Will this cause any problem in any government job document verification process?
Yes, this will cause unneccessary hassles during document verification for Government jobs as most of these exams will have online form submisions and data entry.
Please take this oppourtinity to check you other ID documents like PAn,Aadhaar,Caste certificate if applicable, so that you can get all rectified now itself when you have time.
First, please check your father's actual ID proofs ( PAN, Aadhaar etc) that has his name spelt correctly. In some centers, this would be enough to get the mistake rectified. But in some places you will need an affidavit to authenticate the correct name of father.
You need to approach a Notary in your town or city, thye would guide you to get an affidavit done stating your father's proper name.
Then please approach two newspaper offices ( one-English, one-vernacular) and publish the name change. Please keep both original copies.
for the 10th and 12th markssheet, please draft a request letter mentioning the service requested and forward it through the school and college principal to the appropirate state(10) and college(12) boards with the supporting doucments. There would be a small fee payable for the service.
For the graduation certificate and markcard, please check the university website under whose auspices your college comes under. Some universities offer this service online/by post, some in person. Once you meet the adminstrative staff of the unversity with the prescibed form and fees, this would be done.
Any documents you are sending by post, please keep an acknowledgement/ registered post slip so that you can follow it up later on if there is a delay.
If you have no time to do this before applying of the UPSC Indian Statistical Service Exam, please do not waste the chance, apply, prepare and simultaneously get the name change done, you can explain this when the document verification date comes up.