If the absence of the in your third example is intended, then that example is different from the first two, but let us not get sidetracked. We'll put it back in.
Managed to implies, to me, that there was more expectation in the mind of the speaker (or of the person taking the exams) that failure might be possible. Similarly, was able to implies more chance of expected failure than could.
And that's about it for me.
NOTE: The above comments are about the expressions in your three example. In other situations there are much clearer differences, including times when only one, or two, of them may be used but not the other(s).
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