1. Find dimensions of each quantity in LIST-I

We match each physical quantity with its dimensional formula.

2. A. Boltzmann constant

Boltzmann constant $k_B$ has unit: $\frac{\text{energy}}{\text{temperature}}$

Since energy has dimensions: $[E] = ML^2T^{-2}$

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

This matches III.

So, $A \to III$

3. B. Coefficient of viscosity

Coefficient of viscosity $\eta$ has dimensions: $[\eta] = ML^{-1}T^{-1}$

This matches IV.

So, $B \to IV$

4. C. Planck's constant

Planck's constant $h$ has unit: $\text{energy} \times \text{time}$

Thus, $[h] = (ML^2T^{-2})(T) = ML^2T^{-1}$

This matches I.

So, $C \to I$

5. D. Thermal conductivity

Thermal conductivity $k$ comes from Fourier’s law: $\frac{Q}{t} = kA\frac{\Delta T}{L}$

So, $k = \frac{Q/t \cdot L}{A\Delta T}$

Now,

• $Q/t$ is power $\Rightarrow ML^2T^{-3}$
• Multiply by $L$ $\Rightarrow ML^3T^{-3}$
• Divide by area $L^2$ $\Rightarrow MLT^{-3}$
• Divide by temperature $K$ $\Rightarrow MLT^{-3}K^{-1}$

Thus, $[k] = MLT^{-3}K^{-1}$

This matches II.

So, $D \to II$

6. Final matching

$A \to III, \quad B \to IV, \quad C \to I, \quad D \to II$

This corresponds to Option A.