As you have passed your schooling from Bihar board, it doesn't matter from which board you have passed your schooling. It just needs to be a recognised board. But for participating in MP-BE, you should have a domicile in Madhya Pradesh. If you can show them the domicile certificate that you have a home in Madhya Pradesh, then only you will be able to participate in MP BE, otherwise you will not be able to participate in it. 
Also you have to clear your JEE Mains for participating in it.

Regards.