Yes, it is possible to get admission to NITs (National Institute of Technology) and IITs (Indian Institute of Technology ) even if you have taken an improvement exam in 3 subjects but there are conditions that such institutions have in place which one must comply with.
As simple as that!
IITs (Joint Entrance Examination – JEE Advanced):
Eligibility Criteria: It's a requirement for candidates to sit for JEE Main, pass it and sit for JEE Advanced. If you take further exams, the last results after improvements, in this case, are used to count upon your eligibility exploitation. Of course, you must also fulfil all other requirements that are in place – for example, there is a threshold aggregate score that must be achieved in standard 12 in most instances, if not always, above 75% for those belonging to the general category.
Important Rule: Improvement exam marks are counted only when they override the original scores. If the new marks after improvement are not an improvement on the performance, the previous scores can be used for JEE Advanced eligibility (the Entrance Examination - JEE Main)*: Eligibility Criteria: For NITs, the eligibility rules are almost the same as that of the IITs regarding the improvement exams taken before the main examination. The final 12th-grade marks after improvement will be considered. If the improvement exams result in a higher aggregate, these new grades can be substituted for admissions. As a condition, however, every applicant must satisfy the minimum cut-off required for JEE Main Examination.
Cut-off: Most NITs and other government-aided technical institutions require an aggregate of at least 75% in 12th Standard (or be among the top 20 percentile in their respective boards) to be eligible for admission through JEE Main.
Important ions:
Date of Improvement: Similarly Ensure that the improvement examination should be sat on the stated period by your concerned board and results should be out before you sit for JEE Main or Advanced.
Percentage of candidates in the top 20 percentile: Some NITs and IITs also allow candidates who rank within the first 20 percentile of their respective boards and improvement exams enhance your rank then you may still be considered in case of improvement in most exams.
Also, keep in mind respective institutions (IITs and NITs) to know any changes to the eligibility criteria for the candidates who intend to take improvement examinations as it may change every academic year.