Consider exactly at 9 : 00, minute hand is at position '12' and hour hand is at '9' hence angle between these two hands at 9 : 00 is 270°
Since relative speed between them is 11/2° per minute or every minute hand will travel 11/2° more than the hour hand.
Then will be perpendicular to each other when the minute hand will travel 270° more than the hour hand. And time taken for this is (270)/(11/2) = 540/11 or 49 1/11 minute
Hence they will be perpendicular to each other at 9 : 49 minute : 0.09 second

Since in 1 year we have one odd day while in one leap year we have two odd days.
Number of odd days are 1 (for 2006)
1 days more than Friday means Saturday.

Since a day of the week repeat after every 7 days, divide 59 by 7, remainder is 3 so number of odd days is 3, and 3 days after Monday is Thursday.

Since in 1 year we have one odd day while in one leap year we have two odd days .
Number of odd days are: 1(for 2006) + 1(for 2007) + 2 (For leap year 2008) + 1 (for year 2009) so total number of odd days are 1 + 1 + 2 + 1 = 5 days. 5 days more than Friday means Wednesday.

Number of days in June is 30 -2 = 28, and number of days in July is 7, total number of days is 28 + 7 = 35, when we divide 35 by 7 remainder is 0, or number of odd days is 0 hence 7th July must be the same day as that of 2nd June i.e. Saturday in this case.

Number of years between 2001 to 2013 is 12 years (2013 - 2001 = 12 years)
Number of leap years between 2001 to 2013 is 3, since a year has 1 odd days and a leap year has 2 odd days , so total number of odd days is 12 + 3 = 15 odd days.
When we divide 15 by 7 remainder is 1 or in total we have 1 odd day so 1st jan 2013 is Tuesday.

The century divisible by 400 is a leap year. 2100 is not divisible by 400 hence is not a leap year.

The century divisible by 400 is a leap year. 2100 is not divisible by 400 hence is not a leap year so 2100 is not a leap year, out of remaining 1128 is divisible by 4, so 1128 is a leap year.

On 31^st December, 2005 it was Saturday.
Number of odd days from the year 2006 to the year 2009 = (1 + 1 +2 + 1) = 5 days.
? On 31^st December, 2009, it was Thursday.
Thus, on 1^st Jan, 2010 it is Friday.

When 69 is divided by 7 we will get remainder 6 so the day will be 6 days more than Sunday i.e Saturday.