She can definitely pass, compared to the other mages, her power is unparalleled as an elf like Serie, but it all boils down to how Frieren and Serie view things differently. Serie is a warmonger as Flamme said, so Serie values power and strength to fight, but Frieren who shared and inherited Flammes values believes that there is more to magic than just strength and power. That’s why Serie asked Frieren what magic Frieren wants to learn, if her answer is different from when she was asked before, then she might have passed, because it means that Frieren had given up the “ideal” views that she has. Frieren is more of a dreamer type in the eyes of Serie and she views that as weak or futile that’s why she failed Frieren.
Ironic since it was frieren and the Hero's party who managed to kill the demon king and in earlier chapters serie asked flamme how could this child (frieren) become more powerful than her.
So, Freiren can pass but the examiner won't let her to or is she unqualified?