GATE

Just go through this link and fill out the required details:
https://www.shiksha.com/college-predictor

As per the eligibility for M.Tech, you must have done either Btech or (Bsc+Msc), then only you can apply for M.Tech program. Since you have done B.Sc in Biotechnology, you further need to pursue Msc in any specialisation of Biotechnology
Yes, you can do m.tech after b.sc in maths. sit for gate, score good and get admission to the college of your choice.

Just go through this link and fill out the required details:
https://www.shiksha.com/college-predictor

Go for GATE.
For IAS, you should appear in UPSC CSE.

You have to apply for counselling process after getting score in GATE to get a college.

GATE CUTOFF Mechanical Engineering
34.1 for general
30.7 for obc
22.7 for sc,sT

You can find it either on JoSAA or CSAB official website. Otherwise, look for CBSE official website. You can find it on any of the sites.

There are many books for GATE exam. But first of all you go through the syllabus and you can read any book which cover the syllabus.

There are many books for GATE exam. But first of all you go through the syllabus and you can read any book which cover the syllabus.

NITs, BMS-Bangalore and RV College-Bangalore are some of them where you don't need GATE scores other than Spans and CEPT.