Hello, For your 12th exam details you have to to give in your details form your 12th marksheet. So in your case you will have to specify that you are a private student. This will not affect in any manner in your selection process further in your government exams as long as qualify the written and interview rounds of the Government examination. There are lots of inspirational stories of candidates who struggled in their school exams but later came out as toppers in the government exams so be positive and prepare well.

All the Best.