If in a particular year 12th January is a Sunday, then which one of the following is correct?

The correct option is: 12th July is a Sunday if the year is a leap year.

Let us analyze the problem step-by-step to find the day of the week for July based on the given information that 12th January is a Sunday.

First, we determine the number of days remaining in January after the 12th:
Remaining days in January = 31 - 12 = 19 days.

Next, we consider the case where the year is a leap year:

- Days remaining in January = 19 days
- Days in February (leap year) = 29 days
- Days in March = 31 days
- Days in April = 30 days
- Days in May = 31 days
- Days in June = 30 days
- Days up to 12th July = 12 days

Now, let us calculate the total number of days from 12th January to 12th July in a leap year:

$\text{Total days}=19+29+31+30+31+30+12=182\text{days}$

To find the day of the week for 12th July, we divide the total number of days by 7 to determine the number of odd days (remainder):

$182÷7=26\text{weeks and}0\text{remainder}$

Since the remainder (number of odd days) is 0, the day of the week for 12th July is exactly the same as 12th January.

Therefore, 12th July is a Sunday if the year is a leap year.