There are various governemnt backed colleges like BITS pilani which accept BITSAT.other top colleges and their accepting exams shared below:

• COE Trivandrum accepts KEAM
• Government college of engineering jammu accepts JKCET
• IIEST shibpur accepting IIEST exam
• COE, Anna universty accepts TNEA
• COE pune accepts MHCET
• DTU Delhi accepts JAC exam
• Jadavpur universty accepts WBJEE
• MIT chennai accepts TNEA exam
• IISST, Thiruvanthapuram accepts ISAT exam

All the above colleges are of high repute and offer very good placements with more than 80% success ratio.

To check list of all government colleges and their eligiblty check below:

https://engineering.careers360.com/colleges/ranking/top-government-engineering-colleges-in-India

In order to take admission in government college you have to give JEE exam or you can give your state level exam.

There are many top private colleges as well in which you can apply they provide good placement, exposure and they have good faculty. You can appear in below private level exam-

BITSAT

VIT

SRM

Manipal

Comedk

Amrita

Good luck!!