A ray of light passes from vacuum into a medium of refractive index $n.$ If the angle of incidence is twice the angle of refraction, the angle of refraction is

Two plane mirrors $M_1$ and $M_2$ intersecting at point $O$ forming an angle of $60°.$ The distance between the two direct images of a point object $\text{'}P\text{'}$ formed by them is $10 \mathrm{cm}.$ The distance $OP$ is

A hemispherical glass paper weight of radius $10 \mathrm{cm}$ contains a small air bubble located at a distance of $4 \mathrm{cm}$ from its flat surface on its axis of symmetry. What is apparent shift in the position of air bubble when it is looked by observer from top.
(Refractive index of glass $=\frac{4}{3}$)

A glass $\left({\mu }_g=\sqrt{2}\right)$ cube of edge length $5 \mathrm{cm}$ placed on horizontal surface has a point source of light $\text{'}S\text{'}$ placed at centre of face. What should be minimum area to be covered on side faces so that no direct light comes out of the side faces.

A convex lens is held $60 \mathrm{cm}$ above the bottom of an empty tank. The image of a point $\text{'}P\text{'}$ (on tank) is formed $30 \mathrm{cm}$ above the lens. Image of same point $\text{'}P\text{'}$ (on tank) shifts by $15 \mathrm{cm}$ when tank is filled with liquid up to height of $40 \mathrm{cm}.$ Refractive index of liquid, is

For the system shown in figure, for what value of $d$ image of point object $\text{'}O\text{'}$ is formed on object itself

For the system shown, if the speed of image formed by the plane mirror is minimum then speed, $v$ of the plane mirror is