IIT JEE is the National Level Exam for the admission to Engineering Programme, As per the eligibility to apply for IITs you should have minimum 75% in 10+2.

You can check the further eligibility to apply for the exam with the given below link.

https://engineering.careers360.com/articles/jee-main-eligibility-criteria

After qualify JEE Main you need to apply for JEE Advance to get admission in IITs.

There are other University level exam also, You can apply like SRM, Manipal, Here are the some of the institutes you can prefer given below:

Indian Institute of Technology Madras

Indian Institute of Technology Bombay

Indian Institute of Technology Kharagpur

Indian Institute of Technology Kanpur

SRM University, Chennai

RV College of Engineering, Bangalore

Manipal Institute of Technology, Manipal

Good Luck!