Starting premise: Chazal were not fond of people asking questions solely to waste their time. We see this from, for instance, Bava Basra 23b:
בעי ר' ירמיה רגלו אחת בתוך חמשים אמה ורגלו אחת חוץ מחמשים אמה מהו ועל דא אפקוהו לרבי ירמיה מבי מדרשא
R' Yirmiya asked, "If one leg was within fifty amos and one leg beyond fifty amos, what is the halacha?" For this they expelled R' Yirmiya from the Beis Midrash.
Critically, Tosfos there note that the issue with R' Yirmiya's question had nothing to do with its improbability, but rather the fact that he was questioning a basic premise of how measurements work that showed he was just wasting their time:
ועל דא אפקוהו לרבי ירמיה מבי מדרשא. אין לפרש משום דבעי מילתא דלא שכיחא כלל דהיכי איתרמי דרגלו אחת תוך חמשים ורגלו אחת חוץ לחמשים בצמצום דהא אשכחנא דמתניתין נמי בכהאי גוונא איירי דקתני מחצה על מחצה יחלוקו ונראה לרבינו תם דמשום הכי אפקוהו משום דמדדה אינו מדדה כלל יותר מחמשים אמה אפילו רגלו אחת דכל מדות חכמים כן הוא
"For this they expelled R' Yirmiya from the Beis Midrash" – Don't explain that it's because he asked a question which never occurs, as how can it be so precise that one leg is within fifty amos and one leg is beyond fifty amos, because we find that our Mishnah as well discusses a similar case, as it teaches, "Half and half – they split." It appears to Rabbeinu Tam that because of this did they expel him: because the measurement is not a measurement at all beyond fifty amos, even if it's just one foot, as all measurements of the Chachamim are like this [i.e. this precise].
(If I may add to Tosfos, this isn't the first time R' Yirmiya's questioned this premise; cf. Rosh Hashanah 13a.)
It's implicit in Tosfos that the issue is specifically wasting time; were it simply an impossible question, there would have been no problem with it.
Yet we find in Menachos 37a that Rebbi threatened to exile or excommunicate someone for asking an impossible question until he found that question was not only possible but had actually happened:
בעא מיניה פלימו מרבי מי שיש לו שני ראשים באיזה מהן מניח תפילין א"ל או קום גלי או קבל עלך שמתא אדהכי אתא ההוא גברא א"ל איתיליד לי ינוקא דאית ליה תרי רישי כמה בעינן למיתב לכהן
Plimo asked of Rebbi, "Someone who has two heads – on which one should he place Tefillin?" He said to him, "Either go into exile or accept upon yourself excommunication." At that time, a man came and asked him, "A child was born to me with two heads – how much do I need to give to the Kohen [for Pidyon HaBen]?"
Rashi clarifies Rebbi's complaint against Plimo:
או עמוד וצא בגלות או קבל עליך שמתא דאחוכי חייכת בי
Either get up and go into exile or accept upon yourself excommunication, because you are mocking me.
If impossible questions are okay, what was it about Plimo's question which Rebbi felt was mockery? And if impossible questions are not okay, then what of Tosfos' proof that that's not why R' Yirmiya was expelled?
