Teacher

Yes. you can sdo both the courses.

LExturer banne ke lie nahin, lecturer banke aage tarakkee karne ke lie doctorate zaroori hai,

Dear Mahendradas,
You need to complete your Bachelor and master in math and than go for NET exam to become eligible to apply for professor job in colleges.Or you can go for Phd level after your master to apply as professor in colleges.
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Universities advertise their teaching opening at a regular interval. Nowadays minimum qualification for Asst Prof is PhD. Some universities and college s have relaxed rules on education qualifications. Asst Prof are hired in pay scale of Rs 15600-39100 known as PB3 with AGP of Rs 6000. In addition you will be entitled for central DA, HRA (30% of Basic) and other allowances. Private universities also provide similar scales to Asst Prof.

In promotional matters of jobs, there are quotas. Quotas are there for BEd admissions, if sponsored by institutional heads.

You can do any of these as with b.tech you can pursue m.tech and teach in engineering colleges. With bsc physics + msc physics + net you can teach in other colleges too.
if you wish to teach in school, then go for b.ed.