Yes, Its really a matter of concern. I will suggest you to apply for the correction of the name in your all mark sheets of 10th and intermediate
You have to applied for the correction of names in the mark sheets and certificates. For the correction you have to submit a form online /offline with certain fees and some documents like photocopy of scholar book of school and photo copy of mark sheet along with original mark sheet. After 10-15 days your board will issue you a new certificate and mark sheet.
In most of the boards, you can submit online application. If you apply within three months of declaration of result you will get new mark sheet free of charge. If you are applying after time periods, they will charge some nominal charges.
You can apply online yourselves of through any Kiosk Center.
So, go for the correction and it very easy process. I will suggest you to apply as soon as possible.
Honesty is the best policy.