Given: Original average monthly income of the family = Rs. $10,000$ Number of persons in the family = $5$ Income increase of one person = Rs. $1,20,000$ per year

Step-by-Step Calculation: Original total monthly income of the family: Total income = $10,000 * 5$ = Rs. $50,000$

Annual income increase for one person: Increase per month = Rs. $1,20,000 / 12$ = Rs. $10,000$

New total monthly income of the family: New total income = $50,000 + 10,000$ = Rs. $60,000$

New average monthly income: New average income = $60,000 / 5$ = Rs. $12,000$

Answer: A (Rs. $12,000$)