Let's break down the dimensional analysis of each physical quantity:

P. Boltzmann Constant ($k_B$)

The Boltzmann constant relates energy to temperature. Its dimensional formula is derived from the relationship: $E = k_B T$.

Energy ($E$) has dimensions of [ML²T^-2]. Temperature ($T$) has dimensions of [K]. Therefore:

$$[k_B] = \frac{[E]}{[T]} = \frac{[ML^2T^{-2}]}{[K]} = [ML^2T^{-2}K^{-1}]$$

Q. Coefficient of Viscosity ($\eta$)

Viscosity is a measure of a fluid's resistance to flow. It is defined as the ratio of shear stress to shear rate. Shear stress has dimensions of [ML^-1T^-2] (force per unit area), and shear rate has dimensions of [T^-1] (velocity gradient). Therefore:

$$[\eta] = \frac{[ML^{-1}T^{-2}]}{[T^{-1}]} = [ML^{-1}T^{-1}]$$

R. Plank Constant ($h$)

Planck's constant relates energy to frequency. Its dimensional formula is derived from the relationship: $E = h\nu$.

Energy ($E$) has dimensions of [ML²T^-2]. Frequency ($\nu$) has dimensions of [T^-1]. Therefore:

$$[h] = \frac{[E]}{[\nu]} = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$$

S. Thermal Conductivity ($k$)

Thermal conductivity is a measure of a material's ability to conduct heat. It is defined as the rate of heat transfer per unit area per unit temperature gradient. Heat transfer rate has dimensions of [ML²T^-3]. Area has dimensions of [L²], and temperature gradient has dimensions of [K/L]. Therefore:

$$[k] = \frac{[ML^2T^{-3}]}{[L^2][K/L]} = [MLT^{-3}K^{-1}]$$

Matching the Dimensions:

Based on the dimensional analysis, we can match the quantities in List I with the dimensions in List II as follows:

Therefore, the correct answer is Option C.