Can my mother set a time for me to play PlayStation, knowing that she swore I wouldn't play it until the end of exams for the purpose of studying? And is this oath of hers considered a valid oath or a vain one?

Your mother's oath is binding, and you must obey her in refraining from what she swore you to abandon, due to the obligation to obey her when she commands a good deed, and because it is the right of a Muslim upon another Muslim to fulfill their oath. If your mother wishes to grant you permission, she may expiate her oath if she chooses, based on the Prophet's (peace be upon him) saying: "Whoever takes an oath and then sees something better than it, let him do what is better and expiate his oath." However, if her intention was for you to refrain from playing most of the time or for specific hours, then this intention is valid and specifies the oath, and you would not break your oath by playing in a way that does not harm your studies.