What is the ruling on continuing studies based on an exam score provided by a teacher, and is that considered permissible or forbidden?

Raising students' grades without them deserving it is cheating and treachery. If this happens without the student's effort, then the sin is upon the one who facilitated it. The student should seek Allah's forgiveness and repent to Him, and strive to study the materials diligently to become deserving of the grades, while being keen on working hard in their studies and comprehending the curricula in the future.