To determine which of the options are correct, let's analyze each statement step by step.
### Step 1: Analyze Option 1
**Statement:** If the center of mass of three particles is at rest and it is known that two of them are moving along different lines, then the third particle must also be moving.
**Solution:**
- Consider three particles A, B, and C.
- Let particle A be moving in the upward direction (y-direction) and particle B be moving in the forward direction (x-direction).
- For the center of mass (CM) of the system to remain at rest, the motion of particle C must counterbalance the motions of particles A and B.
- Therefore, particle C must move in the negative x-direction and negative y-direction to keep the center of mass stationary.
- Thus, this statement is **correct**.
### Step 2: Analyze Option 2
**Statement:** If the center of mass remains at rest, then the net work done by the forces acting on the system must be zero.
**Solution:**
- If the center of mass is at rest, it implies that the net external force acting on the system is zero (from Newton's first law).
- Since work done is defined as the product of net force and displacement, if the net force is zero, the work done must also be zero.
- Therefore, this statement is **correct**.
### Step 3: Analyze Option 3
**Statement:** If the center of mass remains at rest, net external force must be zero.
**Solution:**
- This is a direct consequence of Newton's first law of motion.
- If the center of mass of a system is at rest, it means that there is no net external force acting on the system.
- Hence, this statement is **correct**.
### Step 4: Analyze Option 4
**Statement:** If the velocity of the center of mass is changing, then there must be some net work being done on the system.
**Solution:**
- A change in the velocity of the center of mass indicates that there is an acceleration present.
- According to Newton's second law, acceleration is produced by a net force acting on the system.
- If there is a net force, then work must be done on the system (since work is the product of force and displacement).
- Therefore, this statement is **correct**.
### Final Conclusion
All four options (1, 2, 3, and 4) are correct.
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