The correct answer is (A).
It seems like Christine is one who avoids her responsibilities, so let's look for a word that means just that.
(A) "malingering" (which means "evading responsibility by pretending to be ill") looks promising, so don't eliminate this. You can eliminate (B) "fomenting" because it means "treating pain," and we know that she only pretends to have headaches. (C) "propitiating" means "winning favor," so this is incorrect because it doesn't mean to avoid work. (D) "fawning" or "flattering" can also be eliminated, as can (E) "repudiating," which means "disowning" someone or something. Choice (A), "malingering," is the best answer.
