Exercise : 1


1. If the two hands in a clock are 3 minutes divisions apart,then the angle between them is
(a) 3°
(b) 18°
(c) 24°
(d) 60°
(e) None of these
Ans.b

In a clock, each minute makes 6°

∴ 3 minutes will make 6 × 3 = 18°

2. At what approximate time between 4 and 5 am will the hands of a clock be at right angle?
(a) 4 : 40 am
(b) 4 : 38 am
(c) 4 : 35 am
(d) 4 : 39 am
(e) None of these
Ans.b

Here H × 30 = 4 × 30 = 120°.

(Since initially the hour hand is at 4. ∴ H = 4).

Required angle A = 90° and since, H × 30 > A° so, there will be two timings.

Required time T = 211(H × 30 ± A) minutes past H.

∴ One timing = 211(4 × 30 + 90) minutes past 4

= 38211 minutes past 4.

Or 4 : 38 approx.

3. What will be the acute angle between hands of a clock at 2 : 30?
(a) 105°
(b) 115°
(c) 95°
(d) 135°
(e) None of these
Ans.a

At 2'O Clock, Minute Hand will be 10 × 6 = 60°

behind the Hour Hand.

In 30 minutes, Minute Hand will gain (512)° × 30

= 150 + 15 = 165°

∴ Angle between Hour Hand and Minute Hand

= 165 – 60 = 105°

4. In 16 minutes, the minute hand gains over the hour hand by
(a) 16°
(b) 80°
(c) 88°
(d) 96°
(e) None of these
Ans.c

In 1 hour, the minute hand gains 330° over the hour hand.

i.e. in 60 minute, the minute hand gains 330° over the hour hand.

∴ In 16 minutes, the minute hand gains over the hour hand by 330°60 × 16° = 88°

5. A clock is set right at 1 p.m. If it gains one minute in an hour, then what is the true time when the clock indicates 6 p.m. in the same day?
(a) 55561 minutes past 5
(b) 5 minutes past 6
(c) 5 minutes to 6
(d) 59164 minutes past 5
(e) None of these
Ans.a

Time interval indicated by incorrect clock

= 6 p.m – 1 p.m = 5hrs.

Time gained by incorrect clock in one hour

= + 1 min. = +160 hr.

Using the formula, \(\frac{True \; time\; interval}{Time \; interval \; in \; incorrect \; clock}\)

= \(\frac{1}{1 + hour \; gained \; in \; 1 \; hour \; by \; incorrect \; clock}\)

\(\frac{True \; time \; interval}{5} = \frac{1}{1 + \frac{1}{60}}\)

⇒ True time interval = \(\frac{5 \times 60}{61} = 4\tfrac{56}{61}\)

∴ True time = 1 p.m. + \(4\tfrac{56}{61}\) hrs.

= 5 p.m. + 5661 hrs. = 5 p.m. + 5661 × 60 min.

= 55561 minutes past 5.

6. Two clocks were set right at noon on Sunday. One gains 2 min and the other loses 3 min in 24 hours. What will be the true time when the first clock indicates 3 pm on Wednesday?
(a) 2 : 38 pm
(b) 2 : 54 pm
(c) 2 : 23 pm
(d) 2 : 48 pm
(e) None of these
Ans.b

Time from noon on Sunday to 3 pm on Wednesday = 75 hours.

24 hours 2 minutes of the first clock = 24 hours of the correct one.

⇒ 1 hour of the first clock = 24 × (30/721) hours of correct one.

⇒ 75 ours of the first clock

= 24 × 30 × (75/721) hours of correct one

= 54000/721 hours = 74 hours 53.7 min.

Hence the answer is 2 : 54 pm.

7. At what time between 9’o clock and 10’o clock will the hands of a clock point in the opposite directions?
(a) 16411 minutes past 9
(b) 16411 minutes past 8
(c) 55561 minutes past 7
(d) 55561 minutes past 8
(e) None of these
Ans.a

At 9’o clock, the Minute Hand is ahead of Hour Hand by 45 minutes. The hands will be opposite to each other when there is a space of 30 minutes between them.

This will happen when the Minute Hand gains 15 minutes space over Hour Hand.

Time taken by Minutes Hand to gain 15 minutes

= 15 × (1 + 111) = 15 + 1511 = 15 + 1411 + 16411 minutes.

Hence the Hands are opposite to each other at 16411 minutes past 9.

8. A clock gains 15 minutes per day. It is set right at 12 noon. What time will it show at 4.00 am, the next day?
(a) 4 : 10 am
(b) 4 : 45 am
(c) 4 : 20 am
(d) 5 : 00 am
(e) None of these
Ans.a

The clock gains 15 min in 24 hours.

Therefore, in 16 hours, it will gain 10 minutes.

Hence, the time shown by the clock will be 4.10 am.

9. What is the angle between the 2 hands of the clock at 8:24 pm?
(a) 100°
(b) 107°
(c) 106°
(d) 108°
(e) None of these
Ans.d

Required angle = 240 – 24 × (11/2)

= 240 – 132 = 108°

10. In a watch, the minute hand crosses the hour hand for the third time exactly after every 3 hrs., 18 min., 15 seconds of watch time. What is the time gained or lost by this watch in one day?
(a) 14 min. 10 seconds lost
(b) 13 min. 50 seconds lost
(c) 13 min. 20 seconds gained
(d) 14 min. 40 seconds gained
(e) None of these
Ans.b

In a watch than is running correct the minute hand should cross the hour hand once in every 65 + 511 min.

So they should ideally cross 3 times once in

\(3 \times \left ( \frac{720}{11} \right )\frac{-2060}{11} \; min = 196.36 \; minutes.\)

But in the watch under consideration, they meet after every 3 hr,18 min and 15 seconds,

i.e, (3 × 60 + 18 + 1560) = 7934 min.

Thus, our watch is actually losing time (as it is slower than the normal watch). Hence when our watch elapsed

\(\left ( 1440 \times \frac{196.36}{198.25} \right ) = 1426.27.\)

Hence the amount of time lost by our watch in one day = (1440 - 1426.27) = 13.73 i.e. 13 min and 50s (approx).