Hi Boris,
So what you are saying is that the question's answer can determine if the student actually is allowed in the course, or perhaps the answer allows the student to stay in the course. Here is a wild idea.
If students can enter but not proceed until they answer the question, then there is a contributed conditional activity code called Activity Locking. Here the student would enter the course, have only one thing they can do (Answer your question). It could be an assignment question, Quiz with a single question, an essay question in Lesson, or one of the other survey, feedback or Choice methods already presented. For that matter you make them enter a Wiki page. You pick the activity or resource.
You would then "lock" everything you did not want the student to use unless they completed your initial task. Completion of a task could mean achieve a minimum score, spend a certain amount of time in a task, or merely viewing it.
I assume you want what is called internal enrolment, the student initiated enrolments? There just might be an external Enrolment plugin that can handle that.
Hope this helps. Chris
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