A symmetrical converging convex lens of focal length $10 \mathrm{cm}$ & diverging concave symmetrical lens of focal length $- 20 \mathrm{cm}$ are cut from the middle and perpendicularly and symmetrically to their principal axis. The parts thus obtained are arranged as shown in the figure. Find the focal length $(\mathrm{in} \mathrm{cm})$ of this arrangement.

 

In the given figure, a stationary observer $O$ looking at a fish $F$ (in water of, $\mu =\frac{4}{3}$) through a converging lens of focal length $90 \mathrm{cm}$. The lens is allowed to fall freely from a the height $62.0 \mathrm{cm}$ with its axis vertical. The fish and the observer are on the principal axis of the lens. The fish moves up with constant velocity $100 \mathrm{cm} {\mathrm{s}}^{-1}$. Initially, it was at a depth of $44.0 \mathrm{cm}$. The velocity with which the fish appears to move to the observer at $t=0.2 \mathrm{s}$ is, $\left(x+\frac{3}{4}\right) \mathrm{m} {\mathrm{s}}^{-1}$. Find the value of $x. (g\mathit{=}10 \mathrm{m} {\mathrm{s}}^{-2} )$

A glass rod of the refractive index $1.50$of rectangular cross-section, $\left\{\mathrm{d}\times l\right\}$ is bent into a $“\mathrm{U}”$ shape see figure (A). The cross-sectional view of this rod is shown in figure (B).

The bent portion of the rod is semi-circular with inner and outer radii $R$and $R+d$, respectively. A parallel monochromatic beam of light is incident normally on face $ABCD$.

(a) Consider two monochromatic rays $r_1$ and $r_2$ in figure(B). State whether the following statements are True or False.

(b) Consider the ray $r_1$ whose point of incidence is very close to the edge $\mathrm{BC}$. Assume it undergoes total internal reflection at $p_1$. In cross-sectional view below, draw the trajectory of this reflected ray beyond the next glass-air boundary that it encounters.

(c) Obtain the minimum value of the ratio $\frac{R}{d}$ for which any light ray entering the glass normally through the face $ABCD$ undergoes at least one total internal reflection.

(d) A glass rod with the above computed minimum ratio of $\frac{R}{d}$, is fully immersed water of refractive index $1.33$. What fraction of light flux entering the glass through the plane surface $ABCD$ undergoes at least one total internal reflection?

The following figure shows the section $ABC$ of an equilateral triangular prism. A ray of light enters the prism along$\mathit{ }LM$ and emerges along $QD$. If the refractive index of the material of the prism is $1.6$, angle $LMN$ is