Statement 1: Two spheres undergo a perfectly elastic collision. The kinetic energy of system of both spheres is always constant. [There is no external force on system of both spheres].
Statement 2: If net external force on a system is zero, the velocity of centre of mass remains constant.

Options

A. Statement `-1` is True, Statement `-2` is True, Statement`-2` is a correct explanation for Statement`-1`
B. Statement `-1` is True,Statement`-2` is True, Statement`-2` is NOT a correct explanation for Statement`-1`
C. Statement `-1` is True, Statement `-2` is False
D. Statement `-1` is False, Statement `-2` is True.

Explanation

To analyze the two statements given in the question, we will evaluate each statement separately and determine their validity.

### Step 1: Evaluate Statement 1
**Statement 1**: "Two spheres undergo a perfectly elastic collision. The kinetic energy of the system of both spheres is always constant. [There is no external force on the system of both spheres]."

In a perfectly elastic collision, both momentum and kinetic energy are conserved. However, the statement claims that the kinetic energy of the system remains constant throughout the collision.

1. **Initial Kinetic Energy**: Before the collision, the total kinetic energy (KE_initial) of the system can be expressed as:
\[ KE_{\text{initial}} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 \]
where \(m_1\) and \(m_2\) are the masses of the spheres, and \(u_1\) and \(u_2\) are their initial velocities.

2. **Final Kinetic Energy**: After the collision, the total kinetic energy (KE_final) can be expressed as:
\[ KE_{\text{final}} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \]
where \(v_1\) and \(v_2\) are the final velocities of the spheres after the collision.

3. **Conservation of Kinetic Energy**: In a perfectly elastic collision, it is true that:
\[ KE_{\text{initial}} = KE_{\text{final}} \]
However, the statement implies that kinetic energy remains constant at all times, which is misleading. The kinetic energy before and after the collision is equal, but it is not constant during the collision itself. Therefore, Statement 1 is **false**.

### Step 2: Evaluate Statement 2
**Statement 2**: "If the net external force on a system is zero, the velocity of the center of mass remains constant."

1. **Understanding Center of Mass**: The velocity of the center of mass (V_cm) of a system is given by:
\[ V_{\text{cm}} = \frac{\sum m_i v_i}{\sum m_i} \]
where \(m_i\) and \(v_i\) are the masses and velocities of the particles in the system.

2. **Conservation of Momentum**: If there are no external forces acting on the system, the total momentum of the system is conserved. This means:
\[ \text{Total momentum before} = \text{Total momentum after} \]
Since momentum is conserved, the velocity of the center of mass must also remain constant.

Thus, Statement 2 is **true**.

### Conclusion
- **Statement 1**: False (Kinetic energy is not constant during the collision).
- **Statement 2**: True (Velocity of the center of mass remains constant if no external force acts on the system).

### Final Answer
- Statement 1 is false.
- Statement 2 is true.