A complete list of IITs, IISERs, NITs and other institutes is given in the MHRD website.
Previous years' question papers of various competitive exams including NBHM, CSIR-UGC NET-JRF, TIFR, GATE and JAM may be found on the following pages. Please note that I'm not responsible for the (copyrighted) contents shared on these websites; visit at your own risk.
You may join this google group to subscribe to email alerts regarding Math Conferences/Workshops, Ph.D./Post Doctoral positions etc in India and abroad.