$A$ glass beaker is filled with water up to $5 \,cm$. It is kept on top of a $2 \,cm$ thick glass slab. When a coin at the bottom of the glass slab is viewed at normal incidence from above the beaker,its apparent depth from the water surface is $d \,cm$. The value of $d$ is close to ........ $cm$ (the refractive indices of water and glass are $1.33$ and $1.5$,respectively).

$A$ paraxial beam is incident on a glass $(n = 1.5)$ hemisphere of radius $R = 6\, cm$ in air as shown. The distance of the point of convergence $F$ from the plane surface of the hemisphere is......$cm$.

$A$ composite slab consisting of different media is placed in front of a concave mirror of radius of curvature $150 \, cm$. The whole arrangement is placed in water (refractive index $\mu = 4/3$). An object $O$ is placed at a distance $20 \, cm$ from the slab. The refractive indices of the different media are given in the diagram. Find the final image formed by the system.

How much water should be filled in a container $21 \ cm$ in height,so that it appears half-filled when viewed from the top of the container? (Given that refractive index of water $\mu = 4/3$)

$A$ narrow,paraxial beam of light is converging towards a point $I$ on a screen. $A$ plane parallel plate of glass of thickness $t$ and refractive index $\mu$ is introduced in the path of the beam. The convergence point is shifted by

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. . . . .