The answer for Part A is C.
"Abandon" means to quit or lose, so the word "quit" would be the best answer.
The answer for Part B is B.
Since "abandon" means to "quit", "lose", or "give up", the phrase "given up trying" would fit best for the answer.
This is a sentence because it has a subject and a predicate. a fragment is not a complete thought/idea.
He rejects vengeance. I've took the test last week. ;)