$A$ beaker contains water up to a height $h_{1}$ and kerosene of height $h_{2}$ above water. The total height of (water $+$ kerosene) is $(h_{1} + h_{2})$. The refractive index of water is $\mu_{1}$ and that of kerosene is $\mu_{2}$. The apparent shift in the position of the bottom of the beaker when viewed from above is:

$A$ diverging beam of light from a point source $S$ having divergence angle $\alpha$,falls symmetrically on a glass slab as shown. The angles of incidence of the two extreme rays are equal. If the thickness of the glass slab is $t$ and the refractive index is $n$,then the divergence angle of the emergent beam is

$A$ beaker containing liquid is placed on a table,underneath a microscope which can be moved along a vertical scale. The microscope is focused,through the liquid,onto a mark on the table when the reading on the scale is $a$. It is next focused on the upper surface of the liquid and the reading is $b$. More liquid is added and the observations are repeated,the corresponding readings are $c$ and $d$. The refractive index of the liquid is

The figure shows a transparent slab of length $1 \, m$ placed in air,whose refractive index in the $x$-direction varies as $\mu = 1 + x^2$ for $0 \leq x \leq 1$. The optical path length of ray $R$ will be:

Consider a configuration of $n$ identical units,each consisting of three layers. The first layer is a column of air of height $h=\frac{1}{3} \text{ cm}$,and the second and third layers are of equal thickness $d=\frac{\sqrt{3}-1}{2} \text{ cm}$,and refractive indices $\mu_1=\sqrt{\frac{3}{2}}$ and $\mu_2=\sqrt{3}$,respectively. $A$ light source $O$ is placed on the top of the first unit,as shown in the figure. $A$ ray of light from $O$ is incident on the second layer of the first unit at an angle of $\theta=60^{\circ}$ to the normal. For a specific value of $n$,the ray of light emerges from the bottom of the configuration at a distance $l=\frac{8}{\sqrt{3}} \text{ cm}$,as shown in the figure. The value of $n$ is. . . . .

Three immiscible transparent liquids with uniform refractive indices $\frac{3}{2}, \frac{4}{3}$ and $\frac{6}{5}$ are arranged one on top of another. The depths of the liquids are $3 \ cm, 4 \ cm$ and $6 \ cm$ respectively. The apparent depth of the vessel is (in $cm$)