Hello Student, 

IIITs generally have closing rank for general candidates around 40-45k (last round cutoff). 

So,  I am sorry to say but with a rank of 60k it's not possible for to get any IIIT. 

Yes, you can apply for some government colleges of your state. 

Also, there are some private institutes that accept Jee Score for admissions like; IPU, Thappar, Jaypee etc. 

Hope this will help. 

All the best for future.