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Usually when something is said to be done with no questions asked, it implies that the task should be completed no matter what, even if one has to employ methods they wouldn't normally use.
However, this phrase can also just mean that no further information will be asked for.
This expression has been around since at least the 1700s, probably longer, and it was often used in its early recordings in conjunction with a reward. Take The London Magazine from 1757 as an example, where a reward is being offered to find a suspected thief:
"Whoever brings her to Mrs. Trolly's above-mentioned, or to the guard-room at E Whitehall, shall have fifty
guineas reward, and no questions asked."
However, given the context of that quote, it appears as though the idiom is being used to show that the people with authority will not ask for any details to whoever brings in the suspected thief.
* Harold, I need you to acquire a very delicate piece of delicious turkey for me from the grocery store, no questions asked, and it needs to be done first thing tomorrow morning.
Note: The origins for most common idioms cannot be said with a certainty. What's provided are theories that may be plausible to how a phrase originated, but not necessarily so.
In addition, quotes that contain a phrase may be taken from old newspapers, poems, or books that were written centuries ago, but this by no means shows they originated from these. In all likelihood, if an expression is being used in a newspaper, it's probably already well known, and thus, from an older period of time.