Angle traced by hour hand in 12 hours = 360o.

Angle traced by hour hand in 5 hrs 10 min = 31/6 hrs = (360/12) x (31/6) = 155 Degree

The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o’clcok only).

So, in a day, the hands point in the opposite directions 22 times.

Angle traced by hour hand in 13/3 hrs = (360/12) x (13/3) = 130o.

Angle traced by minute in 20 minutes =(360/12) x (20) = 120o.

Required angle is = 130o – 120o = 10 Degree

Time elapsed from 2 p.m. to 8.30 a.m. = 18 hours 30 minutes = 1110 minutes.
Now, 3 minutes and 5 seconds of the given clock is 3 minutes of the correct clock.
Therefore, 1110 minutes of this clock is (1110 * 3) / (37/12) minutes of the correct clock.
= 1080 minutes = 18 hours of the correct clock
Hence, the correct time is 8 a.m

Angle traced by the hour hand in 6 hours = (360/12) x 6 = 180 Degree

The watch gains 5 seconds in 3 minutes => 1 minute in 36 minutes

From 8 AM to 5:15, the incorrect watch has moved 9 hours and 15 minutes = 555 minutes.

When the incorrect watch moves for 37 minutes, the correct watch moves for 36 minutes.

=> When the incorrect watch moves for 1 minute, correct watch moves for 36/37 minutes

=> When the incorrect watch moves for 555 minutes, correct watch moves for (36/37)*555 = 36*15 minutes = 9 hours

=> 9 hours from 8 AM is 5 PM.

=> The correct time is 5 PM.