In India there are some tutorials which offer coaching to students for competitive examinations. Students initially pay them and procure study material (correspondence tutorials). After results, if student secures 1st, 2nd, 3rd or rank up to certain limit, then the tutorials offers them money for example for 1st rank holder - Rs. 2 lakh, for 2nd rank holder- Rs. 1 lakh, for 3rd rank holder - Rs. 50,000/- like that. If the student has not secured any rank he doesn't get anything. I think this type of monetary gift is for motivation purpose. My question is, can we accept this money?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

If the situation is as you described in your question, then this is a prize and it is lawful as long as it is for encouragement and provided those who participated were students whose primary aim was to study. Those who paid such fees to these centers only for the aim of getting the prize, then what they did was unlawful as this is like gambling.
So, the Shari'a rule depends on the students' intention and objective. One indication that what is done is gambling (not prizes) is the high fees in these centers compared to other centers of similar quality and similar capabilities.
Allah knows best.