Clock
Name: _____________________
Date: _____________________
Instructions: Answer all questions. Write your answers clearly in the space provided.
Question 1:
A clock strikes once at 1 o'clock, twice at 2 o'clock, thrice at 3 o'clock and so on. What is the total number of strikings in a day = ?
A.
136
B.
146
C.
156
D.
166
Answer: _________
Question 2:
Through what angle does the minute hand of a clock turn in 5 minutes ?
A.
$${30^ circ }$$
B.
$${32^ circ }$$
C.
$${35^ circ }$$
D.
$${36^ circ }$$
Answer: _________
Question 3:
It is between 3 pm and 4 pm and the distance between the hour hand and the minute hand of clock is 18 minutes spaces. What time does the clock show ?
A.
3 : 12 pm
B.
3 : 27 pm
C.
3 : 31 pm
D.
3 : 36 pm
Answer: _________
Question 4:
In an accurate clock, in a period of 2 hours 20 minutes the minute hand will move over = ?
A.
140°
B.
320°
C.
520°
D.
840°
Answer: _________
Question 5:
What is the area of the face of a clock described by its minutes hand between 9 am and 9 : 35 am, if the minutes hand is 10 cm long ?
A.
$${ ext{36}}frac{2}{3}{ ext{c}}{{ ext{m}}^2}$$
B.
$${ ext{157}}frac{1}{7}{ ext{c}}{{ ext{m}}^2}$$
C.
$${ ext{183}}frac{1}{3}{ ext{c}}{{ ext{m}}^2}$$
D.
None of these
Answer: _________
Question 6:
The angle between the hands of a clock when the time is 4 : 25 am is = ?
A.
$${ ext{13}}{frac{1}{2}^ circ }$$
B.
$${ ext{17}}{frac{1}{2}^ circ }$$
C.
$${ ext{14}}{frac{1}{2}^ circ }$$
D.
$${ ext{12}}{frac{1}{2}^ circ }$$
Answer: _________
Question 7:
Imagine that your watch was correct at noon, but then it began to lose 30 minutes each hour. It now show 4 pm but it stopped 5 hours ago. What is the correct time now = ?
A.
9 : 30 pm
B.
11 pm
C.
1 am
D.
1 : 30 am
Answer: _________
Question 8:
How many rotations will the hour hand of a clock complete in 72 hours ?
A.
3
B.
6
C.
9
D.
12
Answer: _________
Question 9:
At 8 : 30, the hour hand and the minute hand of clock form an angle of = ?
A.
$${ ext{8}}{{ ext{0}}^ circ }$$
B.
$${ ext{7}}{{ ext{5}}^ circ }$$
C.
$${ ext{7}}{{ ext{0}}^ circ }$$
D.
$${ ext{6}}{{ ext{0}}^ circ }$$
Answer: _________
Question 10:
At 9 : 38 A.M. through how many degrees the hour hand of a clock moved since noon the previous day ?
A.
323°
B.
612°
C.
646°
D.
649°
Answer: _________
Question 11:
The hands of a clock are 10 cm and 7 cm respectively. The difference between the distance traversed by their extremities in 3 days 5 hours is = ?
A.
4552.67 CM
B.
4555.67 CM
C.
4557.67 CM
D.
4559.67 CM
Answer: _________
Question 12:
There are two clocks, both set to show 10 pm on 21 st January 2010. One clock gains 2 minutes in an hour and the other clock loses 5 minutes in an hour. Then by how many minutes do the two clocks differ at 4 pm on 22 nd January 2010 ?
A.
126 minutes
B.
136 minutes
C.
96 minutes
D.
106 minutes
Answer: _________
Question 13:
In every 30 minutes the time of a watch increases by 3 minutes. After showing the correct time at 5 am , what time will the watch show after 6 hours ?
A.
10 : 54 am
B.
11 : 30 am
C.
11 : 36 am
D.
11 : 42 am
Answer: _________
Question 14:
A watch is 1 minute slow at 1 pm on Tuesday and 2 minutes fast at 1 pm on Thursday. When did it show the correct time = ?
