One thousand small water drops of equal radii combine to form a big drop. The ratio of final surface energy to the total initial surface energy is

If $u = x y^2 \tan^{-1}\left(\frac{y}{x}\right)$,then $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$ is equal to

An insect crawls up a hemispherical surface very slowly (see figure). The coefficient of friction between the insect and the surface is $\mu = 1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical,the maximum possible value of $\alpha$ is given by

The radiation emitted by a star $A$ is $10000$ times that of the sun. If the surface temperatures of the sun and the star $A$ are $6000 \ K$ and $2000 \ K$ respectively,the ratio of the radii of the star $A$ and the sun is : (in $: 1$)

$y=\log \left\{\left(\frac{1+x}{1-x}\right)^{1 / 4}\right\}-\frac{1}{2} \tan ^{-1}(x)$,then $\frac{d y}{d x}$ is equal to

In which of the following reactions is an aromatic aldehyde treated with an acid anhydride in the presence of the corresponding salt of the acid to give an unsaturated aromatic acid?