For any Government or PSU jobs, nowadays the applications are mostly online; the name of the candidate, date of birth and vital details are very important. One has to enter it correctly and have the correct supporting documents.
So, in your case, it could be an issue during document verification, if you are going to produce the 10th marks sheet.
You can get this rectified.
For State Board.
You need to have copies of the present marks sheet, ID proof documents and then draft a letter to the respective state board mentioning that you need to correct the name spelling mistake.
Forward this letter via the principal or the head of the institution (from where you completed your 10th) to the state board authorities and pay the necessary fees as per the board rules.
You should get the corrected certificate in 4 weeks time despatched to your school.
Here apart from the other documents, you need to have an affidavit that states your correct name.
Please get this done via the Notary in your locality or near the civil courts.
Once this is done, then publish the name change in two local papers (one English, one vernacular). Please approach the newspaper offices, they will do this, it's a simple routine service. Keep the original cutouts of the newspaper.
Then fill in the CBSE form for name correction.
Please find the forms in this link (http://cbse.nic.in/faq/MM%202016%20Duplicate%20Form.pdf).
CBSE issues regulations for name correction based on the time elapsed. Please read this document(http://cbse.nic.in/faq/revised_dob_name_corr_rules_2015.pdf).
You can find the charges for the services(http://cbse.nic.in/faq/notice%20of%20fee%20revision_2016.pdf)
Once you have the documents ready, please submit them by speed post or in person based on the office under whose jurisdiction, your school comes under.
Please keep all the correspondence and reference numbers handy so that you can follow-up if you do not get the certificates in time.