Correct Option (2)

The price of petrol on the $m^{t h}$ day of the year is given by $80 + 0.1 m$ Rupees per litre for $m = 1, 2, . . ., 100$ . After the $100^{t h}$ day, the price remains constant.

On the $100^{t h}$ day, the petrol price is calculated as $80 + 0.1 \times 100 = 80 + 10 = 90$ Rupees per litre.
Therefore, for any day $n > 100$ , the petrol price is 90 Rupees per litre.

The price of diesel on the $n^{t h}$ day of the year is given by $69 + 0.15 n$ Rupees per litre for any $n$ .

To determine the day when the prices of these two fuels are equal, we consider the period after the $100^{t h}$ day, as the petrol price stabilizes then. If we were to equate the initial formulas $80 + 0.1 n = 69 + 0.15 n$ , it would yield $11 = 0.05 n$ , resulting in $n = 220$ . Since $220 > 100$ , this indicates that the equality does not occur during the period when the petrol price is still increasing according to its initial formula. Thus, the equality must occur when the petrol price is constant at 90 Rupees per litre.

Setting the constant petrol price equal to the diesel price formula:

$90 = 69 + 0.15 n$

$21 = 0.15 n$

$n = \frac{21}{0.15} = \frac{2100}{15} = 140$

The prices of the two fuels are equal on the $140^{t h}$ day of the year 2021.

To convert the $140^{t h}$ day into a specific date in 2021 (which is not a leap year):

January: 31 days
February: 28 days
March: 31 days
April: 30 days
Total days up to April 30th = $31 + 28 + 31 + 30 = 120$ days.
The remaining days to reach the $140^{t h}$ day are $140 - 120 = 20$ days.

Therefore, the $140^{t h}$ day of 2021 is May 20th.

Incorrect Options:

The prices of the two fuels are equal specifically on the $140^{t h}$ day of the year, which corresponds to May 20th. Other dates would result in different diesel prices compared to the constant petrol price of 90 Rupees per litre.

For 18th May (the $138^{t h}$ day): Diesel price = $69 + 0.15 \times 138 = 69 + 20.7 = 89.7$ Rupees.
For 19th May (the $139^{t h}$ day): Diesel price = $69 + 0.15 \times 139 = 69 + 20.85 = 89.85$ Rupees.
For 21st May (the $141^{s t}$ day): Diesel price = $69 + 0.15 \times 141 = 69 + 21.15 = 90.15$ Rupees.

None of these dates result in the prices of both fuels being equal to 90 Rupees per litre.