Which of the following is/are correct

If the change in the position of the two particles complement each other the centre of mass will remain at rest,  And displacement vectors will be anti parallel vectors but need not to be on the same line.

If center of mass remains at rest, Then net work done by the forces acting on the system need not to be zero. The if the forces are in opposite direction and have same magnitude the acceleration will be zero but the change in kinetic energy of individual particles need not to be zero.

If center of mass remains at rest, then the net external force must be zero   

If speed of center of mass is changing then there must be some net work being done on the system from outside. On considering a two particle system and if a force is acting on only one particle in such a way that it balances the internal force(the particle is at rest), in this case the velocity of centre of mass in changing and the work done by external force is zero.