In 6 minutes, how many degrees minute hand traverses?
Answer: D
The minute hand covers \( 360^{\circ} \) in sixty minutes:
\[ \frac{360}{60} = 6^{\circ} \text{ per minute} \]
\[ 6 \times 6 = 36^{\circ} \]
The 6° option is the rate rather than the total, and 30° is the hour hand's hourly figure.
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A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours will be the true time when the clock indicates 1 p.m. on the following day?
Answer: A
The clock gains, so it reaches 1 pm before the true time does.
From 8 am to 1 pm the next day is 29 hours on the clock. The clock's 24 h 10 min corresponds to 24 true hours:
\[ \text{true time} = 29 \times 60 \times \frac{1440}{1450} = 1740 \times \frac{144}{145} \]
\[ = 1728 \text{ minutes} = 28 \text{ h } 48 \text{ min} \]
\[ 8 \text{ a.m.} + 28 \text{ h } 48 \text{ min} = 12{:}48 \text{ p.m.} \]
So the true time is 48 minutes past 12.
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At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?
Answer: D
The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is
\[ \theta = \left| 30H - 5.5M \right| \]
Opposite directions means \( \theta = 180 \), with \( H = 4 \):
\[ \left| 120 - 5.5M \right| = 180 \]
\[ 5.5M = 300 \quad\Longrightarrow\quad M = \frac{600}{11} = 54\tfrac{6}{11} \]
The other root, \( 5.5M = -60 \), is negative and discarded.
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An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?
Answer: B
The hour hand sweeps the full \( 360^{\circ} \) in twelve hours, so it turns \( 30^{\circ} \) an hour.
\[ 6 \text{ hours} \times 30^{\circ} = 180^{\circ} \]
Only the hour hand is asked about; the minute hand would have turned \( 6 \times 360^{\circ} \) over the same period.
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A watch which gains uniformly ,is 5 min,slow at 8 o'clock in the morning on sunday and it is 5 min 48 sec.fast at 8 p.m on following sunday. when was it correct?
Answer: B
From 8 am Sunday to 8 pm the following Sunday is
\[ 7 \times 24 + 12 = 180 \text{ hours} \]
\[ \text{total gain} = 5 + 5\tfrac{4}{5} = 10.8 \text{ minutes} \]
\[ \text{rate} = \frac{10.8}{180} = 0.06 \text{ min per hour} \]
Starting 5 minutes slow, it becomes correct after
\[ \frac{5}{0.06} = 83\tfrac{1}{3} \text{ h} = 83 \text{ h } 20 \text{ min} \]
\[ 8 \text{ a.m. Sunday} + 83 \text{ h } 20 \text{ min} = 7{:}20 \text{ p.m. on Wednesday} \]
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A clock is set right at 5 a.m. The clock loses 16 minutes in 24 hours.What will be the true time when the clock indicates 10 p.m. on 4th day?
Answer: A
Time from 5 am. on a day to 10 pm. on 4th day = 89 hours.
Now 23 hrs 44 min. of this clock = 24 hours of correct clock.
356/15 hrs of this clock = 24 hours of correct clock
89 hrs of this clock = (24 x 31556 x 89) hrs of correct clock.
= 90 hrs of correct clock.
So, the correct time is 11 p.m.
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At what time between 2 and 3 o'clock will the hands of a clock be together?
Answer: B
The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is
\[ \theta = \left| 30H - 5.5M \right| \]
Together means \( \theta = 0 \), with \( H = 2 \):
\[ 5.5M = 60 \quad\Longrightarrow\quad M = \frac{120}{11} = 10\tfrac{10}{11} \]
So the hands meet at \( 10\tfrac{10}{11} \) minutes past 2.
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At what time between 5 and 6 o'clock are the hands of a clock 3 minutes apart?
Answer: A
The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is
\[ \theta = \left| 30H - 5.5M \right| \]
Three minute-divisions is \( 3 \times 6 = 18^{\circ} \), with \( H = 5 \):
\[ \left| 150 - 5.5M \right| = 18 \]
\[ 5.5M = 132 \quad\Longrightarrow\quad M = 24 \]
The other root gives \( M = 30\tfrac{6}{11} \), on the far side of the coincidence at \( 27\tfrac{3}{11} \) minutes past 5.
The 24-minute answer is the exact one, which is what makes it the intended reading.
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Find the angle between the hour hand and the minute hand of a clock when the time is 3.25
Answer: A
At 3 O'clock, Minute hand is at 12 while the Hour hand is at 3. Again the minute hand has to sweep through ( 30 x 5 ) ie 150° for reaching the figure 5 to show 25 mins.
Simultaneously the Hour hand will also rotate for 25 mins. Thus starting from the mark, 3 the hour hand will cover an angle = (25 x 30) / 60 = 12.5°
Hence, Angle between Hour and the Minute hand = ( 60 - 12.5 ) = 47.5°
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At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but,not together?
Answer: A
The hour hand moves \( 0.5^{\circ} \) a minute and the minute hand \( 6^{\circ} \), so at \( H \) hours \( M \) minutes the angle between them is
\[ \theta = \left| 30H - 5.5M \right| \]
"Same straight line but not together" means \( \theta = 180 \), with \( H = 7 \):
\[ \left| 210 - 5.5M \right| = 180 \]
\[ 5.5M = 30 \quad\Longrightarrow\quad M = \frac{60}{11} = 5\tfrac{5}{11} \]
The other root, \( 5.5M = 390 \), gives \( M = 70.9 \), beyond the hour.
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