A ray of light is incident at an angle of ${45}^{\circ }$ from air to a glass slab at point $A,$ the angle of incidence being ${45}^{\circ }$. The minimum value of the refractive index of the material of the slab so that TIR takes place at $B$ is

The refractive index of the material of the prism and liquid are $1.56$ and $1.32$, respectively. What will be the value of $\theta$ for the following refraction?

A plano-convex thin lens is used to obtain the image of a point object $O$ on the screen $S$ as shown in figure. The thickness of the lens in the middle is $0.5 \mathrm{cm}$ and refractive index of the material of the lens is $1.5.$ If the separation $D$ between the object and the screen is to be minimum, then the aperture diameter of the lens (in $\mathrm{cm}$ ) should be (consider lens as thin).

The image for the converging beam after refraction through the curved surface (in the given figure) is formed at: