1. Write dimensions of each quantity

We compare the dimensional formulas of the two quantities in each option.

Basic dimensions used:

• Mass $= [M]$
• Length $= [L]$
• Time $= [T]$

2. Option A: Angular momentum and Planck's constant

(i) Angular momentum

Angular momentum $L$ is $L = r \times p$ where momentum $p = mv$.

So, $[L] = [r][p] = [L]\,[M L T^{-1}] = [M L^2 T^{-1}]$

(ii) Planck's constant

From $E = h\nu$ we get $h = \frac{E}{\nu}$

Now, $[E] = [M L^2 T^{-2}], \qquad [\nu] = [T^{-1}]$

Therefore, $[h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$

So both have the same dimensions.

3. Option B: Torque and energy

(i) Torque

Torque $= \text{force} \times \text{distance}$ $[\tau] = [M L T^{-2}] [L] = [M L^2 T^{-2}]$

(ii) Energy

Energy (work) $= \text{force} \times \text{distance}$ $[E] = [M L T^{-2}] [L] = [M L^2 T^{-2}]$

So both have the same dimensions.

4. Option C: Surface tension and impulse

(i) Surface tension

Surface tension $= \dfrac{\text{force}}{\text{length}}$ $[S] = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}]$

(ii) Impulse

Impulse $= \text{force} \times \text{time}$ $[J] = [M L T^{-2}] [T] = [M L T^{-1}]$

Thus, $[S] = [M T^{-2}], \qquad [J] = [M L T^{-1}]$

These are not the same.

5. Option D: Pressure and Young's modulus

(i) Pressure

Pressure $= \dfrac{\text{force}}{\text{area}}$ $[P] = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}]$

(ii) Young's modulus

Young's modulus $= \dfrac{\text{stress}}{\text{strain}}$

Strain is dimensionless, so dimensions of Young's modulus are same as stress.

Stress $= \dfrac{\text{force}}{\text{area}}$ $[Y] = [M L^{-1} T^{-2}]$

So both have the same dimensions.

6. Conclusion

Only Option C has quantities with different dimensions.

Therefore, the correct answer is: $\boxed{\text{C}}$

7. Comparison with stored correct answer

Stored correct answer: C

My derived answer: C

They match.