Yes, Anyone can give exam in whichever stream he or she is interested in GATE. So you being from Production and Industrila background can give mechanical branch paper in gate, but you will only get M.Tech admission in that particular branch in which you have written your gate.
Although if your target is some PSU and other recruitments through GATE then it could be a barrier to give GATE from other stream as opposed to your graduation stream .These companies have some branch specific criteria along with GATE stream paper which you have appeared in. Same goes for some courses in IIT’s & NITs.
So if you are writing gate in mechanical engineering to get a job as mechanical engineer in PSUs then please don't do that.
Hope this clears your doubt.
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