In an inertial frame of reference, a series of experiments is conducted. in each experiment, two or three forces are applied to an object. the magnitudes of these forces are given. no other forces are acting on the object. in which cases may the object possibly remain at rest?

The objects will remain at rest if net force acting on it is zero if their magnitude is same and they are acting in opposite direction then according to Newton's 2nd law the net force acting on the system is zero. Since the net force on the system is zero, the object will remain at rest.