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

The question asks us to determine the correct statement given that 12th January is a Sunday in a particular year. We need to calculate the number of odd days between 12th January and the target dates in July, considering whether the year is a leap year or not.

Let's calculate the number of days from 12th January to 12th July: Remaining days in January = 31 - 12 = 19 days Days in February = X (29 for a leap year, 28 for a non-leap year) Days in March = 31 Days in April = 30 Days in May = 31 Days in June = 30 Days in July (up to 12th) = 12 days

Total days = 19 + X + 31 + 30 + 31 + 30 + 12

Case 1: The year is a leap year (February = 29 days) Total days = 19 + 29 + 31 + 30 + 31 + 30 + 12 = 182 days To find the day of the week, we divide the total days by 7 and look at the remainder (odd days). 182 / 7 = 26 weeks and 0 odd days. Since there are 0 odd days, if 12th January is a Sunday, then 12th July will also be a Sunday.

Now let's evaluate the options:

A) 15th July is a Sunday if the year is a leap year If 12th July is a Sunday (as calculated above for a leap year), then: 13th July is Monday 14th July is Tuesday 15th July is Wednesday. So, option A is incorrect.

B) 15th July is a Sunday if the year is not a leap year If the year is not a leap year, February has 28 days. Total days = 19 + 28 + 31 + 30 + 31 + 30 + 12 = 181 days Number of odd days = 181 / 7 = 25 weeks and 6 odd days. If 12th January is a Sunday, then 12th July will be Sunday + 6 days = Saturday. If 12th July is a Saturday, then: 13th July is Sunday 14th July is Monday 15th July is Tuesday. So, option B is incorrect.

C) 12th July is a Sunday if the year is a leap year As calculated in Case 1, if the year is a leap year, there are 0 odd days between 12th January and 12th July. Therefore, if 12th January is a Sunday, 12th July will also be a Sunday. So, option C is correct.

D) 12th July is not a Sunday if the year is a leap year. This statement contradicts our finding for Case 1 and option C. Since C is correct, D must be incorrect.

The final answer is C.