To find the pair with non-identical dimensions, we analyze the dimensional formula for each quantity using the NCERT Appendix:

1. Angular Momentum and Planck's Constant: Angular Momentum ($L = mvr$): Dimensions are $[ML^2T^{-1}]$ . Planck's Constant ($h = E/\nu$): Dimensions are $[ML^2T^{-1}]$ .

• Result: Identical.

2. Moment of Inertia and Moment of a Force:

• Moment of Inertia ($I = mr^2$): Dimensions are $[M][L]^2 = [ML^2]$ .
• Moment of a Force (Torque, $\tau = r \times F$): Dimensions are $[L][MLT^{-2}] = [ML^2T^{-2}]$ .
• Result: Not Identical ($[ML^2] \neq [ML^2T^{-2}]$).

3. Work and Torque: Work ($W = F \cdot d$): Dimensions are $[ML^2T^{-2}]$ . Torque: Dimensions are $[ML^2T^{-2}]$ .

• Result: Identical.

4. Impulse and Momentum: Impulse ($I = F \times t$): Dimensions are $[MLT^{-2}][T] = [MLT^{-1}]$ . Momentum ($p = mv$): Dimensions are $[M][LT^{-1}] = [MLT^{-1}]$ .

• Result: Identical.