The rearrangement of phenyl esters to $o-$ and $p-$ hydroxy ketones in the presence of $AlCl_3$ is known as:

Kepler's second law (law of areas) is nothing but a statement of

In the $HCl$ molecule,the separation between the nuclei of the two atoms is about $1.27 \ \mathring{A}$. The approximate location of the centre of mass of the molecule from the hydrogen atom is ....... $\ \mathring{A}$. (Assuming the chlorine atom to be about $35.5$ times as massive as the hydrogen atom.)

An insect crawls up a hemispherical surface very slowly (see the 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

$\sin ^{-1} \frac{4}{5} + 2 \tan ^{-1} \frac{1}{3}$ is equal to

The set $\{x \in R: \cos 2x + 2 \cos^2 x = 2\}$ is equal to