A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^2$$/sec.

The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

The curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect each other orthogonally, then $$a^2=$$

The surface area of a spherical balloon is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the balloon is , when the radius of the balloon is $$6 \mathrm{~cm}$$.