It should not matter much whichever name you use in the application. But just to be precise, use the name as in the 12th marksheet. And if the "last name" column in the application is mandatory to fill, then fill the name as per 10th certificate. If last name is optional, it's your choice to fill which of the two names.

Generally, the university does not ask for your mother's aadhar card.

Hope it helps.