1. Have you seen Jane lately?

2. Have you seen Jane recently?
3. Did you see Jane lately?
4. Did you see Jane recently?
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Are they all grammatical? Which one is not?

All are correct.

Well, well, well! Let's take a closer look at these questions, shall we?

1. Have you seen Jane lately?
2. Have you seen Jane recently?
3. Did you see Jane lately?
4. Did you see Jane recently?

All of these questions are grammatically correct, my dear human friend. However, if we want to be quite nitpicky, the one that is a bit eyebrow-raising is the first one. Why? Because "Have you seen Jane lately?" suggests that the speaker is asking if you have seen Jane sometime in the near past. "Lately" implies a more recent time frame, so if Jane disappeared into thin air a while ago, you might want to opt for "recently" instead.

But hey, don't you worry, my friend! Jane's whereabouts are not as important as livening up this conversation with some good ol' humor, right?

All four sentences are grammatically correct.

Yes, all four sentences are grammatically correct. There is no sentence that is incorrect. However, the usage of "lately" and "recently" may vary depending on the context. Let's break down each sentence to understand their meanings:

1. Have you seen Jane lately? (Grammatically correct)
This sentence is asking if the person being addressed has seen Jane at some point during a recent period of time. The word "lately" suggests that the speaker is referring to a period of time that is relatively close to the present.

2. Have you seen Jane recently? (Grammatically correct)
This sentence has the same meaning as the previous one, but it uses the word "recently" instead of "lately." Both words convey a similar idea of a period of time that is not long ago.

3. Did you see Jane lately? (Grammatically correct)
In this sentence, the word "lately" is used in the past tense ("did you see" instead of "have you seen"). This suggests that the speaker is referring to a time before the present. The question is asking if the person being addressed had seen Jane at some point during a recent period of time in the past.

4. Did you see Jane recently? (Grammatically correct)
Similar to the previous sentence, this one uses "recently" in the past tense ("did you see"). It also refers to a period of time in the past relatively close to the present. The question is asking if the person being addressed had seen Jane at some point during that recent period.

So, all the sentences are grammatically correct, but the difference lies in the choice of "lately" and "recently," and whether they are used in the present tense or the past tense.