Решение:
- a. I knew him to be a just prosecutor. (Конструкция 'know someone to be').
- b. My parents like you to behave like that. (Конструкция 'like someone to do').
- c. The old women saw that robber enter the room. (После глаголов восприятия 'see', 'hear', 'watch' используется инфинитив без 'to').
- d. The noise caused her to awake. (Глагол 'cause' требует инфинитив с 'to').
Ответ: a. to be, b. to behave, c. enter (без to), d. to awake.