A quarter of a cylinder of $\mu =\sqrt{3}$ and radius $R=2 \mathrm{cm}$ is placed as shown. The surface $OB$ is silvered. A ray of light making angle $30°$ is incident on cylinder as shown in figure. If the ray retraces its path back, the $x$-coordinate of point $P$ is

A bulb is at the centre of base of a hollow cube of side length $a.$ It is half filled with water $\left(\mu =\frac{4}{3}\right).$ The area on the ceiling illuminated by light is

A concave mirror of radius of curvature $R=40 \mathrm{cm}$ is placed inside water $\left(\mu =\frac{4}{3}\right)$ of depth $30 \mathrm{cm}.$ A parallel beam of light is incident on water surface as shown. The distance of final image is

A point object is placed in the arrangement shown. The distance of image formed by lens-$2$ from its optical centre is

An equiconvex lens $(\mu =1.5)$ of focal length $f=20 \mathrm{cm}$ is placed on a plane mirror. A liquid of $\mu =\frac{4}{3}$ is filled between the gap as shown. At what distance from the lens an object should be placed so that a virtual image of double the size of object is formed?