Which of the following pairs of physical quantities does not have same dimensional formula?

The correct option is Tension and surface tension.

To understand why this is the correct answer, we can analyze the dimensional formulas of each pair of physical quantities given in the options.

1. Work and Torque:
Work done is defined as force multiplied by displacement:
$$\mathrm{Work}=\mathrm{Force}\times \mathrm{Displacement}$$
The dimensional formula of force is [MLT^-2] and displacement is [L]. Therefore, the dimensional formula of work is:
$$\left[\mathrm{Work}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[L\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$$
Torque is defined as the cross product of position vector and force:
$$\mathrm{Torque}=\mathrm{Force}\times \mathrm{Perpendicular\; Distance}$$
Thus, the dimensional formula of torque is:
$$\left[\mathrm{Torque}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[L\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$$
Since both have the dimensional formula [ML²T^-2], they have the same dimensions.

2. Angular Momentum and Planck's Constant:
Angular momentum (L) is given by:
$$L=mvr$$
where m is mass [M], v is velocity [LT^-1], and r is distance [L]. The dimensional formula is:
$$\left[L\right]=\left[M\right]\times \left[{\mathrm{LT}}^{-1}\right]\times \left[L\right]=\left[{\mathrm{ML}}^2{T}^{-1}\right]$$
Planck's constant (h) is related to energy (E) and frequency (ν) by E = hν:
$$h=\frac{E}{\nu }$$
The dimensional formula of energy is [ML²T^-2] and frequency is [T^-1]. Thus, the dimensional formula of Planck's constant is:
$$\left[h\right]=\frac{\left[{\mathrm{ML}}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[{\mathrm{ML}}^2{T}^{-1}\right]$$
Since both have the dimensional formula [ML²T^-1], they have the same dimensions.

3. Tension and Surface Tension:
Tension is a pulling force exerted by a string, cable, or chain. Since it is a force, its dimensional formula is:
$$\left[\mathrm{Tension}\right]=\left[{\mathrm{MLT}}^{-2}\right]$$
Surface tension is defined as force per unit length:
$$\mathrm{Surface\; Tension}=\frac{\mathrm{Force}}{\mathrm{Length}}$$
Thus, the dimensional formula of surface tension is:
$$\left[\mathrm{Surface\; Tension}\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[L\right]}=\left[{\mathrm{MT}}^{-2}\right]$$
Comparing the two, [MLT^-2] is not equal to [MT^-2]. Therefore, tension and surface tension do not have the same dimensional formula.

4. Impulse and Linear Momentum:
Impulse is defined as force multiplied by time duration:
$$\mathrm{Impulse}=\mathrm{Force}\times \mathrm{Time}$$
Thus, the dimensional formula of impulse is:
$$\left[\mathrm{Impulse}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[T\right]=\left[{\mathrm{MLT}}^{-1}\right]$$
Linear momentum is defined as mass multiplied by velocity:
$$\mathrm{Momentum}=\mathrm{Mass}\times \mathrm{Velocity}$$
Thus, the dimensional formula of linear momentum is:
$$\left[\mathrm{Momentum}\right]=\left[M\right]\times \left[{\mathrm{LT}}^{-1}\right]=\left[{\mathrm{MLT}}^{-1}\right]$$
Since both have the dimensional formula [MLT^-1], they have the same dimensions.

Consequently, the only pair that does not share the same dimensions is Tension and surface tension.