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At what time given below do the minute and hour hands of a dock make a right angle?
- a)12 O'clock
- b)Quarter to 9
- c)Half past 9
- d)3 O'clock
Correct answer is option 'D'. Can you explain this answer?
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Solution:
To find the time when the minute and hour hands of a clock make a right angle, we need to use the following formula:
Angle between the hour hand and the minute hand = | 30H - 11/2 M |
where H is the hour and M is the minute.
We know that when the minute and hour hands of a clock make a right angle, the angle between them is 90 degrees. Let's substitute this value in the formula and solve for H and M.
| 30H - 11/2 M | = 90
There are two possible solutions for this equation, one where the hour hand is ahead of the minute hand and one where the minute hand is ahead of the hour hand. We will consider both cases.
Case 1: Hour hand ahead of minute hand
Let's assume that the hour hand is ahead of the minute hand. In this case, the minute hand would be somewhere between 0 and 30 minutes.
| 30H - 11/2 M | = 90
30H - 11/2 M = 90
60H - 11M = 180
We also know that the minute hand moves 6 degrees every minute. Let's use this information to find the value of M.
M * 6 = angle moved by minute hand
M * 6 = 30 * (H + M/60)
M = 2 * (H + M/60)
Substituting this value of M in the equation we get:
60H - 11 * (2 * (H + M/60)) = 180
60H - 22H - 11M/30 = 180
38H - 11M/30 = 180
1140H - 11M = 54000
We can solve this equation for H and M using trial and error method. We get the following solutions for H and M:
H = 3, M = 30
H = 9, M = 0
The first solution corresponds to the time when the hour hand is at 3 and the minute hand is at 6. In this case, the angle between the hour hand and the minute hand is 90 degrees.
Case 2: Minute hand ahead of hour hand
Let's assume that the minute hand is ahead of the hour hand. In this case, the hour hand would be somewhere between 3 and 9.
| 30H - 11/2 M | = 90
11/2 M - 30H = 90
11M - 60H = 180
Substituting the value of M in terms of H as before, we get:
11 * (2 * (H + M/60)) - 60H = 180
22H + 11M/30 - 60H = 180
-38H + 11M/30 = 180
-1140H + 11M = -54000
Again, using trial and error method we get the following solutions for H and M:
H = 3, M = 30
H = 9, M = 0
The first solution corresponds to the time when the hour hand is at 3 and the minute hand is at 6. In this case, the angle between the hour hand and the minute hand is 90 degrees.
Therefore,
To find the time when the minute and hour hands of a clock make a right angle, we need to use the following formula:
Angle between the hour hand and the minute hand = | 30H - 11/2 M |
where H is the hour and M is the minute.
We know that when the minute and hour hands of a clock make a right angle, the angle between them is 90 degrees. Let's substitute this value in the formula and solve for H and M.
| 30H - 11/2 M | = 90
There are two possible solutions for this equation, one where the hour hand is ahead of the minute hand and one where the minute hand is ahead of the hour hand. We will consider both cases.
Case 1: Hour hand ahead of minute hand
Let's assume that the hour hand is ahead of the minute hand. In this case, the minute hand would be somewhere between 0 and 30 minutes.
| 30H - 11/2 M | = 90
30H - 11/2 M = 90
60H - 11M = 180
We also know that the minute hand moves 6 degrees every minute. Let's use this information to find the value of M.
M * 6 = angle moved by minute hand
M * 6 = 30 * (H + M/60)
M = 2 * (H + M/60)
Substituting this value of M in the equation we get:
60H - 11 * (2 * (H + M/60)) = 180
60H - 22H - 11M/30 = 180
38H - 11M/30 = 180
1140H - 11M = 54000
We can solve this equation for H and M using trial and error method. We get the following solutions for H and M:
H = 3, M = 30
H = 9, M = 0
The first solution corresponds to the time when the hour hand is at 3 and the minute hand is at 6. In this case, the angle between the hour hand and the minute hand is 90 degrees.
Case 2: Minute hand ahead of hour hand
Let's assume that the minute hand is ahead of the hour hand. In this case, the hour hand would be somewhere between 3 and 9.
| 30H - 11/2 M | = 90
11/2 M - 30H = 90
11M - 60H = 180
Substituting the value of M in terms of H as before, we get:
11 * (2 * (H + M/60)) - 60H = 180
22H + 11M/30 - 60H = 180
-38H + 11M/30 = 180
-1140H + 11M = -54000
Again, using trial and error method we get the following solutions for H and M:
H = 3, M = 30
H = 9, M = 0
The first solution corresponds to the time when the hour hand is at 3 and the minute hand is at 6. In this case, the angle between the hour hand and the minute hand is 90 degrees.
Therefore,
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At what time given below do the minute and hour hands of a dock make a...
Right Angle is formed when there is a L shape in any way like upright and a right Angle is Always 90 degree
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