Hello Vishal,

When you apply for JEE, they ask you for both your board exam as well as improvement exam roll number and while you go to give JEE, you need to carry board the admit cards.

Other than JEE, no other engineering exam asks for the same (as far as I know).

Yes, you will be eligible for admission into IITs if you crack JEE advanced with improvement exam(75%+ is the minimum requirement this year).

If you are aiming for NITs or other government institutions, then better apply for improvement exam and try to score at least 90%+ to fetch you a decent rank

For BITS, you need to get at least 75% in PCM to remain eligible.

Good luck!