An object is placed on the principal axis of a convex lens of focal length $10\,cm$ as shown. $A$ plane mirror is placed on the other side of the lens at a distance of $20\,cm$. The final image produced by the plane mirror is $5\,cm$ behind the mirror. The distance of the object from the lens is $............cm$.

The figure shows a glowing mercury tube. The illuminances at points $A$, $B$, and $C$ are related as

The dispersive power of the material of a lens of focal length $20 \; cm$ is $0.08$. The longitudinal chromatic aberration of the lens is ...... $cm$.

$A$ ray of light falls on the surface of a spherical glass paperweight,making an angle $\alpha$ with the normal,and is refracted in the medium at an angle $\beta$. What is the angle of deviation of the emergent ray from the direction of the incident ray?

An 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,a convex mirror is placed such that the image formed by the combination coincides with the object itself. If the distance between the lens and the mirror is $20\,cm$,find the focal length of the convex mirror in $cm$.

In the diagram shown,all the velocities are given with respect to the earth. What is the relative velocity of the image in mirror $(1)$ with respect to the image in mirror $(2)$? The mirror $(1)$ forms an angle $\beta$ with the vertical.