Which one of the following pairs of quantities has the same dimension ?

The correct option is momentum and impulse.

To find which pair of physical quantities has the same dimensions, we can derive the dimensional formula for each quantity step-by-step.

1. Momentum and Impulse:
Linear momentum is defined as the product of mass and velocity:

$\text{Momentum}=\text{Mass}\times \text{Velocity}$

The dimensional formula for mass is $\left[M\right]$ and for velocity is $\left[{\mathrm{LT}}^{-1}\right]$. Thus, the dimensional formula for momentum is:

$\left[\text{Momentum}\right]=\left[{\mathrm{MLT}}^{-1}\right]$

Impulse is defined as the product of force and the time interval for which it acts:

$\text{Impulse}=\text{Force}\times \text{Time}$

The dimensional formula for force is $\left[{\mathrm{MLT}}^{-2}\right]$ and for time is $\left[T\right]$. Thus, the dimensional formula for impulse is:

$\left[\text{Impulse}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[T\right]=\left[{\mathrm{MLT}}^{-1}\right]$

Since both momentum and impulse share the same dimensional formula $\left[{\mathrm{MLT}}^{-1}\right]$, they have the same dimensions.

Let us verify the other options to show why they do not match:

2. Force and Work done:

$\left[\text{Force}\right]=\left[{\mathrm{MLT}}^{-2}\right]$

Work done is the product of force and displacement:

$\left[\text{Work done}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[L\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$

These two have different dimensions.

3. Pressure and Force:
Pressure is defined as force per unit area:

$\left[\text{Pressure}\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[L^2\right]}$

Since Force has dimensions $\left[{\mathrm{MLT}}^{-2}\right]$, their dimensions are different.

4. Surface tension and Stress:
Surface tension is defined as force per unit length:

$\left[\text{Surface tension}\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[L\right]}$

Stress is defined as force per unit area:

$\left[\text{Stress}\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[L^2\right]}$

These two also have different dimensions.