$A$ large glass slab $\left( \mu = \frac{5}{3} \right)$ of thickness $8 \, cm$ is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius $R \, cm$. What is the value of $R$? ....... $cm$

$A$ tiny air bubble in a glass slab $(\mu = 1.5)$ appears from one side to be $6 \, cm$ from the glass surface and from the other side,$4 \, cm$. The thickness of the glass slab is......$cm$

$A$ glass hemisphere of radius $0.04\, m$ and refractive index $1.6$ is placed centrally over a cross mark on a paper:
$(i)$ with the flat face in contact with the paper;
$(ii)$ with the curved face in contact with the paper.
In each case,the cross mark is viewed directly from above. The positions of the images will be:

When a convex lens is placed above an empty tank,the image of a mark at the bottom of the tank,which is $45 \ cm$ from the lens,is formed $36 \ cm$ above the lens. When a liquid is poured in the tank to a depth of $40 \ cm$,the distance of the image of the mark above the lens is $48 \ cm$. The refractive index of the liquid is

The bottom of a container is a glass slab of thickness $4 \ cm$ and refractive index $\mu = 1.5$. The container contains two immiscible liquids $A$ and $B$ of depths $6 \ cm$ and $8 \ cm$ respectively. When a crack at the bottom surface of the glass slab is viewed from above,by what distance (in $cm$) does it appear to be shifted? The refractive indices of $A$ and $B$ are $1.4$ and $1.3$ respectively.

$A$ parallel-sided block of glass with a refractive index of $1.5$ and a thickness of $36 \, mm$ rests on the floor of a tank filled with water (refractive index $= 4/3$). The difference between the apparent depth of the floor at $A$ (where the glass block is present) and $B$ (where only water is present) when seen from vertically above is equal to $....... \, mm$.