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.

Time from Monday noon (12 o' clock) to following Monday 2 pm = 7 day 2 hours
= 7×24+2
= 168+2
= 170 hours 
Total time gained Monday noon (12 o' clock) to following Monday 2 pm = 2 min + 4 min 48 sec

The given watch gains 34/5 mins in 170 hours
Given that the said watch was 2 minutes low at Monday noon. Hence when it gained 2 minutes, the time was correct.
The given watch gains 34/5 mins in 170 hours
⇒ The given watch gains 2 mins in  
= 5 * 5 * 2 hours = 50 hours
Hence the time was correct after 50 hours from Monday noon = after 2 days 2 hours from Monday noon = 2 pm on Wednesday

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.