$A$ convex lens of focal length $20 \, cm$ and a concave mirror of focal length $f$ are placed coaxially at a distance of $5 \, cm$ apart. $A$ parallel beam of light incident on the convex lens emerges from the system as a parallel beam. Then $f = \dots \, cm$.

$A$ point source is placed $20 \,cm$ to the left of a concave lens of focal length $10 \,cm$.
$(a)$ Where is the image formed?
$(b)$ Where to the right of the lens would you place a concave mirror of focal length $5 \,cm$,so that the final image is coincident with the source?
$(c)$ Where would the final image be formed,if the concave mirror is replaced by a plane mirror at the same position?

$A$ 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 $cm$?

$A$ planoconvex lens,when silvered at its plane surface,is equivalent to a concave mirror of focal length $28 \ cm$. When its curved surface is silvered and the plane surface is not silvered,it is equivalent to a concave mirror of focal length $10 \ cm$. The refractive index of the material of the lens is:

$A$ symmetric double convex lens is cut into two equal parts by a plane perpendicular to the principal axis. If the power of the original lens was $4 \ D$,what will be the power of each cut lens (in $D$)?

$A$ 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.