A

To solve the matching question, we need to understand the concepts related to temperature, equipartition law, translational kinetic energy (K.E.), and internal energy of an ideal gas.
For Column-I:
(i) Temperature: The first image likely represents the formula for temperature in a thermodynamic context.
(ii) Equipartition law: This law states that energy is distributed equally among all degrees of freedom.
(iii) Translational K.E. for one mole of an ideal gas: The translational kinetic energy can be expressed as \( \frac{3}{2}RT \) for one mole of an ideal gas.
(iv) Internal energy of an ideal gas: The internal energy for an ideal gas depends only on the temperature and is given by \( \frac{3}{2}RT \) for monatomic gases.

For Column-II:
[A] represents the formula \( \frac{3}{2}RT \), which aligns with the definition for temperature or internal energy of an ideal gas.
[B] relates to the equipartition of energy, confirming energy distribution among degrees of freedom.
[C] would relate to the description of translational kinetic energy, which is equally distributed among available degrees of freedom.
[D] likely denotes the relation for internal energy consistent with the ideal gas behavior.
Thus, the correct matching is:
(i) - [A], (ii) - [C], (iii) - [D], (iv) - [B].
Hence, option A is correct.