Hii Hitesh
Yes,the private candidates are also eligible for appearing in NEET and AIIMS exam but you need to full the eligibility criteria for both exams.For NEET exam,if you belong to general category then you should have 50% aggregate marks in PCB and if you belong to obc/sc/St candidate then you should have 40% aggregate marks in PCB while for AIIMS it is 60% in PCBE for general/obc candidate and 50% for sc/St.i hope this information helps you.
Thank you

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