If $$P(1)=0$$ and $${{dp\left( x \right)} \over {dx}} > P\left( x \right)$$ for all $$x \ge 1$$ then prove that
$$P(x)>0$$ for all $$x>1$$.

If $$f$$ is an even function then prove that
$$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $$

A right circular cone with radius $$R$$ and height $$H$$ contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant $$ = k > 0$$. Find the time after which the come is empty.

$$A$$ is targeting to $$B, B$$ and $$C$$ are targeting to $$A.$$ Probability of hitting the target by $$A,B$$ and $$C$$ are $${2 \over 3},{1 \over 2}$$ and $${1 \over 3}$$ respectively. If $$A$$ is hit then find the probability that $$B$$ hits the target and $$C$$ does not.