Was his action correct after he himself completed the admission examinations, having previously sent someone to act on his behalf, without informing the college of that?

The questioner has done well in seeking truth and honesty, and in re-examining what is required, fleeing from deception and falsehood. He has also done well in concealing his own shortcomings, which is a noble and honorable religious requirement, especially since the faculty will not benefit from disclosure nor be harmed by the incident.