Question

When no current is passed through a conductor, (a) the free electrons do not move

Answer 1

to move the electron ,some excitation or deexcitation is needed

Answer 2

When there is no current in the conductor, the free electrons in the conductor do move but the average of the velocities possessed by all the free electrons in the conductor at any instance of time will be zero due to the fact of absence of electric field which is determined by the presence of the current flowing through any conductor.

Therefore, the absence of current does not mean the absence of movement of free electrons.