Electrical and Electronics Engineering

Hello Nikhil,
Ye,s you are eligible for Automobile Engineering after completing B.Tech in any stream. You have to give GATE in ME subject for that. You can take Admission in Colleges which offer M.Tech in Automobile Engineering. List of Colleges offering M.Tech Automobile Engineering is in the link below. https://www.shiksha.com/engineering/colleges/m-tech-automobile-engineering-colleges-india
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If you choose Civil, Mechanical, Electrical, then it is better, because subject which has been taught in these branch are available as a main subject for IAS main exam.

Hello Karthik, Aah! You are in a typical Shakespearean dilemma! Calicut as well as Nagpur, both NIT' s have quite good reputation. Therefore I would like to advise you that you can take admission in the NIT which is convenient to access and commute from your home.

Placements of EEE at NIT Nagpur are very good. In fact, if you are very good at coding, you can sit for placements along with CSE people for the companies that come to hire them. However, the curriculum is very strenuous and for a good pointer, you'll have to study really hard. But all that will be worth in the long run.
All the best!

https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
Go to the college predictor link I am sharing here to know about the colleges you can get.
Good luck.

Hi Kunal,
Both the colleges are equally good and rank among the top engineering institutions of India. However, the kind of flexibility in education, zero attendance policy and campus life that BITS offers is simply unmatchable. Other than that, academics, placements and infrastructure are at par for both the institutes. The fee of NITT is much lesser than BITS. So, choose wisely as per your preference and constraints.
Best wishes.

You can go to join via lateral entry. It will be the right choice for your career.
Good luck.