How many times will minute hand and hour handa) coincide in one dayb) perpendicular in one dayc) opposite in one day
The hands will be opposite to each other 22 times in 24 hours actually (kindly ignore the typing mistake in the last line of my earlier answer).
Here is another method to solve the problem:
The minute hand travels an angular distance of 360 degrees in 60 minutes. So, it covers 6 degrees in one minute.
The hour hand covers 2*360 degrees in 24 hours, i.e 1440 minutes. So, in one minute, it covers 1440/720 = 1/2 degree. Difference in angular distance travelled by the minute hand and the hour hand in one minute is thus 6-1/2 = 11/2 degrees. So, on a full rotation (360ᵒ), any similar event between them will be repeated every `360/(11/2)` = `65 5/11` minutes.
In twenty four hours, i.e. 24*60 = 1440 minutes, the number of times angle between the minute hand and hour hand will be same, can be calculated by dividing 1440 minutes with this value, thus n = `1440/(720/11)` = 22.
So, a) 0ᵒ angle (event of coincidence) will be obtained = 22 times.
and similarly, c) 180ᵒ angle (event of opposition) will be obtained = 22 times.
b) 90ᵒ angle (event of perpendicular disposition) will be obtained = 22*2 = 44 times (as this situation arises twice on a complete rotation).
a) 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).
The hands overlap about every 65 minutes, not every 60 minutes.
Thus the minute hand and the hour hand coincide 22 times in a day.
(b) In 12 hrs. the hands of the clock will be perpendicular to each other for 22 times as in 1 hr. time the hands of the clock wil be at rt. angles for 2 times but between 1` `1'o clock and 1'o clock they will be at rt. angles only twice.
Therefore, in 24 hrs. they will be at rt. angles 44 times.
(c) Eleven times in 12 hrs.. The hands will be directly opposite each other at the following times:
12:32 and 8/11 of a minute
1:38 and 2/11 of a minute
2:43 and 7/11 of a minute
3:49 and 1/11 of a minute
4:54 and 6/11 of a minute
7:05 and 5/11 of a minute
8:10 and 10/11 of a minute
9:16 anfd 4/11 of a minute
10:21 and 9/11 of a minute
11:27 and 3/11 of a minute.
Therefore, the hands of the clock will be opposite to each other 44 times.