Engineering Entrance Exam

You should have pass with 60% and have to give entrance test for PU.

Anna University is a big university so first of all you need to study details of admission procedure because it has some admission rules.

Colleges like VJTI, COEP, SPCE are government / government aided autonomous colleges. They fill 100% of their seats through MHTCET state ranks and not through JEE. Unlike other private engineering college, which has 15% quota through JEE, these colleges do not have any JEE quota.

Your question is not clear but by gist I could guess you are talking about not giving any entrance exam for engineering. Yes sure you can, there are many private college where you can get admission without any entrance exam but mostly they are not so good, if you have adequate money you can also opt

If that university requires an entrance exam then you have to give entrance and fill the form, but nowadays, many universities are offering admission on the basis of JEE Mains rank or marks, so you have to apply for counselling of that university. Like, if u want to get any college under PTU or PU t

Almost, all the colleges including the private one's consider JEE Mains marks for admission. The cut-off generally varies from college to college and also on the basis of that year's JEE cut-off.
Some colleges also offer direct admission on the basis of class 12th marks.

It depends on the college, if you want a private collection then you can get admission by getting 60 marks in JEE Mains but if you want NITs you need to get 250+ marks.

If you're qualified in JEE mains, your application will be accepted. But, it's compulsory to have studied Mathematics in PUC to apply any stream of engineering in Presidency University, Bangalore.

You can find all the information here:
Shiksha.com