$A$ 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 $f = \dots \, cm$.

$A$ luminous object is placed at a distance of $30 \,cm$ from a convex lens of focal length $20 \,cm$. On the other side of the lens,at what distance from the lens should a convex mirror of radius of curvature $10 \,cm$ be placed in order to have an upright image of the object coincident with it (in $,cm$)?

$A$ 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.

The figure shows an equiconvex lens (of refractive index $1.50$) in contact with a liquid layer on top of a plane mirror. $A$ small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be $45.0 \; cm$. The liquid is removed and the experiment is repeated. The new distance is measured to be $30.0 \; cm$. What is the refractive index of the liquid?

$A$ plano-convex lens,when silvered on the plane side,behaves like a concave mirror of focal length $30 \ cm$. However,when silvered on the convex side,it behaves like a concave mirror of focal length $10 \ cm$. The refractive index of its material is:

$A$ point-like object is placed at a distance of $1\, m$ in front of a convex lens of focal length $0.5\, m$. $A$ plane mirror is placed at a distance of $2\, m$ behind the lens. Find the position and nature of the final image formed by the system.