1. Intensity after first polaroid

   The incident light is unpolarised with intensity $2I_0.$

   When unpolarised light passes through a polaroid, its intensity becomes half: $I_1 = \frac{1}{2}(2I_0) = I_0.$

2. Intensity after second polaroid

   The second polaroid $Q$ makes an angle of $30^\circ$ with the first polaroid $P$.

   By Malus' law, $I_2 = I_1 \cos^2 \theta$ where $\theta = 30^\circ$.

   So, $I_2 = I_0 \cos^2 30^\circ.$

   Since $\cos 30^\circ = \frac{\sqrt{3}}{2},$ we get $\cos^2 30^\circ = \frac{3}{4}.$

   Therefore, $I_2 = I_0 \cdot \frac{3}{4} = \frac{3I_0}{4}.$

3. Option check

   • A: $\frac{I_0}{4}$ ❌
   • B: $\frac{I_0}{2}$ ❌
   • C: $\frac{3I_0}{4}$ ✅
   • D: $\frac{3I_0}{2}$ ❌

Hence, the intensity of the emergent light is $\boxed{\frac{3I_0}{4}}.$