A.
1 : 00 am on Wednesday
B.
5 : 00 am on Wednesday
C.
1 : 00 pm on Wednesday
D.
5 : 00 pm on Wednesday
Answer: _________
Question 15:
Henry started a trip into the country between 8 am and 9 am when the hand of clock were together, He arrived at his destination between 2 pm and 3 pm when the hands of the clock were exactly 180° apart. How long did he travel ?
A.
6 hours
B.
7 hours
C.
9 hours
D.
11 hours
Answer: _________
Question 16:
Between 5 and 6, a lady looked at her watch and mistaking the hour hand for the minute hand, she thought that the time was 57 minutes
earlier than the correct time. The correct time was = ?
A.
12 minutes past 5
B.
24 minutes past 5
C.
36 minutes past 5
D.
48 minutes past 5
Answer: _________
Question 17:
How many times are the hour hand and the minute hand of a clock of a right angles during their motion from 1 : 00 pm to 10 : 00 pm ?
A.
9
B.
10
C.
18
D.
20
Answer: _________
Question 18:
Wall clock gains 2 minutes in 12 hours, while a table clock loses 2 minutes every 36 hours. Both are set right at 12 noon on Tuesday.
The correct time when both show the same time next would be = ?
A.
12:30 at nights, after 130 days
B.
12 noon, after 135 days
C.
1:30 at nights, after 130 days
D.
12 midnight, after 135 days
Answer: _________
Question 19:
A clock is displaying correct time at 9 am on Monday. If the clock loses 12 minutes in 24 hours, then the actual time when the clock indicates 8 : 30 pm on Wednesday of the same week is = ?
A.
8 pm
B.
7 pm
C.
9 pm
D.
8 : 59 : 45 pm
Answer: _________
Question 20:
A wall-clock takes 9 seconds in tinging at 9 o'clock. The time, it will take in tinging at 11 o'clock, is = ?
A.
10 seconds
B.
1.80 seconds
C.
11 seconds
D.
11.25 seconds
Answer: _________
Question 21:
A mechanical grandfather clock is at present showing 7 hours 40 minutes 6 seconds. Assuming that it loses 4 seconds in every hour, what time will it show after exactly $$6frac{1}{2}$$ hours ?
A.
14 hours 9 minutes 34 seconds
B.
14 hours 9 minutes 40 seconds
C.
14 hours 10 minutes 6 seconds
D.
14 hours 10 minutes 32 seconds
Answer: _________
Question 22:
How many times are the hands of a clock at right angle in a day?
A.
22
B.
24
C.
44
D.
48
Answer: _________
Question 23:
The angle between the minute hand and the hour hand of a clock when the time is 8:30, is:
A.
80°
B.
75°
C.
60°
D.
105°
Answer: _________
Question 24:
How many times in a day, are the hands of a clock in straight line but opposite in direction?
A.
20
B.
22
C.
24
D.
48
Answer: _________
Question 25:
At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?
A.
45 min. past 4
B.
40 min. past 4
C.
$$50frac{4}{{11}}$$ min. past 4
D.
$$54frac{6}{{11}}$$ min. past 4
Answer: _________
Question 26:
At what time between 9 and 10 o'clock will the hands of a watch be together?
A.
45 min. past 9
B.
50 min. past 9
C.
$$49frac{1}{{11}}$$ min. past 9
D.
$$49frac{11}{{1}}$$ min. past 9
Answer: _________
Question 27:
At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other?
A.
$$5frac{1}{{11}}$$ min. past 3
B.
$$12frac{4}{{11}}$$ min. past 3
C.
$$13frac{4}{{11}}$$ min. past 3
D.
$$16frac{4}{{11}}$$ min. past 3
Answer: _________
Question 28:
How many times do the hands of a clock coincide in a day?
A.
20
B.
21
C.
22
D.
24
Answer: _________
Question 29:
How many times in a day, the hands of a clock are straight?
A.
22
B.
24
C.
44
D.
