The reflex angle between the hands of a clock at 10.25 is:

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	❨	60	x 40	❩min	=	43	7	min.
		55					11

∴ Required time = 43	7	min. past 5.
	11

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°.

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

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.