To be a good project manager do you need to do homework?

yes

Of course not only for a Good project manager but every one in his / her life must always have a (( DO TO IT LIST )) in order to know what finished and which is not or even partially finished , in order to update yourself either in personal life or career to be on the track of life racing and not out of the line or even behind it.

Not really, jus maintain a daily srum board adding all your tasks on it and seggregating them under "to do, inprogress , pending with some one else, closed".
Its also recomended to think about the status of small tasks pending/closed before going to bed which helps you remeber better before going to office next morning.

Definitely - homework for me means also staying up to date with what is happening in global economics, technology etc in the field that I am busy in - this has always added value for me.
Planning is key and crucial right from the word go and checking that the project is still in line with the objective.

Certainly yes, but working honestly with projects, he should know each and everything of projects even if you wake him from rest at any time he should be answerable by all mean.
Like this Projects Manager dose not need home work.
According to me home work always good for decision and action to promote your good work.

As like as any other work we should home work.

why not?

Yes most Definitely!