Light enters at an angle of incidence in a transparent rod of refractive index $n$. For what value of the refractive index of the material of the rod the light once entered into it will not leave it through its lateral face whatsoever be the value of angle of incidence

For a ray entering through the end face from air, let the refracted angle inside the rod with the axis be $r$. By Snell's law at the end face, $$\sin r=\frac{\sin i}{n}.$$ Hence the maximum possible value of $r$ is obtained for $\sin i=1$, so $$r_{\max}=\sin^{-1}\!\left(\frac{1}{n}\right).$$ At the lateral surface, the angle of incidence is $$90^\circ-r.$$ For the ray to never emerge from the lateral surface, this must always be at least the critical angle $c$, where $$\sin c=\frac{1}{n}.$$ In the worst case, $$90^\circ-r_{\max}\ge c.$$ Using $r_{\max}=c$, this becomes $$90^\circ-c\ge c.$$ So $$c\le 45^\circ.$$ Therefore, $$\frac{1}{n}=\sin c\le \sin 45^\circ=\frac{1}{\sqrt{2}}.$$ Thus $$n\ge \sqrt{2}.$$ Now checking the options, the only listed value consistent with this is that values less than $\sqrt{2}$ do not satisfy the condition, while $1, 1.1,$ and $1.3$ are all less than $\sqrt{2}$. Therefore the intended matching option is $\sqrt{2}$ as the threshold, so the marked choice is $\text{A}$ based on the given list.