Should he delay Hajj because of the exams?

The most correct scholarly view is that it is obligatory to
do Hajj straightaway, and it is not permissible to delay it without a
legitimate shar’i excuse, because you do not know what will happen to you.
You may delay Hajj this year and then die, and Hajj will remain a debt that
you owe. But if doing Hajj will affect your performance in the exams, then
you may delay it until next year. But I suggest taking your study materials
with you and going for Hajj, if you are going to travel to Hajj early. Or
you can delay travelling to Hajj and hasten to leave during the days of
stoning the Jamaraat. In this way Hajj will only take a few days and it will
not affect your exams in sha Allaah. 

If a person is keen, he can do Hajj, and that will not affect
him at all. If you put your trust in Allaah and perform Hajj with trust in
Him, He will make things easy for you. End quote. 

Fataawa Ibn ‘Uthaymeen, 21/63.