Confusing use of 聞く in this sentence

Question

試験では、ことばを覚えているかどうか聞く問題を減らすように言っています。
(Scientists) are saying to reduce the problem in exams of whether or not you've memorised a word.

I'm not sure I've translated this correctly. I'm confused about the inclusion of 聞く. I'm sure the act of asking whether or not you've memorised the word isn't the problem.

If I'd been asked to write this sentence I'd have replaced 聞く with either:

1) という
2) が
3) nothing

Would any of these options lead to a grammatical, natural sentence? Would my sentence have a different meaning?

Answer 1

問題 here is not a “problem” in the meaning of a “general issue”, but a problem as an “exercise”, a “test item”.

Thus ことばを覚えているかどうか聞く問題 are exercises asking (verifying) if examinees remembered words.

And it’s not the problem itself that needs to be reduced, but the number of such problems (exercises) in exams.

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That said, I’m not sure if the latter part of the question still remains relevant, but:

I'd have replaced 聞く with either:

1) という

This would refer to a direct question: “did you remember the word ...?”

2) が

Ungrammatical. I’m not even sure what the intention was. A possessive が? Anyway, it doesn’t follow an embedded question.

3) nothing

Passable, I guess. No change in meaning.