Hello Aspirant,

Yes, definitely. You can give 12th public exam whenever you want. Failing in an exam does not mean that you cannot give it again. You can prepare for the exams with dedication again and can pass them with flying colors. Some private institutions have their own age regulations  which allows only students below 17 year old to give exam but this is not a case in every institution. Also, there is an option of open boards to give board exams where there is no age limits.

All the best!