It is not Mandatory to complete 10th & 12th in same state.

However at the time of admission through 85% state quota seats, every state has their own set of rules.

For ex. In Karnataka it is compulsory for the student to complete minimum 7 years of education in karnataka Irrespective of whether he has given 10th or 12th Boards Exam in the same state. If not there are other clauses of eligibility.

Please check the respective state Examination Authority website for your eligibility