Correct Option (3)

To determine the day of the week for 12 July, given that 12 January is a Sunday in a leap year, we calculate the total number of days between these two dates.

Remaining days in January: 31 − 12 = 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 in July (up to 12th): 12 days

Total number of days = 19 + 29 + 31 + 30 + 31 + 30 + 12 = 182 days.

To find the number of odd days, we divide the total days by 7:

Number of odd days = $\frac{182}{7}$ = 26 weeks and 0 odd days.

Since there are 0 odd days, 12 July falls on the same day of the week as 12 January. Therefore, if 12 January is a Sunday, 12 July is also a Sunday in a leap year.

Incorrect Options:

Option 1: 15 July is a Sunday if the year is a leap year.

As established, 12 July is a Sunday in a leap year. Therefore, 15 July would be 3 days after Sunday, which is a Wednesday. Hence, this option is incorrect.

Option 2: 15 July is a Sunday if the year is not a leap year.

For a non-leap year, the number of days from 12 January to 15 July is calculated as follows:

Remaining days in January: 19 days
Days in February (non-leap year): 28 days
Days in March: 31 days
Days in April: 30 days
Days in May: 31 days
Days in June: 30 days
Days in July (up to 15th): 15 days

Total number of days = 19 + 28 + 31 + 30 + 31 + 30 + 15 = 184 days.

Number of odd days = $\frac{184}{7}$ = 26 weeks and 2 odd days.

If 12 January is a Sunday, then 15 July would be Sunday + 2 days = Tuesday. Hence, this option is incorrect.

Option 4: 12 July is not a Sunday if the year is a leap year.

As demonstrated in the explanation for the correct option, 12 July is a Sunday if the year is a leap year. Therefore, this statement is incorrect.