What is the difference between these two?
out of question
out of the question
Does "out of question" mean "undoubtedly" and "out of the question" mean "impossible"?
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It's been discussed on ELU, here.
Apparently, "out of question" meant
Beyond doubt, undoubtedly
but it is obsolete. This says it is obsolete.
Out of the question is still in common use, in my opinion
out of the question
Not worth considering because of being too difficult or impossible:
Out of the question is the term in use today. It means beyond consideration; unthinkable or impossible (Collins dictionary).
Out of question is obsolete, which means no longer used. When it was used, it had two meanings.
The first one was unquestionably or doubtless. See Definition 5 in Universal Dictionary of the English Language. Today, for this meaning we use without question.
As you can see from Definition 6 from the same dictionary, it also meant out of the question or not to be thought of. So it was just a variation of the same phrase with the definite article. (As an aside, this shows how idiomatic the use of the definite article can be.)
The Oxford English Dictionary gives an example of out of question, as a variation of out of the question, as recently as 1940:
Flight tests have shown that more than one glider can be towed by an aeroplane, so that a glider train is not out of question.
(from the magazine Aeronautics).
So, in general, you won't see out of question in use nowadays, but in older uses (that are now obsolete), it could mean either the same thing as out of the question or it could mean unquestionably.
I think it is like this :
out of the question : Not worth considering because of being too difficult or impossible
Your answer is out of the question.
In other word,
"out of the question" do not relate to the question.
And the other hand :
out of questions : You no longer have any questions at all.