- (A) No matter what he tried to cover up the truth, it came out at last.
- (B) Whatever he tried to cover up the truth, it came out at last.
- (C) No matter how he tried to cover up the truth, it came out at last.
- (D) However hard he tried to cover up the truth, it came out at last.
Each of these four sentences is grammatically valid. Each is reasonably natural. I see no good reason within the question as given to prefer any one of the four, except possibly personal taste. Perhaps there was some context in the original quiz that made one of these preferable. Perhaps the question was poorly written, it seems that a significant number of ESL quizzes and tests are poorly written.
In any case, I cannot advise how to pass a particular quiz, but I can confidently assert that all four are grammatically valid.