An exam dream is an extremely common anxiety dream across the world, and it resonates with particular force in Korea, where the 수능 (national college entrance exam) looms so large in the culture that "exam anxiety" is close to a shared national experience, well beyond students who are actually taking tests at the time.
This dream is generally read as being evaluated or tested in some area of waking life, whether or not an actual exam is involved — a work review, a major decision others are watching, or simply a period where you feel like you're being judged on your performance.
Failing the exam within the dream is read as a fear of being judged inadequate, often out of proportion to the actual real-world stakes — the dream tends to amplify the fear well past what the real situation would justify. Realizing you're completely unprepared, or that you forgot to study, is read as feeling caught off guard by a real responsibility that's arrived faster than you feel ready for.
Arriving late to the exam, or being unable to find the room, is read as anxiety about missing an important opportunity or deadline — the exam itself almost secondary to the fear of not even making it to the starting line in time.