Test: Clocks- 1 - UPSC MCQ

In 12 hours, the hands coincide or are in opposite direction 22 times.
∴ In 24 hours, the hands coincide or are in opposite direction 44 times a day.

Angle traced by hour hand in 125/12 hrs = 
Angle traced by minute hand in 25 min = 
∴ Reflex angle = 360⁰ - 

Angle traced by hour hand in 12 hrs = 360º.
Angle traced by hour hand in 5 hrs 10 min. i.e., 31/6 hrs 
= (360/12)*(31/6)°
= 155º

Time from 7 a.m. to 4.15 p.m. = 9 hours 15 min = 37/4 hours
3 min 5 sec of this clock = 3 min. of the correct clock.
⇒ 37/720 hours of this clock = 1/20 hours of the correct clock
⇒ 37 hours of this clock = (1/20 x 720/37 x 37/4) hours of the correct clock.
= 9 hours of the correct clock.
∴ The correct time is 9 hours after 7 a.m. i.e., 4 p.m

When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.
At 7 o'clock, they are 25 min. spaces apart.
 Minute hand will have to gain only 5 min. spaces.
55 min. spaces are gained in 60 min.
5 min. spaces are gained in 
∴ Required time =  min. past 7.

At 5 o'clock, the hands are 25 min. spaces apart.
To be at right angles and that too between 5.30 and 6, the minute hand has to gain (25 + 15) = 40 min. spaces.
55 min. spaces are gained in 60 min.
40 min. spaces are gained in 
∴ Required time =  min. past 5.

The angle between hour and minute hand in 4:20 is 10 degrees.
For a minute, the hour hand rotates by 30/60 = 1/2 degrees.
hence, for 20 minutes it rotates by an angle of 20*1/2 = 10 degrees.

15 past 5 means quarter past 5 = (5:15)
⇒ (11*15 - 60*5)/2 = 67.5

Time from 12 p.m. on Monday to 2 p.m. on the following Monday = 7 days 2 hours = 170 hours.
∴ The watch gains  in 170 hrs.
Now, 34/5 min. are gained in 170 hrs.
∴ 2 min. are gained in 
 Watch is correct 2 days 2 hrs. after 12 p.m. on Monday i.e., it will be correct at 2 p.m. on Wednesday.

At 11 O'clock, the hour hand is 5 spaces apart from the minute hand.

During the next 60 minutes, i.e. between 11' O clock and 12' O clock the hour hand will move five spaces [integral values as denoted by the 56 minute, 57 minute, 58 minute, 59 minute and 60 minute positions].

For each of these 5 positions, the minute hand will be at the 12th minute, 24th minute, 36th minute, 48th minute and 60th minute positions.

Hence the difference between the positions of the hour hand and the minute hand will have an integral number of minutes between them, i.e., 5 positions.