Mechanical energy and momentum conservation

[yolo123]

Mechanical energy is conserved whenever momentum is conserved. T/F?

My guess is true:
If momentum is conserved, no external forces are applied on the system. Hence mechanical energy can't change! (No change in PE, no change in KE).

BUT: My answer booklet says FALSE! Why?

 

[guitarphysics]

Momentum is conserved when there's an inelastic collision, but kinetic energy isn't. That's one typical example.