Does the total kinetic energy of an isolated system necessarily have to be conserved when its total momentum is conserved?

2 answers

Momentum is conserved period.

total kinetic energy conserved? No, where did you get that idea?
Think about a bullet hitting a block and getting embedded. I am sure you have done that for homework. You solve it using conservation of momentum. However
(1/2)m V^2 of bullet
is not equal to
(1/2)(m + M) v^2 of bullet and block after
during the penetration, mechanical energy escaped as heat.
Thank for the explanation, I knew about momentum but the fact that kinetic energy was mentioned in the question just threw me off.