1. Use standard formulas for a uniform rod

For a rod of length $l$ and mass $m$:

• About an axis perpendicular to the rod through its midpoint: $I_{\text{center}}=\frac{1}{12}ml^2$

• About an axis perpendicular to the rod through one end: $I_{\text{end}}=\frac{1}{3}ml^2$

2. Evaluate each item in List-I

(a) Rod of length $L$, mass $M$, axis through midpoint

Using $I=\frac{1}{12}ML^2$ So, $(a) \to \frac{ML^2}{12}$ This matches (iii).

(b) Rod of length $L$, mass $2M$, axis through one end

Using $I=\frac{1}{3}(2M)L^2=\frac{2ML^2}{3}$ So, $(b) \to \frac{2ML^2}{3}$ This matches (iv).

(c) Rod of length $2L$, mass $M$, axis through midpoint

Using $I=\frac{1}{12}M(2L)^2=\frac{1}{12}M\cdot 4L^2=\frac{ML^2}{3}$ So, $(c) \to \frac{ML^2}{3}$ This matches (ii).

(d) Rod of length $2L$, mass $2M$, axis through one end

Using $I=\frac{1}{3}(2M)(2L)^2=\frac{1}{3}\cdot 2M \cdot 4L^2=\frac{8ML^2}{3}$ So, $(d) \to \frac{8ML^2}{3}$ This matches (i).

3. Final matching

Thus,

• $(a) \to (iii)$
• $(b) \to (iv)$
• $(c) \to (ii)$
• $(d) \to (i)$

This corresponds to Option C.

4. Comparison with stored answer

Stored correct answer = C

My derived answer = C

So, the derived answer agrees with the stored answer.