Consider the following two statements:
A. Kinear momentum of the system remains constant.
B. Centre of mass of the system remains at rest

Options

A. A implies B and B implies A
B. A does not imply B and B does not imply A.
C. A implies B but B does not imply A.
D. B implies A but A does not imply B.

Answer

To solve the problem, we need to analyze the two statements regarding linear momentum and the center of mass of a system.

### Step-by-Step Solution:

1. **Understanding Linear Momentum**:
- Linear momentum (p) of a system is defined as the product of its mass (m) and its velocity (v): \( p = mv \).
- For the momentum to remain constant, either the mass must remain constant and the velocity must be constant, or the system must be isolated (no external forces acting on it).

2. **Understanding Center of Mass**:
- The center of mass (CM) of a system is the point where the total mass of the system can be considered to be concentrated.
- The center of mass remains at rest if its velocity is zero. This means that the system is either at rest or the net external force acting on the system is zero.

3. **Analyzing Statement A**:
- Statement A claims that the linear momentum of the system remains constant. This can occur when the system is isolated from external forces or if the system's velocity is constant.

4. **Analyzing Statement B**:
- Statement B claims that the center of mass of the system remains at rest. This implies that the velocity of the center of mass is zero.

5. **Implications of the Statements**:
- If the center of mass remains at rest (Statement B), it means that the total linear momentum of the system must also be constant (since \( p = mv \) and \( v = 0 \)).
- However, if the linear momentum is constant (Statement A), the center of mass could still be moving with a constant velocity (not necessarily at rest). Thus, Statement A does not imply Statement B.

6. **Conclusion**:
- From the analysis, we conclude that Statement B implies Statement A, but Statement A does not imply Statement B. Therefore, the correct option is:
- **Option 4: B implies A, but A does not imply B.**