$A$ light beam is traveling from Region $I$ to Region $IV$ (Refer Figure). The refractive indices in Region $I, II, III$ and $IV$ are $n_0, \frac{n_0}{2}, \frac{n_0}{6}$ and $\frac{n_0}{8}$,respectively. The angle of incidence $\theta$ for which the beam just misses entering Region $IV$ is

$A$ $Ge$ specimen is doped with $Al$. The concentration of acceptor atoms is $\sim 10^{21} \, \text{atoms}/m^3$. Given that the intrinsic concentration of electron-hole pairs is $\sim 10^{19}/m^3$, the concentration of electrons in the specimen is:

What is the figure given below showing in particular?

Which of the following reagents would you use to separate $SO_4^{2-}$ ions from a solution containing a mixture of $SO_4^{2-}$ and $Cl^-$ ions?

The angle between the lines represented by $y^2 \sin^2 \theta - xy \sin^2 \theta + x^2(\cos^2 \theta - 1) = 0$ is