The correct answer is Option A : $$A$$ does not imply $$B$$ and $$B$$ does not imply $$A$$.

Here's why :

Statement $$A:$$ The linear momentum of a system of particles being zero does not mean that the kinetic energy is also zero. For example, consider two equal mass particles moving with the same speed but in opposite directions. The linear momentum of the system will be zero because momentum is a vector quantity and the two momenta will cancel out. However, kinetic energy is a scalar quantity and does not cancel out in this way. Each particle has kinetic energy due to its motion, so the total kinetic energy of the system is not zero.

Statement $$B:$$ The kinetic energy of a system of particles being zero also does not imply that the linear momentum is zero. If the kinetic energy is zero, it means that all the particles are at rest (since kinetic energy is associated with motion). However, the linear momentum will also be zero in this case because momentum depends on both mass and velocity, and the velocity of each particle is zero.

But remember that zero linear momentum doesn't exclusively mean all particles are at rest. As explained above, it could be the scenario where particles have equal and opposite momenta, thereby cancelling each other. Hence, $$B$$ doesn't imply $$A$$.

So, neither statement implies the other. Hence, Option A is correct.