Hello Rajesh,

If your name is Rajesh Garg in JEE admit and same in your other Government issued identity card like Aadhaar, PAN, Driving Licence, etc.

You can appear for JEE Advance exam as far your name were same as your identity cards.

As on the examination centre they only check for identity card and Hall ticket.

As far as your mistake of name in your 12th marksheet you can change it by an application to your 12th board with some of required documents that they will ask for and you must possess a affidavit from the court of the mistake of your name.

Hope the information helps you..

Best of luck..