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At what time between $6$ a. m. and $7$ a. m. will the minute hand and hour hand of a clock make an angle closest to $60^{\circ}$?

  1. $6: 22$ a.m.
  2. $6: 27$ a.m.
  3. $6: 38$ a.m.
  4. $6: 45$ a.m.
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9 Answers

2 votes
2 votes

Minute hand rotates with the speed:   360°/(60min) = 6° per minute.

Hour Hand rotates with the speed:   360°/((24*60)mins) = 0.25° per minute.

At 6 o' clock, minute hand is vertical up & hour hand is vertical down. 

Let us assume, minute hand is at 0°, this means hour hand is at 180°. 

So, angle between these two hands after t minutes from 6 a.m is either (180° + 25t) - 6t or 6t - (180° + 25t)

Angle given in the question is 60°.

(180° + 25t) - 6t = 60

   -> t = 120/5.75    

   -> t= 20.86min i.e. 6:20am

OR

6t - (180° + 25t) = 60°

-> t= 41.73min i.e 6:41am

6:22am is most closer to the actual answer .

Therefore, Option A is correct

2 votes
2 votes

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Pic courtesy : @amarVashishth

Angle made by hour hand$=12-1-2-3-4-5-6=180^{\circ}$

$+$

$22^{\circ}\times \frac{1}{2}=11^{\circ}$

$180^{\circ}+11^{\circ}=191^{\circ}$

Angle made by minute hand$=22^{\circ}\times 6^{\circ}=132^{\circ}$

minute hand and hour hand of a clock make an angle closest to 60°?

$191^{\circ}-132^{\circ}=59^{\circ}$

Correct Answer $: A$

1 votes
1 votes
option a
1 votes
1 votes

Speed of  minutes hand :   360°/60min = 6° per minute.

Speed of hour’s hand : 360°/12hr  = 30$^{\circ}$ per hour = 30/60min =0.5$^{\circ}$ per min

At 6 o' clock, minute hand is vertical up & hour hand is vertical down. 

Let us assume, minute hand is at 0°, this means hour hand is at 180°. 

So, angle between these two hands after t minutes from 6 a.m is

case 1 : Minute hand is behind hour hand than angle between them is (180° + 0.5t) - 6t

      Angle given in the question is 60°.

      (180° + 0.5t) - 6t = 60

      t = 120/5.5    

      t=21.81

case 2 :Minute hand passes Hour hand  than angle between them is  6t - (180° + 0.5t)

      6t - (180° + 0.5t) = 60°

       t= 240/5.5= 43.63

6:22am is most closest to the actual answer .

Therefore, answer is (A)

Answer:

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