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(A) The moment of inertia $(MI)$ of a rod of mass $M$ and length $L$ is given by:$1$. About an axis passing through the center and perpendicular to th

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$(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)$

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(A) The moment of inertia $(MI)$ of a rod of mass $M$ and length $L$ is given by:$1$. About an axis passing through the center and perpendicular to the rod: $I = \frac{ML^2}{12}$.$2$. About an axis passing through one end and perpendicular to the rod: $I = \frac{ML^2}{3}$.Applying these formulas:$(a)$ For mass $M$,length $L$,axis through midpoint: $I = \frac{ML^2}{12}$ (Matches $iii$).$(b)$ For mass $2M$,length $L$,axis through end: $I = \frac{(2M)L^2}{3} = \frac{2ML^2}{3}$ (Matches $iv$).$(c)$ For mass $M$,length $2L$,axis through midpoint: $I = \frac{M(2L)^2}{12} = \frac{4ML^2}{12} = \frac{ML^2}{3}$ (Matches $ii$).$(d)$ For mass $2M$,length $2L$,axis through end: $I = \frac{(2M)(2L)^2}{3} = \frac{2M(4L^2)}{3} = \frac{8ML^2}{3}$ (Matches $i$).Thus,the correct matching is $(a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)$.

List-$I$	List-$II$
$(a)$ $MI$ of the rod (length $L$,mass $M$,about an axis $\perp$ to the rod passing through the midpoint)	$(i) \frac{8ML^2}{3}$
$(b)$ $MI$ of the rod (length $L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends)	$(ii) \frac{ML^2}{3}$
$(c)$ $MI$ of the rod (length $2L$,mass $M$,about an axis $\perp$ to the rod passing through its midpoint)	$(iii) \frac{ML^2}{12}$
$(d)$ $MI$ of the rod (length $2L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends)	$(iv) \frac{2ML^2}{3}$

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