JEE mains is a entrance exam you can get seat in a private college without appearing in JEE mains. Students with negative marks also get admission in private colleges. JEE mains gives you a chance to score good and get through a good government college with less fee load on your shoulders.

You can predict college you are eligible for based on your jee mains marks here

https://engineering.careers360.com/jee-main-college-predictor