NIT

Can you make your question more clear please?

No, if you use state quota in some lower NIT, then you can get only. Otherwise, on the basis of this rank, you will not get any branch at any NIT. Closing rank of all NIT closed on less than 40000 rank.

It will depend upon the college cut-off or rank. Every NIT has their own cutoff for taking admission into various branches. https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
For getting knowledge that which college you will get, use the above link for predicting the college.

Unfortunately No. The chances are very less. This score is not enough for NIT. Look below link for selecting the campus:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=4479470.

You can get an NIT but I am not sure that you will get your favourable branch or no:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
You can use the above link for college prediction. Best of luck. Hope for the best.

For getting college, you can use the college predictor. Please find the link in below details:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
If you have qualified in your respective Category with high margins between the cut-off and the marks you scored then you will definitely get the college like NIT. I cannot tell at the time that which college you are getting so also the fee structure I can't tell you.

Hello,
You can use the given link to predict your colleges and branches as well. Thank you!
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=7589611.

Hello joins,
You may use the link below to know your chances of getting admission into NITs:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
All the best!

It depends on your home state. For All of India, I would recommend VIT, SRM, KIIT, ITER, BIT, NSIT and many more. You can go to https://shiksha.com and find great colleges in your home state or any part of the country.
Good luck.

As far as I know, no colleges in India concentrate their entire Masters course in pure mathematics. But they do provide abstract papers as a part of the syllabus. Further if you like any paper, you can go ahead and start reading more about that paper. Almost all the NIT's have very similar course structure. National Institute of Technology Calicut will be your best BET. Regards.