When the plane surface of a plano-convex lens is silvered,it behaves as a concave mirror of focal length $60 \ cm$. However,when the convex surface is silvered,it behaves as a concave mirror of focal length $20 \ cm$. What is the refractive index of the lens?

$A$ glass slab of thickness $6 \, cm$ is silvered at one surface. An object is placed at a distance of $8 \, cm$ from the first surface. The image is formed at a distance of $12 \, cm$ behind the silvered surface. Find the refractive index of the glass.

$A$ point source of light is placed at $2f$ from a converging lens of focal length $f$. $A$ flat mirror is placed on the other side of the lens at a distance $d$ such that rays reflected from the mirror are parallel after passing through the lens again. If $f=30 \, cm$,then $d$ is equal to ............. $cm$.

The two surfaces of a biconvex lens have the same radii of curvature. This lens is made of glass of refractive index $1.5$ and has a focal length $10 \ cm$ in air. The lens is cut into two equal halves along a plane perpendicular to its principal axis to yield two plano-convex lenses. The two pieces are glued such that the convex surfaces touch each other. If this combination lens is immersed in water (refractive index $= 4/3$), its focal length (in $cm$) is:

$A$ concave mirror of focal length $f_1$ is placed at a distance of $d$ from a convex lens of focal length $f_2$. $A$ beam of light coming from infinity and falling on this convex lens-concave mirror combination returns to infinity. The distance $d$ must equal:

$A$ plano-convex lens has a focal length of $20\,cm$. If its plane surface is silvered,it behaves as a converging mirror. What will be its focal length in $cm$?