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Many people say that NCERT is enough to crack JEE mains but I'll say it is partially true only for chemistry you can fully rely on NCERT but for physics and maths choose reference books like HC Verma, Rd Sharma etc. Also I advise you to fully solve and read your coaching materials and notes then only NCERT will help as it has written in a much compact way.
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NDA syllabus is same as your 11th and 12th. You will not have to do anything extra just focus on your maths very much. You can study from Rd Sharma for physics HC Verma will be more than sufficient and for chemistry NCERT. Also you should solve previous year NDA question bank it will help you a lot.
For gate either you should be passed or in final year of graduation in engineering/technology. Gate is an exam you can take any number of times. This program was made for students who wanted to study further but now PSUs are recruiting through this exam and so it has become so popular.
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