$A$ luminous object is placed $20 \, cm$ from the surface of a convex mirror and a plane mirror is adjusted in such a way that the virtual images formed by the two mirrors coincide. If the focal length of the convex mirror is $5 \, cm$,then the distance between the plane mirror and the object will be......$cm$.

During a cloudy day,a primary and a secondary rainbow may be created,then the:

$A$ 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:

$A$ ray of light strikes a plane mirror $M$ at an angle of $45^{\circ}$ as shown in the figure. After reflection,the ray passes through a prism of refractive index $1.5$ whose apex angle is $4^{\circ}$. The total angle through which the ray is deviated is......$^{\circ}$

$A$ camera lens is set at $f/2$ with an exposure time of $1/100 \ s$. If the lens is set at $f/4$,what will be the correct exposure time?

$A$ solid hemisphere of refractive index $\mu_1 = 3/2$ and radius $R = 20\ cm$ is placed as shown in the figure. Another special cylindrical glass of refractive index $\mu_2$,radius $R$,and thickness $t = 2R/3$ is placed adjacent to the flat surface of the solid hemisphere. There is a very small luminous spot at $P$ just inside the solid hemisphere,and the $MN$ surface of the special glass is silvered. Considering only paraxial rays,if the final image of point $P$ for an observer to the left of the hemisphere is at the center $C$ of the hemisphere,find the refractive index $\mu_2$ of the special glass.