Here's the completed exercise:
Try to avoid making him angry.
- We use the gerund "making" here because "avoid" is followed by a gerund.
Three months is a long time to wait.
- "Three months" is considered a singular unit of time, so we use "is".
John or his brothers is/are going to help me.
- Both "is" and "are" can be correct here. It depends on whether you want to emphasize John individually or the group of brothers. If you're emphasizing John, use "is". If you're emphasizing the group, use "are".
It's no use crying over spilt milk.
- "It's no use" is followed by a gerund.
It's difficult to get used to eating with chopsticks.
- "Get used to" is followed by a gerund.
His doctor advised him to take up jogging.
- "Advise someone to do something" is a common structure. "Take up" is a phrasal verb meaning to start a new hobby or activity.