A water tank has a shape of inverted right circular cone whose semi-vertical angle is $$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of water is rising, at the instant when depth of water in the tank is $$10 \mathrm{~m}$$ is

The differential equation of all circles which pass through the origin and whose centres lie on $$\mathrm{Y}$$-axis is

If $$x^{\mathrm{k}}+y^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}}(\mathrm{a}, \mathrm{k}>0)$$ and $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$$, then $$\mathrm{k}$$ has the value

The area (in sq. units) bounded by the curve $$y=x|x|, \mathrm{X}$$-axis and the lines $$x=-1$$ and $$x=1$$ is