Match List-I with List-II.

Choose the correct answer from the options given below :

The dimensional formula of a physical quantity is obtained from the defining equation of that quantity in terms of the fundamental mechanical quantities $$M$$ (mass), $$L$$ (length) and $$T$$ (time).

Case (A) : Mass density

Mass density $$\rho$$ is mass per unit volume:
$$\rho = \frac{\text{mass}}{\text{volume}}$$.
Mass has dimension $$[M]$$ and volume has dimension $$[L^3]$$, so

$$[\rho] = \frac{[M]}{[L^3]} = [M L^{-3} T^{0}]$$.
Thus (A) corresponds to (IV).

Case (B) : Impulse

Impulse $$J$$ is defined as force multiplied by the time interval:
    $$J = F \, \Delta t$$.
Force has dimension $$[M L T^{-2}]$$ and time has dimension $$[T]$$, so

$$[J] = [M L T^{-2}] \,[T] = [M L T^{-1}]$$.
Thus (B) corresponds to (II).

Case (C) : Power

Power $$P$$ is work done per unit time:
    $$P = \frac{W}{t}$$.
Work (or energy) has dimension $$[M L^{2} T^{-2}]$$ and time has dimension $$[T]$$, hence

$$[P] = \frac{[M L^{2} T^{-2}]}{[T]} = [M L^{2} T^{-3}]$$.
Thus (C) corresponds to (I).

Case (D) : Moment of inertia

For a point mass $$m$$ at a distance $$r$$ from the axis, the moment of inertia $$I$$ is:
    $$I = m r^{2}$$.
Mass has dimension $$[M]$$ and distance squared has dimension $$[L^{2}]$$, so

$$[I] = [M] [L^{2}] = [M L^{2} T^{0}]$$.
Thus (D) corresponds to (III).

Collecting the matches:

(A) → (IV), (B) → (II), (C) → ( I ), (D) → (III).

This set of matches is given in Option C. Hence the correct answer is Option C.