It's an interesting question. As an experienced test-taker I would definitely choose D, "sharp," because "a sharp mind" is a very frequent phrase. However, I believe it is a BAD TEST QUESTION, because I believe "an acute mind" is perfectly correct, and not very rare. I'm going to test that with a Google search of Project Gutenberg: "the fact seemed to him extremely probable when looked at all round by his acute mind." "A far less alert and acute mind than his must have seen that the Reform troubles of 1866..." And for some modern examples, Google Books: " And I'm grateful to my eldest daughter, who lent her acute mind and sharp pencil to this manuscript, too." A 2002 book even says: "One might perhaps illustrate this from ordinary expression about the mind, e.g. 'He has an acute mind' means that he sees the point of arguments quickly, spots mistakes or solves problems easily, and so on." I would go further. I would say "acute" is just a synonym for "sharp" and that you could use _any_ synonym for sharp: "a keen mind," "an incisive mind," even a "well-honed mind." My guess is that the person who wrote the exam question is familiar with the phrase "an astute mind" and thinks that because "astute" is right, the similar word "acute" must be wrong!

Yes, you probably could. It's not common, but the meaning should be clear to any reasonably intelligent person.

I wouldn't use an acute mind as it would suggest you didn't have a capacity for it.