(A) She was very ____ with him for his offensive remarks on women in politics.
We need a word that means she was very upset. The right one is cut up (means emotionally hurt). → (III)
(B) I have plenty of interesting books to ____ to book lovers.
This means you give something for free. The correct phrasal verb is give away. → (I)
(C) Strange that people easily ____ temptation and ruin their reputation!
Here, it means they can’t resist or stop themselves. So, the right one is give in. → (IV)
(D) Rahul is trying to ____ on his smoking.
This is about doing something less or reducing it. The phrasal verb is cut down. → (II)