Out of the following pair, which one does NOT have identical dimensions is

The dimensions of a physical quantity are expressed in powers of the fundamental quantities $$M$$ (mass), $$L$$ (length) and $$T$$ (time). To check whether two quantities are dimensionally identical, compare the exponents of $$M, L$$ and $$T$$ in their dimensional formulas.

Angular momentum $$L = mvr$$.
Dimensions: $$[L] = M^1L^2T^{-1}$$.

Planck’s constant $$h$$ is the quantum of action (energy × time).
Energy has $$M^1L^2T^{-2}$$, so $$h = \text{energy} \times \text{time}$$ gives $$M^1L^2T^{-1}$$.
Both match ⇒ identical.

Impulse $$J = F \times t$$.
Force $$F$$ is $$M^1L^1T^{-2}$$, multiply by $$T$$ to get $$M^1L^1T^{-1}$$.

Linear momentum $$p = mv$$.
Velocity is $$LT^{-1}$$, so $$p$$ is also $$M^1L^1T^{-1}$$.
Both match ⇒ identical.

Moment of inertia $$I = mr^{2}$$.
Dimensions: $$M^1L^{2}T^{0}$$.

Moment of a force (torque) $$\tau = F \times r$$.
Force $$F$$ is $$M^1L^1T^{-2}$$, multiply by $$L$$ to get $$M^1L^{2}T^{-2}$$.

Exponents of $$T$$ differ (0 vs −2) ⇒ NOT identical.

Work (energy) $$W = F \times s$$ has dimensions $$M^1L^{2}T^{-2}$$.

Torque $$\tau$$ already found above: $$M^1L^{2}T^{-2}$$.
Both match ⇒ identical.

Hence, the only pair with non-matching dimensions is moment of inertia and moment of a force.

Answer: Option C which is: moment of inertia and moment of a force