(A) The keyword here is "with his plan". We need a phrasal verb that means "to go along with" or "cooperate with" a plan.
"Play along" means to cooperate or pretend to agree with someone's plan. So, (A) = (II) play along
(B) The sentence talks about someone who has enhanced authority and tends to minimize or understate it.
"Play down" means to make something seem less important. So, (B) = (IV) play down
(C) The keyword is "to his vanity". We need a phrasal verb that means to flatter or cater to someone's ego.
"Play up to" means to flatter someone or cater to their ego. So, (C) = (III) play up
(D) The keywords are "weaknesses" and "relentlessly". We need a phrasal verb that means to exploit or take advantage of.
"Play on" means to exploit something like weaknesses or emotions. So, (D) = (I) play on