An object is placed $20 \ cm$ in front of a $4 \ cm$ thick plane mirror. The image of the object is finally formed at $45 \ cm$ from the object itself. The refractive index of the material of the unpolished side of the mirror is (considering near normal incidence):

$A$ vessel of depth $2d \ cm$ is half filled with a liquid of refractive index $\mu_1$ and the upper half with a liquid of refractive index $\mu_2$. The apparent depth of the vessel seen perpendicularly is

Explain the lateral shift for the refraction of light through a rectangular glass slab.

$A$ ray of light is incident on a surface of a glass slab at an angle of $45^{\circ}$. If the lateral shift produced per unit thickness is $\frac{1}{\sqrt{3}}$,the angle of refraction produced is

Consider three media $P$, $Q$, and $R$ with refractive indices $1$, $1.25$, and $1.5$, respectively. The medium $Q$ having a thickness of $5 \text{ cm}$ is placed between extended media $P$ and $R$. An object $O$ is placed at the centre of medium $Q$. If viewed from medium $P$ near the normal direction, the apparent depth of $O$ is $h_1$. For similar observation from medium $R$, the apparent depth is $h_2$. The value of $|h_1 - h_2|$, in $\text{cm}$, is:

The observer at $O$ views two closely spaced spots on a vertical wall through an angled glass slab as shown. As seen by the observer,the spots appear