ^{-1}(1-x)=\frac{\pi}{2}$$ is

The differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}$$ determines a family of circles with

A square plate is contracting at the uniform rate $$4 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate at which the perimeter is decreasing, when side of the square is $$20 \mathrm{~cm}$$, is