Jolin has stopped being Jay's girlfriend since he dated another beautiful young girl.

1. Jolin ______ Jay's girlfriend since he ____ another beautiful young girl.
(A) stopped being; dated
(B) has stopped being; dated
(C) will stop to be; left
(D) would stop to be; left

2. ______ Anna ______ Lisa have stopped exercising at the gym for months.
(A) Both; and
(B) Not only; but also
(C) Either; or
(D) Neither; nor

Source: written by a Taiwanese teacher of English.

According to the designer: the answer to question #1 is option B; the answer to question #2 is option A.
Do native speakers agree with the answers?

I agree with their answer to #2.
For #1, none of the options work for me. I'd expect the continuous in the second half.

emsr2d2 said:

For #1, none of the options work for me. I'd expect the continuous in the second half.

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Is option E acceptable?
1. Jolin ______ Jay's girlfriend since he ____ another beautiful young girl.
(E) has stopped being; has been dating

sitifan said:

Is option E acceptable?
1. Jolin ______ Jay's girlfriend since he ____ another beautiful young girl.
(E) has stopped being; has been dating

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That's OK. In real life I would expect to hear that Jolin stopped being Jay's girlfriend after she found out he was seeing someone else. Or something like that.

For the other one I would probably say that neither Anna nor Lynn have been going to the gym for months.

It wouldn't be normal to include the "exercising" part. Why is that? Well, for one thing, people take it for granted that if you're going to the gym it's to exercise. Also, you could go to the gym and not exercise. (That would be weird. )

sitifan said:

1. Jolin ______ Jay's girlfriend since he ____ another beautiful young girl.
(A) stopped being; dated
(B) has stopped being; dated
(C) will stop to be; left
(D) will stop to be; left

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(C) and (D) are identical.

If you are being compelled to use these very dodgy exercises written (not designed) by a Taiwanese teacher, you have our sympathy.

If not, find some written by English-speaking teachers.

Rover_KE said:

(C) and (D) are identical.

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I am sorry to have made the typo. (D) would stop to be; left

sitifan said:

According to the designer: the answer to question #1 is option B

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Agreed. I don't have a problem with B, and the other answers are obviously wrong.

sitifan said:

the answer to question #2 is option A.

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No. The whole question fails badly, in my opinion, as it doesn't make sense. The part at the end 'for months' makes me think that what the writer of the sentence was trying to say was something like this:

Neither Anna nor Lisa has been to the gym for months.

The subject phrase of this sentence is negative, which completes the idea of 'not for months' to say there has been an absence of an activity.
If you use that sentence as the goal, the question just about works.

jutfrank said:

If you use that sentence as the goal, the question just about works.

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Does the phrase "just about" mean "almost"?

Yes. It comes close to working. Close.

sitifan said:

1. Jolin ______ Jay's girlfriend since he ____ another beautiful young girl.

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I don't find the first clause very natural whatever tense/aspect you use. We don't normally 'stop being someone's girl/boyfriend' in BrE. It's more natural to say we 'stop dating/seeing' them or 'break up with' them.

sitifan said:

Does the phrase "just about" mean "almost"?

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No, quite the opposite. It means it meets the minimum conditions necessary to work.

If you say something 'almost' works, it means it doesn't work. If you say something 'just about' works, it means it does work.

The whole sentence is a very unnatural way of saying "Jolin broke up with/dumped Jay when he (Jay) started dating someone else".

Well, it's not really an argument. I stuck with the OP's original phrasing. More natural is saying she stopped seeing him or "dumped" him.

I hope I didn't give the wrong impression. If you say something "almost" worked you mean it didn't work. If it did work you wouldn't use "almost".

jutfrank said:

No, quite the opposite. It means it meets the minimum conditions necessary to work.
If you say something 'almost' works, it means it doesn't work. If you say something 'just about' works, it means it does work.

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just about​

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Almost, very nearly, as in This job is just about done, or At just about midnight we'll uncork the champagne. This phrase is sometimes used alone, as in Are you finished yet?—Just about. It uses about in the sense of “nearly,” a usage dating from the early 1600s.

I prefer jutfrank's explanation.

sitifan said:

Almost, very nearly, as in This job is just about done, or At just about midnight we'll uncork the champagne. This phrase is sometimes used alone, as in Are you finished yet?—Just about. It uses about in the sense of “nearly,” a usage dating from the early 1600s.

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Is this a question or are you just informing us about the English language?

As I explained, if you say 'something just about works', it means that it meets the minimum conditions of working. If you want to ask about other uses of 'just about', you could start a new thread.