In the diagram shown,the lens is moving towards the object with a velocity $V \, m/s$ and the object is also moving towards the lens with the same speed. What is the speed of the image with respect to the earth when the object is at a distance $2f$ from the lens? ($f$ is the focal length.)

Consider a thin lens placed between a source $(S)$ and an observer $(O)$ (See figure). Let the thickness of the lens vary as $w(b) = w_0 - \alpha b^2$, where $b$ is the vertical distance from the pole. $w_0$ and $\alpha$ are constants. Using Fermat's principle, i.e., the time of transit for a ray between the source and observer is an extremum, find the condition that all paraxial rays starting from the source will converge at a point $O$ on the axis. Find the focal length.
$(ii)$ $A$ gravitational lens may be assumed to have a varying width of the form $W = K_1 \log \left( \frac{K_2}{b} \right)$ (where $b_{\min} < b < b_{\max}$). Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius $\beta = \sqrt{\frac{(n - 1)K_1 u}{v(u + v)}}$.

The refractive index of the material of a concave lens is $\mu$. It is immersed in a medium of refractive index $\mu_1$. $A$ parallel beam of light is incident on the lens. The path of the emergent rays when $\mu_1 > \mu$ is

$A$ film projector magnifies a film onto a screen of area $100 \, cm^2$. If the linear magnification is $4$,what will be the area of the image on the screen in $cm^2$?

An object and a screen are fixed on the uprights of an optical bench. The distance between them is $100 \, cm$. $A$ convex lens is placed in between the object and the screen and the position of the lens is so adjusted that the image of the object is formed on the screen at two conjugate positions of the lens. The distance between these conjugate positions of the lens is $40 \, cm$. What is the focal length of the lens in $cm$?

Two transparent media of refractive indices $\mu_1$ and $\mu_3$ have a solid lens-shaped transparent material of refractive index $\mu_2$ between them as shown in the figures in Column $II$. $A$ ray traversing these media is also shown in the figures. In Column $I$,different relationships between $\mu_1, \mu_2$,and $\mu_3$ are given. Match them to the ray diagram shown in Column $II$.