$A$ glass sphere of refractive index $1.5$ and radius $40\,cm$ has half its hemispherical surface silvered. The point where a parallel beam of light,coming along a diameter,will focus (or appear to) after coming out of the sphere,will be:

$A$ plane mirror is placed $22.5\; cm$ in front of a concave mirror of focal length $10\; cm$. An object is placed between the two mirrors such that the images formed by both mirrors coincide. Find the magnification of the image produced by the concave mirror.

$A$ glass beaker has a solid,plano-convex base of refractive index $1.60$,as shown in the figure. The radius of curvature of the convex surface $(SPU)$ is $9 \ cm$,while the planar surface $(STU)$ acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level $QPR$. If the image of a point object $O$ at a height of $h$ ($OT$ in the figure) is formed onto itself,then which of the following option$(s)$ is (are) correct?
$(A)$ For $n=1.42, h=50 \ cm$
$(B)$ For $n=1.35, h=36 \ cm$
$(C)$ For $n=1.45, h=65 \ cm$
$(D)$ For $n=1.48, h=85 \ cm$

If $\hat{n}_1$ is the unit vector along the incident ray,$\hat{n}_2$ is the unit vector along the normal,and $\hat{n}_3$ is the unit vector along the reflected ray,then which of the following must be true?

$A$ water drop in air refracts the light ray as:

$A$ cubic block of glass with refractive index $n_1$ is in contact with the surface of water with refractive index $n_2$. $A$ ray of light is incident on the vertical face of the block (see figure). After refraction,total internal reflection at the base,and refraction at the opposite vertical face,the ray emerges at an angle $\theta$. The value of $\theta$ is given by: