Hello Nil

Belonging to a general category the chances could be quite tough to get a govt college. However there are still some colleges with cutoff rank above 6000 like haldia institute of technology etc where you can opt for . Here is the link for your better understanding of colleges and their cutoff rank-

https://www.google.co.in/amp/s/engineering.careers360.com/articles/wbjee-cutoff%3famp

Thanku I hope it helps