You are planning to go abroad and you have given you IELTS exams. If you look at any visa application or study enrollment or work application in many foreign countries, the name columns have First name,Middle name and Last name. In the Indian names, the initials we have is often the Last name or the father's name.
Please check your passport, PAN and Aadhaar card first, if they have your father's name correct. If you have to submit the 10th certificate at any place (visa office, education department or employment section) for verification of your date of birth or father's name etc , then there would be a confusion, then you need to correct it. You can correct it after making an affidavit from a Notary, publishing the correct name of your father in two papers and finally forwarding you name correction application letter via the school principal to the 10 th board that covers your school.