Whether or not you need to write a provincial exam depends on the graduation plan that you are working on. There are basically three graduation plans that a BC student will be working toward.
If there is ANY question as to which graduation plan you are on, talk to a counsellor.
Once you're clear on which plan you're on, you can determine your requirements regarding the provincial exams, as below:
Provincial exams are REQUIRED for provincially examinable courses IF you want them to appear on your "official transcript of marks." The key here is whether you would like your mark to appear on your official transcript of marks. OC and UBC-O will accept a Letter of Course Completion from Central School (Signed, Sealed, and Dated, etc) in lieu of an official mark on their transcript.
Students write five Graduation Program exams: Language Arts 10, Science 10, Mathematics 10, a Social Studies 11/12, and a Language Arts 12. Other Grade 12 exams are OPTIONAL - that is, these additional courses will end up on your transcript whether you've written the provincial exam or not. Please talk with your school counsellor as to whether writing an "optional" provincial exam might be beneficial to you.
Provincial exams are optional for students on the Adult Graduation Program. It should be noted that some post-secondary institutions may not accept examinable courses for admission purposes unless the Grade 12 provincial exam has been written.