Two immiscible liquids of refractive indices $\frac{8}{5}$ and $\frac{3}{2}$ respectively are put in a beaker as shown in the figure. The height of each column is $6 \mathrm{cm}$. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $\frac{\alpha }{4} \mathrm{cm}$. The value of $\alpha$ is _______.

Electromagnetic waves emanating from a point $A$ (in air) are incident on a rectangular block of material $M$ and emerge from the other side as shown below. The angles $i$ and $r$ are angles of incidence and refraction when the wave travels from air to the medium. Such paths for the rays are possible,

A cubical vessel has opaque walls. An observer (dark circle in the figure below) is located, such that she can see only the wall, $CD$, but not the bottom. Nearly to what height should water be poured, so that she can see an object placed at the bottom at a distance of, $10 \mathrm{cm}$, from the corner, $C$? Refractive index of water is, $1.33$.

A diver, $30 \mathrm{m}$ below water surface is trying to shine a torch light towards the point O which is horizontally $30 \mathrm{m}$ away. The amount and direction in which diver has to swim (in the figure) so that the light can not travel outside the surface of water is (Refractive index of water $=\frac{4}{3}$)

The path of a light ray inside a transparent slab of variable refractive index is $y\mathit{=}4x^2\mathit{.}$ A light ray travelling in air falls on the surface of the slab at an angle $30°$ as shown in the figure. The refractive index of the slab at $y=0.5 \mathrm{m}$ is