At what time between 2 o'clock and 3 o'clock, will the hands of a clock point in opposite directions?

At 9 o'clock, the hour hand is at 9 and the minute hand is at 12. it means that the two hands are 15 min spaces apart .To be in the same straight line (but not together ), they will be 30 min space apart .
? The minute hand will have to gain (30 - 15) = 15 min space over the hour hand .

As we Know, 55 min spaces are gained in 60 min.
? 15 min will be gained in (60/55 x 15) min = 180/11 = 16⁴ /₁₁ min

Hence, the hands will be in the same straight line but not together at 16⁴/ ₁₁ min past 9.

Time from 7 am on a day to 6 am after 3 days is 95 h.
Now , 23 h 45 min of the clock in question = 24 h of correct clock
i..e., 95/4 h of the clock in question = 24 h of correct clock
? 95 h of the clock in question = (24 x 4/95 x 95) h of correct clock = 96 h
So, the correct time is 96 h from 7 am, Monday, ie., 7 am, Friday.

Since, angle between two number is 360°/12 = 30°
When the time is 3:25 pm, the minute hand will be at 5 and will have moved 60° from 3 and hour hand would be between 3 and 4 it will move 30° in 60 min and so in 25 min it would have moved 30 x 25/60 = 12.5°
So, angle between two hand will be 60 - 12.5 = 47.5°

Director came at 12 : 15.
Steno came at 12 : 40.
Since, steno was late by 30 min to the meeting.
? Meeting time 12 : 10

The angle between hour hand and minute hand at 5 O'clock = 5 x 30° = 150°
Minute hand move in 10 min = 2 x 30° = 60°
Hour hand move in 10 min = 30°/6 = 5°
? Angle between hour hand and minute hand at 5 h 10 min = 150° - 60° + 5deg; = 95°

Time from 6 am on a day to 11 pm on 4th day = 89 h
Now, 23 h 44 min of this clock = 24 h of correct clock
89 h of this clock = 24 x (15/356) x 89 h of correct clock = 90 h of correct clock
? Correct time will be 12 o' clock (mid-night).

Time from 9 pm in a day to 2 pm on the following day = 29 h.
24 h 10 min of this clock = 24 h of the correct clock
or 145/6 h of this clock = 24 h of the correct clock
29 h of this clock = 24 x (6/145) x 29 h of the correct clock
= 28 h 48 min of correct clock
? The correct time is 28 h 48 min after 9 am. This will be 48 min past 1.

Given that, time to overtake N = 70 min

The required result = [(720/11) - N) ] x [(60 x 24)/N) min
= [(720/11) - 70) ] x [(60 x 24)/70) min
= [(720 - 770) / 11] x [(60 x 24) / 70] min
= - (50/11) x [(6 x 24)] / 7 min = -7200/77 min
-ve sign indicates that there is a loss.

If the minute hand of a clock overtakes the hour hand at interval N min of the correct time, then clock losses or gains by
The required result = [(720/11) - N) ] x [(60 x 24)/N) min
Positive for gain and negative for loss Here, N = 64

required time = [(720/11) - 64) ] x [(60 x 24)/64) min
= 32⁸/₁₁ min

The year 2007 is an ordinary year, so it has 1 odd day.
3rd day of the year 2007 was Wednesday.
? 3rd day of the year 2008 will be one day beyond the Wednesday. Hence, it will be Thursday.