durchfallen
to fail
To not pass an exam, test, or class.
Leider ist er bei der Fahrprüfung durchgefallen.
Unfortunately, he failed the driving test.

to fail / fall through
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to fail
To not pass an exam, test, or class.
Leider ist er bei der Fahrprüfung durchgefallen.
Unfortunately, he failed the driving test.

'Durchfallen' (to fail / fall through) is a separable verb. In a main clause, the prefix 'durch' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er fällt oft bei Prüfungen durch.
He often fails exams.
The perfect tense of 'durchfallen' (to fail / fall through) is always formed with 'sein' (to be), not with 'haben' (to have). This applies to all meanings of the word.
Ich bin letztes Mal leider durch die praktische Prüfung gefallen.
Unfortunately, I failed the practical exam last time.
The literal meaning 'physisch durch etwas fallen' (to physically fall through something) helps you understand the figurative meanings. In an exam, you more or less 'fall' through the net of requirements.
Pass auf, der Stuhl ist kaputt, nicht dass du durchfällst!
Watch out, the chair is broken—don't fall through it!
When talking about an exam, you use the preposition 'bei' or 'in', followed by the dative. Both are correct.
Sie ist in der mündlichen Prüfung durchgefallen.
She failed the oral exam.
In the meaning 'abgelehnt werden' (to be rejected), the preposition 'bei' + dative is often used to name the group that rejects something.
Der Vorschlag ist bei der Abstimmung im Team durchgefallen.
The proposal failed in the team vote.
to pass / consist / exist
'bestehen' and 'durchfallen' are opposites for exams, tests, or classes. 'bestehen' means that you are successful and achieve the required performance. 'durchfallen' means that you do not pass the exam or test. Both words are especially typical in school, university, training programs, and official exams. The simple contrast is: someone who passes has succeeded; someone who fails often has to repeat it.

Mein Bruder ist in der theoretischen Fahrprüfung durchgefallen.
My brother failed the theoretical driving test.
Anna besteht die Abschlussprüfung mit einer guten Note.
Anna passes the final exam with a good grade.