A vow does not become binding by mere intention
Question

                I mentally promised that I would fast if I passed my exams, but after that I was not able to fast on those days. What should I do?.
Praise be to Allah.Firstly: 
No one 
    should vow to do anything. Whoever wants to do an act of worship should do 
    it without making a vow, because by making a vow he makes this act of 
    worship obligatory upon himself, and he exposes himself to sin and 
    shortcomings and deserving the punishment if he does not fulfil what he 
    vowed to do. It is proven that the Prophet  (peace and blessings of 
    Allaah be upon him) told people not to make vows. 
See also the 
    answer to question no. 67886. 
Secondly: 
If what you 
    mean is that you vowed to fast if you passed your exams, and you did pass, 
    praise be to Allaah, then you have to fast, because the Messenger  
    (peace and blessings of Allaah be upon him) said: “Whoever vowed to do an 
    act of worship, let him fulfil his vow.” Narrated by al-Bukhaari (6696). If 
    you are unable to do it because of sickness and the like, then you have to 
    offer expiation for breaking an oath (kafaarat yameen), because the 
    Messenger  (peace and blessings of Allaah be upon him) said: “Whoever 
    makes a vow to do something that he is not able for, his expiation is 
    kafaarat yameen.” Narrated by Abu Dawood (3322) from the hadeeth of Ibn 
    ‘Abbaas. Al-Haafiz said in al-Fath: Its narrators are thiqaat, but it 
    was narrated by Ibn Abi Shaybah in a mawqoof report. Ibn Qudaamah said in 
    al-Mughni (10/72): Whoever vows to do an act of worship that he cannot 
    do or is able to do it but fails to do it has to offer kafaarat yameen. 
See: 
    Fataawa al-Lajnah al-Daa’imah (23/201). 
But if what 
    is meant is that you resolved to fast, without saying words such as vowing 
    or promising, then you do not have to do anything in that case, because a 
    vow does not become binding by mere intention, rather it must be spoken. 
It says in
    al-Muhadhdhab: A vow is not valid unless it is spoken. 
Al-Nawawi 
    (may Allaah have mercy on him) said in his Sharh: Does a vow count if 
    it is intended and not spoken? The correct view, according to the consensus 
    of our companions, is that it does not count unless it is spoken. Intention 
    alone does not count (as a vow). End quote from al-Majmoo’ (8/435). 
It says in
    al-Insaaf (11/118): A vow does not count unless it is spoken. If he 
    intends it without speaking, then it is not counted as a vow, and there is 
    no dispute on this point. 
And Allaah 
    knows best.