A

Step 1: For a monoatomic ideal gas, the mean square speed of gas molecules is given by the equation
$$v_{ms} = \frac{3kT}{m}$$
where \(k\) is the Boltzmann constant, \(T\) is temperature, and \(m\) is the mass of a gas molecule. This relates directly to the mean square speed.

Step 2: The root mean square (RMS) speed is calculated using the relationship
$$v_{rms} = \sqrt{v_{ms}} = \sqrt{\frac{3kT}{m}}$$
indicating it is derived from the mean square speed, confirming its relation to the kinetic theory of gases.

Step 3: The average speed of gas molecules is given by the equation
$$v_{avg}= \frac{8kT}{\pi m}$$
which results in a different value compared to RMS and mean square speed.

Step 4: The most probable speed is derived from the distribution of speeds of gas molecules in Maxwell-Boltzmann distribution and is given by
$$v_{mp} = \sqrt{\frac{2kT}{m}}$$

This structured relationship allows us to match column I with column II.

Step 5: The correct matches are:

- (i) Mean square speed [A]
- (ii) RMS speed of gas molecule [B]
- (iii) Average speed of gas molecule [C]
- (iv) Most probable speed of gas molecule [D]

Therefore, the answer is A.