Two particles of mass 1 kg and 3 kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2m/s, their centre of mass has a velocity of 0.5 m/s. When the relative velocity of approach becomes 3m/s, the velocity of the centre of mass is 0.75 m/s.

Explanation

To solve the problem, we will analyze the motion of the two particles and the implications of their velocities, particularly focusing on the center of mass (CM) of the system.

### Step-by-Step Solution:

1. **Identify the Masses and Their Initial Velocities**:
- Let \( m_1 = 1 \, \text{kg} \) (mass of the first particle).
- Let \( m_2 = 3 \, \text{kg} \) (mass of the second particle).
- Let \( v_1 \) be the velocity of the first particle and \( v_2 \) be the velocity of the second particle. Since they are moving towards each other, we can assume \( v_1 \) is positive and \( v_2 \) is negative.

2. **Relative Velocity of Approach**:
- The relative velocity of approach \( v_{rel} \) is given by:
\[ v_{rel} = v_1 + |v_2| \]
- In the first case, when \( v_{rel} = 2 \, \text{m/s} \):
\[ v_1 + |v_2| = 2 \, \text{m/s} \]
- In the second case, when \( v_{rel} = 3 \, \text{m/s} \):
\[ v_1 + |v_2| = 3 \, \text{m/s} \]

3. **Velocity of the Center of Mass**:
- The velocity of the center of mass \( V_{CM} \) of the system is given by:
\[ V_{CM} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \]
- For the first case, we are given \( V_{CM} = 0.5 \, \text{m/s} \):
\[ \frac{1 \cdot v_1 + 3 \cdot v_2}{1 + 3} = 0.5 \]
- For the second case, we are given \( V_{CM} = 0.75 \, \text{m/s} \):
\[ \frac{1 \cdot v_1 + 3 \cdot v_2}{1 + 3} = 0.75 \]

4. **Setting Up the Equations**:
- From the first case:
\[ v_1 + 3v_2 = 2 \cdot 4 = 8 \quad \text{(1)} \]
- From the second case:
\[ v_1 + 3v_2 = 0.75 \cdot 4 = 3 \quad \text{(2)} \]

5. **Analyzing the Results**:
- From equations (1) and (2), we have:
\[ 8 = 3 \quad \text{(which is a contradiction)} \]
- This contradiction arises because the velocities of the center of mass cannot change if no external force is acting on the system. Thus, the assumption that the velocities of the center of mass can vary with changing relative velocities is incorrect.

6. **Conclusion**:
- Since the net external force acting on the system is zero, the velocity of the center of mass should remain constant. Therefore, the statement in the question is incorrect.