Karnataka

There are various good institutes in Bangalore. Christ University is amongst them. You can refer to the below link for the list of good colleges in Bangalore.
https://www.shiksha.com/mba/colleges/mba-colleges-in-india

Here I give you the name of some top college in Karnataka for mechanical engineering who got the admission through COMEDK:
RV College of Engineering, Ramaiah Institute of Technology, PES Uniersity-Bangalore, BMS College of Engineering, The National Institute of Engineering and Shri Sai Ram College of Engineering.

Absolutely not. In fact, the best department in terms of academics and faculty in Manipal is mechanical.

Around 12 - 14 lakhs including everything.
Yes, scholarships are available in this institute given according to its entrance exam score.
For complete information, check Manipal University B.Tech scholarships.