List-I shows four planar structures made of uniform solid rods each of mass $$m$$ and length $$l$$. In the List-II the possible moment of inertia of these structures about an axis $$OCO'$$, which lies in the plane of the structures, are given. Choose the option that describes the correct match between the entries in List-I to those in List-II.

For structure (P):

$$I_{CA} = \frac{1}{3}ml^2\sin^2 45^\circ = \frac{1}{6}ml^2$$

$$I_{CB} = \frac{1}{3}ml^2\sin^2 45^\circ = \frac{1}{6}ml^2$$

$$I_P = \frac{1}{6}ml^2 + \frac{1}{6}ml^2 = \frac{1}{3}ml^2 \implies (\text{P}) \rightarrow (5)$$

For structure (Q):

$$I_{CA} = \frac{1}{3}ml^2\sin^2 60^\circ = \frac{1}{4}ml^2$$

$$I_{CB} = \frac{1}{3}ml^2\sin^2 60^\circ = \frac{1}{4}ml^2$$

$$I_{AB} = I_{\text{cm}} + md^2 = \frac{1}{12}ml^2 + m\left(l\sin 60^\circ\right)^2 = \frac{1}{12}ml^2 + \frac{3}{4}ml^2 = \frac{5}{6}ml^2$$

$$I_Q = \frac{1}{4}ml^2 + \frac{1}{4}ml^2 + \frac{5}{6}ml^2 = \frac{4}{3}ml^2 - \frac{1}{12}ml^2 = \frac{5}{4}ml^2 \implies (\text{Q}) \rightarrow (1)$$

For structure (R):

$$I_{CD} = I_{CB} = I_{AD} = I_{AB} = \frac{1}{3}ml^2\sin^2 45^\circ = \frac{1}{6}ml^2$$

$$I_R = 4 \times \frac{1}{6}ml^2 = \frac{2}{3}ml^2 \implies (\text{R}) \rightarrow (4)$$

For structure (S):

$$I_{CA} = \frac{1}{3}ml^2\sin^2 30^\circ = \frac{1}{12}ml^2$$

$$I_{CB} = \frac{1}{3}ml^2\sin^2 30^\circ = \frac{1}{12}ml^2$$

$$I_S = \frac{1}{12}ml^2 + \frac{1}{12}ml^2 = \frac{1}{6}ml^2 \implies (\text{S}) \rightarrow (2)$$