Is it permissible to break a pledge made to obtain a right, such as an undertaking not to disclose information about a specific exam, especially since the student pays a fee for it?

The fundamental principle regarding conditions is their validity and permissibility unless they contravene Islamic law. The condition mentioned in the question is permissible and there is no harm in it. Therefore, it is not permissible for the student to violate the condition they have committed to, and they must fulfill it and not disclose the exam questions to anyone.