Surname change in India for women from father's name to that of the husband is well accepted and would not cause an issue as long as your marriage is legal and you have evidence for it.
As far as the SSC exams are concerned, these are in various stages. In these exams, it is understood that the admit care issue, written exam, the first step has to passed and then it would go on to the further stages of personal interview and document verification.
When you reach this stage, you are expected to have proper evidence of what you have mentioned in the application form. At this point the authorities would ask about the different surnames. Here you need to provide evidence that you are married. This would be in the form of a valid marriage certificate or an affidavit.
I would suggest, please meet a Notary who would help you to get a legal affidavit stating that you given maiden name is so and so, now after marriage your name is so and so.
In case, the authorities question you while you go for the written exams, then you can show them the marriage registration certificate and the affidavit.
Whether you change all the documents to your new name depends on you. If you get selected and appointed in the Government job wherein you have give you new name, then your signature and new name is already entered. Here you would need to change your identity documents.
In case you are unsuccessful and keep applying for new jobs, you can keep your give name as it is, many married women do that.
Please decide on this before applying for a passport/ if you have options of a job overseas because, here international authorities and visa offices have to have a clear first name and last/surname. You cannot have identity documents or bank accounts with two different surnames submitted for overseas jobs or visa offices.