What was the day of the week on 1st January 2005?

The correct answer is Option (3) → Saturday

Count the odd days up to 31 December 2004. The first 2000 years give 0 odd days, so 31 December 2000 was a Sunday. The years 2001, 2002 and 2003 are ordinary and add 1 odd day each, and 2004 is a leap year and adds 2.

That is 5 odd days after Sunday, bringing 31 December 2004 to a Friday. The next day, 1 January 2005, was therefore a Saturday.