Which pairs do not have equal dimensions?

The correct option is Force and impulse.

To determine which pair of physical quantities does not have equal dimensions, let us analyze the dimensional formulas for each option step-by-step:

1. Energy and Torque:
Energy is the capacity to do work, which is given by:
$\text{Energy}=\text{Force}\times \text{displacement}$
Its dimensional formula is:
$\left[\text{Energy}\right]=\left[ML{T}^{-2}\right]\times \left[L\right]=\left[M{L}^2{T}^{-2}\right]$
Torque ($\tau$) is the rotational equivalent of force, defined as:
$\text{Torque}=\text{Force}\times \text{perpendicular distance}$
Its dimensional formula is:
$\left[\text{Torque}\right]=\left[ML{T}^{-2}\right]\times \left[L\right]=\left[M{L}^2{T}^{-2}\right]$
Since both have the same dimensions, this pair has equal dimensions.

2. Force and Impulse:
Force is defined as mass times acceleration:
$\text{Force}=\text{Mass}\times \text{acceleration}$
Its dimensional formula is:
$\left[\text{Force}\right]=\left[ML{T}^{-2}\right]$
Impulse is defined as the product of force and the time interval over which it acts:
$\text{Impulse}=\text{Force}\times \text{time}$
Its dimensional formula is:
$\left[\text{Impulse}\right]=\left[ML{T}^{-2}\right]\times \left[T\right]=\left[ML{T}^{-1}\right]$
Comparing the two, we see that:
$\left[ML{T}^{-2}\right]\ne \left[ML{T}^{-1}\right]$
Therefore, Force and Impulse do not have equal dimensions.

3. Angular Momentum and Planck's Constant:
Angular momentum ($L$) is given by:
$L=mvr$
Its dimensional formula is:
$\left[L\right]=\left[M\right]\times \left[L{T}^{-1}\right]\times \left[L\right]=\left[M{L}^2{T}^{-1}\right]$
Planck's constant ($h$) is derived from the relation:
$E=h\nu \Rightarrow h=\frac{E}{\nu }$
where $E$ is energy and $\nu$ is frequency (with dimensions of $\left[T^{-1}\right]$).
Its dimensional formula is:
$\left[h\right]=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-1}\right]$
Since both have the same dimensions, this pair has equal dimensions.

4. Elastic Modulus and Pressure:
Elastic modulus is defined as stress divided by strain. Since strain is a dimensionless ratio of lengths, the dimensions of elastic modulus are the same as stress:
$\text{Elastic modulus}=\frac{\text{Stress}}{\text{Strain}}=\frac{\text{Force}/\text{Area}}{\text{dimensionless}}$
Its dimensional formula is:
$\left[\text{Elastic modulus}\right]=\left[M{L}^{-1}{T}^{-2}\right]$
Pressure is also defined as force per unit area:
$\text{Pressure}=\frac{\text{Force}}{\text{Area}}$
Its dimensional formula is:
$\left[\text{Pressure}\right]=\left[M{L}^{-1}{T}^{-2}\right]$
Since both have the same dimensions, this pair has equal dimensions.

Conclusion: The only pair that does not have equal dimensions is Force and impulse.