Good Morning,

Yes, you can give CBSE class 12th exam again alongside JEE next year but when it comes to state boards improvement examinations, some provide these facilities and some not. I would suggest you to go through your state board guidelines to check if they have this rule or not. And since you didn't gave state board examination for class 12th initially (you appeared for CBSE), it means there's no scope for improvement tag here. You have to reappear for state board completely.