Pretty hard but I'll guess......................... A !
C. Both.
I know nothing about anything and I know nothing .
A is correct.
B should read "I know nothing".
Well A is grammatically correct and is used to indicate that the person knows nothing about what has been asked whether he is speaking or writing the sentence.
Occasionally, a person may actually speak (say out loud) the second one with emphasis on the word "nothing."
In that case it's like an idiom that makes it clear to the listener that the speaker knows absolutely nothing about the issue he is referring to.
A. Is correct. B. Contains a double negative
In grammar A is correct, but B has double negation so it's wrong in grammatical writing.