$A$ glass slab of thickness $3 \ cm$ is placed on an ink mark on a piece of paper. For a person looking at the mark from a distance of $5.0 \ cm$ above the top surface of the slab, the mark appears to be at a distance of $4.0 \ cm$ from the observer. The refractive index of the slab is:

$A$ ray of light is incident at an angle of incidence $60^{\circ}$ on a glass slab of refractive index $\sqrt{3}$. After refraction,the light ray emerges out from the other parallel face,and the lateral shift between the incident ray and the emergent ray is $4 \sqrt{3} \, cm$. The thickness of the glass slab is . . . $cm$.

In determining the refractive index of a glass slab using a travelling microscope,the following readings are tabulated:
$(a)$ Reading of travelling microscope for ink mark $= 5.123 \ cm$
$(b)$ Reading of travelling microscope for ink mark through glass slab $= 6.123 \ cm$
$(c)$ Reading of travelling microscope for chalk dust on glass slab $= 8.123 \ cm$
From the data,the refractive index of a glass slab is:

$A$ microscope is focussed on a coin lying at the bottom of a beaker. The microscope is now raised up by $1 \, cm$. To what depth should the water be poured into the beaker so that the coin is again in focus? (Refractive index of water is $\frac{4}{3}$)

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.

$A$ ray of light is incident normally on a glass slab of thickness $5 \ cm$ and refractive index $1.6$. The time taken by a ray to travel from the source of light to the surface of the slab is the same as the time taken to travel through the glass slab. The distance of the source from the surface is: (in $cm$)