Science

Sorry to say, but you can't get admission in DTU for biotechnology unless you crack JEE mains exam with a good rank.

No, it is not necessary. You start a new career after 12th class in a new field.

Yes, it is included in carbon chemistry. I think it is included in the exam.

B.Sc Math Honours: Eligibility:
The normal eligibility criteria for B.Sc. Math Honours degree course is a pass degree in 10+2 or equivalent examination in Science stream with at least 50% marks obtained by candidates in Mathematics subject or in aggregate from a recognised board or institute.
Regards.

Please specify the university name so that I may be able to populate relevant details regarding entrance exams for M.Sc Chemistry.
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There are over 20 B.Tech in Biotechnology Engineering colleges in Karnataka. Please refer to the link provided below for a detailed list of these colleges. You should be able to navigate to a college of your choice using the filters on the web page.
Regards.
https://www.shiksha.com/b-tech/colleges/b-tech-in-biotechnology-engineering-colleges-karnataka

B.Sc Mathematics is currently not in the curriculum of St. Aloysius College, Bangalore. Please refer to the link provided below for a detailed list of all courses offered by this college.
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https://www.shiksha.com/college/st-aloysius-college-magaluru-mangalore-49411

Best courses after 12th for science students with PCM, PCB and PCMB are mentioned in the link provided below.
Students must make an informed decision with good understanding of the course which one is opting for. In this article (mentioned at the link), we will take a look at some of the most popular course options one could opt for after pursuing 10+2 science.
Regards.
https://www.shiksha.com/courses-after-12th/science