$A$ double convex thin lens made of glass of refractive index $1.6$ has radii of curvature $15 \ cm$ each. The focal length of this lens when immersed in a liquid of refractive index $1.63$ is.......$cm$.

$A$ convex lens of refractive index $\frac{3}{2}$ has a power of $2.5 \ D$. If it is placed in a liquid of refractive index $2$, the new power of the lens is: (in $D$)

If the image formed by a thin convex lens of power $P$ has magnification $m$,then the image distance $v$ is:

An object of height $1.5 \ cm$ is placed on the axis of a convex lens of focal length $25 \ cm$. $A$ real image is formed at a distance of $75 \ cm$ from the lens. The size of the image will be........$cm$.

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

$A$ plano-convex lens is made of material having refractive index $1.5$. The radius of curvature of the curved surface is $60 \ cm$. The focal length of the lens is . . . . . . $cm$.