1. We compare the dimensions of each quantity in the given pairs.

2. Option A: velocity gradient and decay constant

• Velocity gradient means rate of change of velocity with distance: $\text{velocity gradient} = \frac{v}{x}$

• Dimensions of velocity: $[v] = [L T^{-1}]$

• Therefore, $\left[\frac{v}{x}\right] = \frac{[L T^{-1}]}{[L]} = [T^{-1}]$

• Decay constant $\lambda$ comes from radioactive decay law: $N = N_0 e^{-\lambda t}$

• Since $\lambda t$ must be dimensionless, $[\lambda] = [T^{-1}]$

So both have the same dimensions: $[T^{-1}]$

Hence, Option A is correct.

3. Option B: Wien's constant and Stefan constant

• Wien's displacement law: $\lambda_{\max} T = b$

• Therefore dimensions of Wien's constant $b$ are: $[b] = [L][\Theta]$ where $\Theta$ denotes temperature.

• Stefan-Boltzmann law: $\frac{P}{A} = \sigma T^4$

• So, $[\sigma] = \frac{[P/A]}{[T^4]}$

• Power per unit area: $\left[\frac{P}{A}\right] = \frac{[M L^2 T^{-3}]}{[L^2]} = [M T^{-3}]$

• Hence, $[\sigma] = [M T^{-3} \Theta^{-4}]$

These are not the same.

So Option B is incorrect.

4. Option C: angular frequency and angular momentum

• Angular frequency: $[\omega] = [T^{-1}]$

• Angular momentum: $L = r p = r(mv)$

• Therefore, $[L] = [L] [M L T^{-1}] = [M L^2 T^{-1}]$

These are not the same.

So Option C is incorrect.

5. Option D: wave number and Avogadro number

• Wave number is: $\bar{\nu} = \frac{1}{\lambda}$

• Therefore, $[\bar{\nu}] = [L^{-1}]$

• Avogadro number is a pure count, so it is dimensionless: $[N_A] = [1]$

These are not the same.

So Option D is incorrect.

6. Final conclusion

Only Option A has the same dimensions in the pair.

$\boxed{A}$

7. Comparison with stored correct answer

Stored correct answer: A

My derived answer: A

They match.