Engineering Entrance Exam

I am not sure where you are studying and what is the context of this question, it certainly sounds bizarre that your professor is asking you to cheat. Having said that, I am also aware that certain institutes allow open book/notes exams where students are allowed to refer to book and/or notes during the examination. I guess the objective being, with availability of all resources at your disposal, how well would you perform. Needless to say, in order to make effective use to books/notes, you should know what and where to refer and that in turn means you would have at least been through the books and notes once, if not more.
I am not sure, but I do hope that, this is the context and background with which your professor asked to cheat.

Yes, there is a quota of JEE Mains in the state colleges of Maharashtra though it is preferable to give MHT-CET for getting into the state government colleges of Maharashtra. If you are seeking admission into COEP you don't have to worry about the branch. Above 200 marks will be sufficient for you.
If you want a core branch such as computer science and engineering, mechanical engineering, electrical
engineering and so on you need marks above 250.
The above cases are for the general student. For an SC and ST student, the cutoff is very low as you do not need to touch even 200 for getting into the CEP as a core streamer.
Same for OBC the cutoff is lower but slightly higher than SC and ST which can be above or around 200 but still have more chances of getting into COEP than a general student.
Now in my personal advice, if you are very much worried and focused about placements, then go for petroleum in COEP. It will make you a millionaire in a very short time but not without hard work.