When a small object is placed at $\mathrm{A}$ on the axis of a convex lens of focal length $\mathrm{f}$, the image is erect. When the object is moved to $\mathrm{B}$, the image is inverted but of the same size as before. If $\mathrm{m}$ be the magnification, show that $\mathrm{AB}=\frac{2\mathrm{f}}{\mathrm{m}}$.

A thin Plano-convex lens of focal length $f$ is split into two halves; one of the halves is shifted along the optical axis [Fig. 14.71]. The separation between object and image is $1.8 \mathrm{~m}$. The magnification of the image formed by one of the half-lenses is $2$. Find the focal length of the lens and the separation between the two halves. Draw the ray diagram for image formation.

What is relation between the refractive index ${\mathrm{\mu }}_1$ and ${\mathrm{\mu }}_2$ If the behaviour of light rays is as shown in the Fig. 

The radius of curvature of the convex face of a Plano convex lens is $12 \mathrm{cm}$ and its $\mathrm{\mu }=1.5$. Find the focal length of the lens. The plane face of the lens is now silvered.