All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
If the case is as explained in the question and you vowed to pray two Rak‘ahs after every obligatory prayer if you passed your exams, then it is incumbent on you to fulfill your vow if you passed the exams. The Prophet, sallallaahu ‘alayhi wa sallam, said: "Whoever vowed to offer an act of obedience to Allaah, the Exalted, must do; and whoever vowed to disobey Allaah, the Exalted, should not do." [Maalik and Al-Bukhari]
Since you cannot remember the exact number of those days and you are almost sure that you did not vow to pray those prayers for more than five days, we believe that it is not incumbent on you to perform the vowed two-Rak‘ah prayers for more than five days. This is because the basic principle is the clarity of your liability, and one is not liable to what is not certainly known. You are positive that you vowed to perform those prayers for five days, so you should fulfill your vow and perform two Rak‘ahs after each obligatory prayer for five days. Moreover, you should make up for what you missed of those vowed prayers (during the five days).
The following is stated Al-Fataawa Al-Hindiyyah, "If someone says 'It is due to Allaah upon me to perform two Rak‘ahs today', yet he fails to do so, then he is obliged to make up for them."
However, you are not obliged to make up for what you miss of the vowed prayers because of menstruation or post-partum bleeding because the obligation to perform those vowed prayers is not as strongly emphasized as that of making up obligatory prayers. It is known that women do not make up for the obligatory prayers that they miss during menstruation and post-partum bleeding. As for the two Rak‘ahs that you vowed to perform after the Fajr and ‘Asr prayers, you are not obliged to perform them or make up for them because in that case you vowed to do a disliked act. When a person vows to do a disliked act, he has the choice either to deliver on his vow or to offer an expiation like that for breaking an oath. It is better to offer the expiation because it is disliked to pray after the Fajr and ‘Asr prayers. Muslim jurists state that it is recommended for the person who vows to do a disliked act to break his vow and offer an expiation like that for breaking an oath. This was underlined by the Hanbali scholar Al-Hijjawi as he said, "When the person vows to do a disliked act such as divorce or anything else, it is recommended for him to break his vow and offer expiation." [Zad Al-Mustaqni‘]
Performing the vowed two-Rak‘ah prayers before the Fajr and ‘Asr prayers is not considered a fulfillment of your vow. This is because they are performed in other than the time specified in the vow (i.e. after the obligatory prayers). Hence, you should offer an expiation like that for breaking an oath for the two Rak‘ahs after the Fajr and ‘Asr prayers that you have missed.
Allaah Knows best.