(a) An object remains in equilibrium not only by net force acting on the object is zero but also the net torque acting on the object about any point must be zero.
Thus it is false.
(b) Any object remains in equilibrium, one of the necessary condition is net force acting on the object is zero.
Thus it is true.
The necessary condition for an object remains in equilibrium; the net external torque about any point must be zero, which means. Thus it is true.
Any object in equilibrium means not at rest. Even in the equilibrium the object can move with constant speed either translational or rotational. Thus it is false.