The magnifications produced by a convex lens for two positions of an object are $4$ and $3$, respectively. If the distance of separation between the two positions of the object is $2 \,cm$, then the focal length of the lens is (in $\,cm$)

The dispersive powers of the glasses of lenses used in an achromatic pair are in the ratio $5 : 3$. If the focal length of the concave lens is $15 \ cm$,then the nature and focal length of the other lens would be:

Consider an equi-convex lens of glass $\left( \mu = \frac{3}{2} \right)$. If the temperature of the lens is increased by $40\, ^\circ C$,its focal length remains the same. The coefficient of linear expansion of glass is $2.5 \times 10^{-4} / ^\circ C$. Calculate the change in the refractive index of the glass on increasing the temperature by $40\, ^\circ C$.

$A$ lens made up of different materials forms two images of a point object $O$ as shown in the figure. Then:

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 $\beta = \sqrt{\frac{(n - 1)K_1 u}{v(u + v)}}$.

$A$ lamp is placed $6.0\; m$ from a wall. On putting a lens between the lamp and the wall at a distance of $1.2\; m$ from the lamp,a real image of the lamp is formed on the wall. The magnification of the image is