It would be understood, but I don't think it is very natural. Normally, in British-style English, we would say 'I took the examination' or 'I sat the examination'. It is rare to use present tense in that situation, because if you are doing it, you are not talking about it!

yes. the only thing it is technically missing is a subject, which could be implied. to fix this you could add "I" as in "I will take part in the exam." or "you" as in "you will take part in the exam."