1) After his illness the boy couldn’t keep …with the class.
2) Keep …it, my boy!
3) You should keep…playing football.
4) The doctor advised him to keep…meat for some time.
5) They were kept …by a heavy rain.
6) You should keep your car…
7) I’ll keep …nothing from you.
8) Keep…with this work.
9) Keep …the topic of the conversation.
10) You’d better keep …the quarrel.​

1) up
The phrase "keep up with" means to maintain the same pace or level as someone or something. In this case, the boy couldn't maintain the same pace or level as the rest of the class after his illness.

2) at
This sentence doesn't provide much context, but "keep at it" is a common phrase which means to continue doing something or to persist in an effort. So, in this case, the speaker is encouraging the boy to continue whatever he is doing.

3) on
"Keep on" means to continue doing something without stopping. So, in this sentence, the person is being advised to continue playing football without stopping.

4) off
In this sentence, the doctor is advising the person to avoid eating meat for some time. "Keep off" means to refrain or stay away from something.

5) in
"They were kept in by a heavy rain" means that the heavy rain prevented them from leaving or going out. "Keep in" means to prevent or confine someone or something within a certain area.

6) clean
"You should keep your car clean" means to make sure the car is tidy and free from dirt or clutter. "Keep" in this case is used with the adjective "clean."

7) nothing
"I'll keep nothing from you" means that the speaker will not hide or withhold anything from the person they are speaking to. "Keep" in this case means to maintain or retain.

8) on
"Keep on with this work" means to continue doing or working on something without quitting. "Keep on" is a phrase used to encourage persistence.

9) to
"Keep to the topic of the conversation" means to stay focused on the specific subject or theme of the conversation. "Keep to" means to adhere or stick to something.

10) out of
"You'd better keep out of the quarrel" means to avoid getting involved or staying away from the disagreement. "Keep out of" means to refrain from getting involved or staying away from something.