A. Your father's name does not match in your 10th,12th/PUC and degree marks card (it is not clear which one has the wrong name). Yes, it would be a problem during document verification but can be rectified.
B.The title says that you are going to take up government exams. I suggest apply and take with exams with what you have because the Government job selection will have many stages and document verification is after the written qualifying rounds and by that time you can get your documents in order.
C.Please check you father's correct name and other details regarding your date of birth, your name etc. Also check your other ID documents (PAN,Aadhaar,DL) and caste certificate if applicable.
D.If only your PU marks card and certificate has the father's name mistach then
1.Please have two ID documents of father that has his name clearly mentioned (PAN, Aadhaar).
2.If you do not have this for father or you have your 12th from CBSE, then you need a correct name affidavit that will be issued by a Notary in the local region.
3.With this, then draft a letter regarding the service you need which would be a name correction service, enquire about the necessary fees and then forward it to the local PU board via the proper channel which would be the principal of the PU college you studied from. The new certificate will be sent to you 12th college in a month's time.
4.For CBSE (http://cbse.nic.in/faq/MM%202016%20Duplicate%20Form.pdf), this is the form and the filling guidelines (http://cbse.nic.in/newsite/dup.html#guidelines)
E. If your 10th and degree certificate has the incorrect name of father then
1. for 10th it would be the same as above in D
2. For the degree, you need to check the university under whose jurisdiction your college falls under. Then check the university website for name correction service or approach the university office/registrar's office, the staff will guide you. It would involve a form, attested copies of your father's documents with the correct name, a prescribed fee, usually in 4-6 weeks time the new certificate would be issued
Please keep a record of the letters and the acknowledgment slips as it would be useful in case there is a delay or if you have questions at the time of document verification.