In theory, different parties could put forward different proposed exams before an election, and then citizens could vote on which exam would be used to filter the candidates. (There would probably have to be a "null exam" option on the ballot, to allow people to oppose the use of this filter, and perhaps all non-voters should be counted as having voted for that option).

Arguably, though, this process would just move the problem of voter apathy back a step, unless there is some reason to expect that voting for exams is more motivating (or more reasonable to make mandatory) than voting for candidates. Also, it's not clear that the exam would be testing anything other than candidates' memory, as presumably the correct answers would have to be published in advance for the citizens to check.

Possibly true, but we also haven’t really attempted it on a society-wide scale. Clearly some government agencies are capable of creating fair tests (NASA, for example.)

In any case, the question isn’t whether an exam would be perfect, but whether it would be better than voting. There are certainly plenty of biases involved in the voting process: appearance, public speaking ability, etc. These biases might already be greater than any which arise from an exam.