At what time between 5.30 and 6 will the hands of a clock be at right angles?

Angle traced by the hour hand in 6 hours =	❨	360	x 6	❩	°	= 180°.
		12

Time from 12 p.m. on Monday to 2 p.m. on the following Monday = 7 days 2 hours = 170 hours.

∴ The watch gains	❨	2 + 4	4	❩min.	or	34	min. in 170 hrs.
			5			5

Now,	34	min. are gained in 170 hrs.
	5

∴ 2 min. are gained in	❨	170 x	5	x 2	❩hrs	= 50 hrs.
			34

∴ 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.

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock).

AM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

PM

12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

The hands overlap about every 65 minutes, not every 60 minutes.

∴ The hands coincide 22 times in a day.

Angle traced by hour hand in	17	hrs =	❨	360	x	17	❩	°	= 255°.
	2			12		2

Angle traced by min. hand in 30 min. =	❨	360	x 30	❩	°	= 180°.
		60

∴ Required angle = (255 - 180)° = 75°.

At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.

To be coincident, it must gain 15 min. spaces.

55 min. are gained in 60 min.

15 min. are gained in	❨	60	x 15	❩min	=	16	4	min.
		55					11

∴ The hands are coincident at 16	4	min. past 3.
	11

Angle traced by hour hand in	125	hrs =	❨	360	x	125	❩	°	= 312	1	°	.
	12			12		12				2

Angle traced by minute hand in 25 min =	❨	360	x 25	❩	°	= 150°.
		60

∴ Reflex angle = 360° -	❨	312	1	- 150	❩	°	= 360° - 162	1	°	= 197	1°	.
			2					2			2

Angle traced by hour hand in 12 hrs = 360°.

Angle traced by hour hand in 5 hrs 10 min. i.e.,	31	hrs =	❨	360	x	31	❩	°	= 155°.
	6			12		6

Angle traced by hour hand in 12 hrs. = 360°.

Angle traced by it in	11	hrs =	❨	360	x	11	❩	°	= 110°.
	3			12		3

Angle traced by minute hand in 60 min. = 360°.

Angle traced by it in 40 min. =	❨	360	x 40	❩	°	= 240°.
		60

∴ Required angle (240 - 110)° = 130°.

To be together between 9 and 10 o'clock, the minute hand has to gain 45 min. spaces.

55 min. spaces gained in 60 min.

45 min. spaces are gained in	❨	60	x 45	❩min or 49	1	min.
		55			11

∴ The hands are together at 49	1	min. past 9.
	11

55 min. spaces are covered in 60 min.

60 min. spaces are covered in	❨	60	x 60	❩min.	= 65	5	min.
		55				11

Loss in 64 min. =	❨	65	5	- 64	❩	=	16	min.
			11				11

Loss in 24 hrs =	❨	16	x	1	x 24 x 60	❩min.	=	32	8	min.
		11		64					11