$A$ thin lens with focal length $f$ is to be used as a magnifying glass. Which of the following statements regarding the situation is true?

$A$ plano-convex lens is made of a material with a refractive index of $1.6$. The radius of curvature of the curved surface is $60 \ cm$. The focal length of the lens is.....$cm$.

The distance between an object and the screen is $100 \ cm$. $A$ lens produces an image on the screen when placed at either of the positions $40 \ cm$ apart. The power of the lens is

$A$ thin lens of material having refractive index $\mu = 1.5$ and focal length of $20 \ cm$ in air has two media of different refractive indices $\mu_1 = 1.2$ and $\mu_2 = 2.5$ covering the upper and lower halves of the lens,respectively,as shown in the figure. If an object is placed on the principal axis,then its two images will be formed,one after refraction from the upper part and the other after refraction from the lower part. Considering the object to be at $\infty$,the separation between the two images formed would be ...... $cm$.

An achromatic telescope objective is to be made by combining the lenses of flint and crown glasses. This proper choice is

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