Correct Option (A)

The initial average monthly income for a family of 5 members is ₹ 10,000. Therefore, the total initial monthly income of the family is calculated as:

Total initial monthly income = 5 × ₹ 10,000 = ₹ 50,000.

The income of one person increases by ₹ 1,20,000 per year. To determine the monthly increase for this person, the annual increase is divided by 12:

Monthly increase = $\frac{1, 20, 000}{12} = ₹ 10,000$

The new total monthly income of the family is the sum of the initial total income and the monthly increase:

New total monthly income = ₹ 50,000 + ₹ 10,000 = ₹ 60,000.

The new average monthly income for the family of 5 members is then calculated by dividing the new total monthly income by the number of family members:

New average monthly income = $\frac{60, 000}{5} = ₹ 12,000.$

Incorrect Options:

Option B (₹16,000): This value would result from an incorrect calculation of the monthly increase or its application. For example, if the annual increase was erroneously divided by 4 instead of 12 (₹1,20,000 / 4 = ₹30,000), the new total income would be ₹50,000 + ₹30,000 = ₹80,000, leading to an average of ₹80,000 / 5 = ₹16,000.

Option C (₹20,000): This outcome would occur if the calculated monthly increase for one person (₹10,000) was directly added to the initial average monthly income (₹10,000), without correctly recalculating the average based on the total income and the number of family members.

Option D (₹34,000): This option reflects a significant misinterpretation where the annual income increase (₹1,20,000) is incorrectly treated as a monthly increase for the entire family's total income, leading to an inflated average calculation.