Hi Aman!

10th and 12th class marksheets are valid everywhere. But as you mentioned that you will be registering for 12th class board exams and this process is generally done by the school. And in most of the cases they follow the name of 10th class, but still to avoit any problems in future you must update your name on the Adhar card by visiting an enrollment centre/update centre and fix the errors using 10th marksheet and birth certificate or any other valid id proof.

Thankyou!