What is the percentile that will ensure a call from IIM Calcutta, Lucknow, Bangalore, and Ahmadabad having 83% in 10th, 83% in 12th and a 6.9 CPGA in graduation, till now having no work experience?

First of all, bear in mind that there is never a 100% certainty of getting a call from a specific interest as there are many factors that determine it. 

Considering your profile that lacks work experience and has witnessed a decline in the graduation marks, I don’t consider anything below 99.9 to fetch you a call from institutes like IIM Calcutta, Lucknow, Bangalore, and Ahmedabad. All these institutes only accept the best profiles and high scores. So unless you score a perfect 100, it will be difficult to get admission here.

A percentile of 99.5% in your CAT exams will give IIM Calcutta, Lucknow, Bangalore, and Ahmedabad significant interest in giving you a chance and calling you up for interviews. If your scores are in Verbal or Quant C then the institute may find you an interesting candidate even if it does not take your percentile in CAT into account.

In all seriousness, however, the applicants will most likely not be called up for interviews by IIM Bangalore if they fail to obtain the given percentile. A decrease in the percentile will significantly hamper your chances of getting called up. The institute also lays stress on your performance in your secondary and higher secondary exams so keeping that in mind is as essential as your percentile in CAT.