Hey Samarth,

Unfortunately not, the eligibility for govt colleges are bellow 5000 ranks (for GEN). But if you can take a risk for the last round then you might get a govt college. But the stream will not be of your choice. 

I suggest you not to do so. There are many good colleges in Kolkata. You can go for NiT(For ECE or CSE), AOT or techno. It totally depends on you which stream you wanna take. 

All the best

Dear Samarth if I say honestly there are very less chance to get any seat but fortunately you get any seat then you have to compromise with the branch and that is not good for your carrier that not help you in future or in corporate life, we can't say directly you get that college because of your rank.

All the best