Is "at outset" grammatically correct to use as in the following
These dormitories were at outset for senior girls
or is "at the outset" better?
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"At the outset" is more modern, so if you want to sound like you are using contemporary English, then "at the outset" would indeed be "better."
- a. The action or fact of setting out upon a journey, course of action, business, etc.; a start, a beginning. Now chiefly in at the outset, from the outset.
(Oxford English Dictionary [emphases OED's])
There seem to be certain contexts in which at outset is a fixed phrase, such as in
French for "on guard"; spoken at outset to warn the participants to take a defensive position. (Wikipedia: Glossary of Fencing)
B in scientific papers, such as this one
The formerly proposed percentage of patients that achieve normal weight (or overweight) is clearly unjust to to patients with more severe obesity at outset
I would recommend using at the outset or from the outset rather than at outset unless you have a special reason not to.