Air India - Boarding not allowed

I had a flight from T* IGI to Chandigarh on **th February **** departing at *:**PM. I had already done self-check-in and got the boarding pass post-checking my baggage in. The security alone took me ** minutes and it was already *:** PM. The lady from airline called me at *:**PM to confirm if I am boarding or not and I specifically told her I'll be there in *-* minutes as I am literally running and gates are far. That was the efficiency of IGI airport security check. So the gate was **C and which was on its own changed to **A and I reached there at *:** PM and they literally ignored the fact that I was asking them about my flight. They said the gates are closed and the flight's about to take off. Boarding pass says that the gates close ** mins prior to departure. That should be *:**PM and not anytime before that. So they blantly ignored me and told me to fetch my luggage by going to gate **. I was made to wait for ** mins to literally just exit the airport as I wished to book the immediate next flight to Chandigarh as I had an important exam on **th at *:**AM and there were only two flights available till *PM. Air India took * hours to deliver me my luggage and at *:**PM , when I received my bag there weren't any flights available and I missed out on my exam which was vital for my career the next morning in Chandigarh because of AirIndia refusing to let me board when I clearly had the time and they had reckoned the fact that I was INSIDE the airport WITH a boarding pass.
This has lead to financial and emotional turmoil for me. I was in time and the fact that they took this long to retrieve my bag that I missed any other possibility of taking any other flight and missing my competitive exam is haunting.