Hi Avinash,
Yes, you would face rejection from the passport office at your appointment because the two documents that are necessarily required while applying for a passport are your 10th standard passing certificate and your Adhaar Card and it is important to note that the name of the candidate applying for passport should be exactly same in both of those documents. If the names in both of those documents are different, like in your case, you should chose the name that is in your legal Adhaar Card and get it published in a newspaper so that you can further use that name everywhere else legally and in the passport office as well. After you have gotten the name published in the newspaper, you will have to take that newspaper, with the declaration of your full name, with you on the date of your appointment at the passport office. Once you show them the newspape with your 10th standard certificate and Adhaar Card, they will accept your published name in your passport and you will not face any further troubles.