I am sorry to say but the query mentioned by you give rise to 2 questions:
i) How did they accept your application form?
ii) How were you able to give the BSNL JAO examination?
The question rising is to know that during acceptance of the form they might have checked the attached documents and a small correction may be overlooked but your name was wrong. When the application form is put for verification, they would have found the wrong details and there itself, your application form would have been rejected.
As the mistake committed cannot be ignored, you can check all your valid documents like Passport, Pan Card, Driving License, Ration card, Mark sheet of school and colleges, Aadhar card and check for correction in all the documents. If you find any mismatch in these documents, kindly do the correction making an affidavit and if required, through a known lawyer make a gazette publication in one local newspaper and the other one in National English newspaper. Keep the paper cutting or the paper for further reference. This will help you to be on a safer side for any further process.
In the meanwhile, you can approach the concerned authority by writing or self-presentation and make them understand the mistake and also provide them the supporting documents and also provide them some reason for such mistake as it shows negligence.
i) Always fill the application form with other documents in front so as to keep it neat and correct.
ii) Always have a double-check of the filled application for correction if any before submission.
iii) Read all the instructions before filling the form as it will help to fill the form correctly without mistakes.
“The most important thing in life is to learn how to give out love, and to let it come in." — Morrie Schwartz