Is success in an exam permissible if I previously failed it due to not studying, and my father paid 100 Riyals to cover the re-examination fees without knowing the real reason for my failure?

Your success in the exam is permissible and a goal for both you and your father. However, your delay in the first exam and your negligence in studying, which led your father to pay a sum of money, is considered a shortcoming on your part. There is no blame on you as long as you told your father the truth and did not lie to him.