The solution of the differential equation $$\mathrm{e}^{-x}(y+1) \mathrm{d} y+\left(\cos ^2 x-\sin 2 x\right) y \mathrm{~d} x=0$$ at $$x=0$$, $$y=1$$ is

Rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant and it is found that the number doubles in 6 hours. The number of bacteria becomes ________ times at the end of 18 hours.

The particular solution of differential equation $$\mathrm{e}^{\frac{d y}{d x}}=(x+1), y(0)=3$$ is

A right circular cone has height $$9 \mathrm{~cm}$$ and radius of base $$5 \mathrm{~cm}$$. It is inverted and water is poured into it. If at any instant, the water level rises at the rate $$\frac{\pi}{\mathrm{A}} \mathrm{cm} / \mathrm{sec}$$. where $$\mathrm{A}$$ is area of the water surface at that instant, then cone is completely filled in