1. Given information

• Three polarizers: $P_1, P_2, P_3$
• Axis of $P_3$ is crossed with respect to $P_1$ $\Rightarrow$ angle between axes of $P_1$ and $P_3$ is $90^\circ$.
• Axis of $P_2$ is inclined at $60^\circ$ to the axis of $P_3$.

Since $P_1$ and $P_3$ are crossed, the angle between $P_1$ and $P_2$ must be $90^\circ - 60^\circ = 30^\circ.$

So the sequence of angles is:

• between $P_1$ and $P_2$: $30^\circ$
• between $P_2$ and $P_3$: $60^\circ$

2. Intensity after first polarizer

Unpolarized light of intensity $I_0$ falls on $P_1$.

After passing through the first polarizer, intensity becomes $I_1 = \frac{I_0}{2}.$

3. Intensity after second polarizer

Apply Malus' law: $I_2 = I_1 \cos^2 30^\circ.$

Since $\cos 30^\circ = \frac{\sqrt{3}}{2}, \qquad \cos^2 30^\circ = \frac{3}{4},$ we get $I_2 = \frac{I_0}{2} \cdot \frac{3}{4} = \frac{3I_0}{8}.$

4. Intensity after third polarizer

Now angle between $P_2$ and $P_3$ is $60^\circ$. Again by Malus' law, $I = I_2 \cos^2 60^\circ.$

Since $\cos 60^\circ = \frac{1}{2}, \qquad \cos^2 60^\circ = \frac{1}{4},$ we get $I = \frac{3I_0}{8} \cdot \frac{1}{4} = \frac{3I_0}{32}.$

5. Required ratio

$\frac{I_0}{I} = \frac{I_0}{3I_0/32} = \frac{32}{3} \approx 10.67.$

6. Option matching

$\frac{I_0}{I} \approx 10.67$ So the correct option is A.

7. Comparison with stored answer

Stored correct answer: A

My derived answer: A

They agree.