Unpolarized light of intensity $I$ passes through an ideal polariser $A$. Another identical polariser $B$ is placed behind $A$. The intensity of light beyond $B$ is found to be $\dfrac I2$. Now another identical polariser $C$ is placed between $A$ and $B$. The intensity beyond $B$ is now found to be $\dfrac I8$. The angle between polariser $A$ and $C$ is:
Answer: (D) $45^\circ$
After $A$ the intensity is $\dfrac I2$; it stays $\dfrac I2$ after $B$, so $A$ and $B$ are parallel.
With $C$ at angle $\theta$ to $A$ (and hence to $B$), Malus's law twice gives
$$\frac I2\cos^2\theta\cos^2\theta = \frac I8 \;\Rightarrow\; \cos^4\theta = \frac14 \;\Rightarrow\; \cos^2\theta = \frac12 \;\Rightarrow\; \theta = 45^\circ$$
Solution by Sreeraj P, M.Sc Physics