Which of the following pairs does not have same dimensions?

To determine which of the given pairs does not have the same dimensions, let's analyze the dimensions of each quantity in each option one by one.

1. Impulse and Momentum:
- Impulse is defined as the product of force and time:
$\text{Impulse}=\text{Force}\times \text{Time}$
The dimensional formula of Force is $\left[{\text{MLT}}^{-2}\right]$ and for Time is $\left[\text{T}\right]$.
Therefore, the dimensions of Impulse are:
$\left[{\text{MLT}}^{-2}\right]\times \left[\text{T}\right]=\left[{\text{MLT}}^{-1}\right]$
- Momentum (linear momentum) is the product of mass and velocity:
$\text{Momentum}=\text{Mass}\times \text{Velocity}$
The dimensional formula of Mass is $\left[\text{M}\right]$ and for Velocity is $\left[{\text{LT}}^{-1}\right]$.
Therefore, the dimensions of Momentum are:
$\left[\text{M}\right]\times \left[{\text{LT}}^{-1}\right]=\left[{\text{MLT}}^{-1}\right]$
Since both have the dimensions $\left[{\text{MLT}}^{-1}\right]$, this pair has the same dimensions.

2. Moment of Inertia and Moment of Force:
- Moment of Inertia ($I$) is defined as the product of mass and the square of perpendicular distance:
$\text{Moment of Inertia}=\text{Mass}\times {\text{Distance}}^2$
Therefore, its dimensions are:
$\left[\text{M}\right]\times \left[{\text{L}}^2\right]=\left[{\text{ML}}^2\right]$
- Moment of Force (also known as Torque, $\tau$) is the cross product of position vector and force:
$\text{Moment of Force}=\text{Force}\times \text{Distance}$
The dimensions of Force are $\left[{\text{MLT}}^{-2}\right]$ and for Distance is $\left[\text{L}\right]$.
Therefore, the dimensions of Moment of Force are:
$\left[{\text{MLT}}^{-2}\right]\times \left[\text{L}\right]=\left[{\text{ML}}^2{\text{T}}^{-2}\right]$
Since $\left[{\text{ML}}^2\right]\ne \left[{\text{ML}}^2{\text{T}}^{-2}\right]$, this pair does not have the same dimensions.

3. Angular Momentum and Planck's Constant:
- Angular Momentum ($L$) is defined as:
$L=r\times p=\text{Distance}\times \text{Linear Momentum}$
Its dimensions are:
$\left[\text{L}\right]\times \left[{\text{MLT}}^{-1}\right]=\left[{\text{ML}}^2{\text{T}}^{-1}\right]$
- Planck's Constant ($h$) is derived from the relation $E=h\nu$, where $E$ is energy and $\nu$ is frequency:
$h=\frac{\text{Energy}}{\text{Frequency}}$
The dimensions of Energy are $\left[{\text{ML}}^2{\text{T}}^{-2}\right]$ and for Frequency are $\left[{\text{T}}^{-1}\right]$.
Therefore, the dimensions of Planck's Constant are:
$\frac{\left[{\text{ML}}^2{\text{T}}^{-2}\right]}{\left[{\text{T}}^{-1}\right]}$
Both have the same dimensions.

4. Work and Torque:
- Work is force times displacement:
$\text{Work}=\text{Force}\times \text{Displacement}=\left[{\text{MLT}}^{-2}\right]\times \left[\text{L}\right]=\left[{\text{ML}}^2{\text{T}}^{-2}\right]$
- Torque is moment of force, which as calculated above has dimensions $\left[{\text{ML}}^2{\text{T}}^{-2}\right]$.
Both have the same dimensions.

Thus, the only pair that does not share the same dimensions is moment of inertia and moment of force.