$\mathop {\lim }\limits_{x \to 0} \frac{{x\tan 2x - 2x\tan x}}{{{{(1 - \cos 2x)}^2}}}$ is

The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If two more normals to this parabola also pass through $(5,0)$,then the centroid of the triangle formed by the feet of these three normals is

$A$ vessel contains $110\,g$ of water. The heat capacity of the vessel is equivalent to $10\,g$ of water. The initial temperature of the water in the vessel is $10\,^{\circ}C$. If $220\,g$ of hot water at $70\,^{\circ}C$ is poured into the vessel,the final temperature,neglecting radiation loss,will be ........ $^{\circ}C$.

Fossil $X$ is older than fossil $Y$ if

The equation $\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos \theta$ represents

$A$ fresh radioactive sample is given at $t = 0$. Its decay fraction is $\frac{1}{5}$ at $t_1$ instant and $\frac{4}{5}$ at $t_2$ instant. Its mean life is