48
Answer: _________
Question 30:
A watch which gains uniformly is 2 minutes low at noon on Monday and is 4 min. 48 sec fast at 2 p.m. on the following Monday. When was it correct?
A.
2 p.m. on Tuesday
B.
2 p.m. on Wednesday
C.
3 p.m. on Thursday
D.
1 p.m. on Friday
Answer: _________
Question 31:
A watch becomes fast by 5 minutes everyday. By what percent does it become fast ?
A.
$$frac{5}{{24}}\% $$
B.
$$frac{1}{{12}}\% $$
C.
5 %
D.
$$frac{{50}}{{144}}\% $$
Answer: _________
Question 32:
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?
A.
144º
B.
150º
C.
168º
D.
180º
Answer: _________
Question 33:
The reflex angle between the hands of a clock at 10.25 is:
A.
180º
B.
$${ ext{192}}{frac{1}{2}^ circ }$$
C.
195º
D.
$${ ext{197}}{frac{1}{2}^ circ }$$
Answer: _________
Question 34:
A clock is started at noon. By 10 minutes past 5, the hour hand has turned through:
A.
145º
B.
150º
C.
155º
D.
160º
Answer: _________
Question 35:
How much does a watch lose per day, if its hands coincide every 64 minutes?
A.
$$32frac{8}{{11}}$$ min.
B.
$$36frac{5}{{11}}$$ min.
C.
90 min.
D.
96 min.
Answer: _________
Question 36:
At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together?
A.
5 min. past 7
B.
$$5frac{2}{{11}}$$ min. past 7
C.
$$5frac{3}{{11}}$$ min. past 7
D.
$$5frac{5}{{11}}$$ min. past 7
Answer: _________
Question 37:
At what time between 5:30 and 6 will the hands of a clock be at right angles?
A.
$$43frac{5}{{11}}$$ min. past 5
B.
$$43frac{7}{{11}}$$ min. past 5
C.
40 min. past 5
D.
45 min. past 5
Answer: _________
Question 38:
The angle between the minute hand and the hour hand of a clock when the time is 4:20, is:
A.
0º
B.
10º
C.
5º
D.
20º
Answer: _________
Question 39:
At what angle the hands of a clock are inclined at 15 minutes past 5?
A.
$$58{frac{1}{2}^ circ }$$
B.
$${64^ circ }$$
C.
$$67{frac{1}{2}^ circ }$$
D.
$$72{frac{1}{2}^ circ }$$
Answer: _________
Question 40:
At 3:40, the hour hand and the minute hand of a clock form an angle of:
A.
120°
B.
125°
C.
130°
D.
135°
Answer: _________
Question 41:
A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is:
A.
$$59frac{7}{{12}}$$ min. past 3
B.
4 p.m.
C.
$$58frac{7}{{11}}$$ min. past 3
D.
$$2frac{3}{{11}}$$ min. past 4
Answer: _________
Answer Key
1:
C
Solution: Total number of strikings $$eqalign{
& = 2left( {1 + 2 + 3 + ..... + 12}
ight) cr
& = 2 imes frac{{12 imes 13}}{2} cr
& = 156 cr} $$
& = 2left( {1 + 2 + 3 + ..... + 12}
ight) cr
& = 2 imes frac{{12 imes 13}}{2} cr
& = 156 cr} $$
2:
A
Solution: Angle traced by the minute hand in 5 minutes. $$eqalign{
& = {left( {frac{{360}}{{60}} imes 5}
ight)^ circ } cr
& = {30^ circ }{ ext{ }} cr} $$
& = {left( {frac{{360}}{{60}} imes 5}
ight)^ circ } cr
& = {30^ circ }{ ext{ }} cr} $$
3:
D
Solution: At 3 o'clock, the minute hand is 15 minute spaces behind the hour hand. Thus, the minute hand has to gain (15 + 18) = 33 minute spaces 55 minutes are gained in 60 minutes. 33 minutes are gained in $$left( {frac{{60}}{{55}} imes 33}
ight)$$ xa0 = 36 minutes ∴ The hands will be 18 minutes spaces apart at 3 : 36 pm.
ight)$$ xa0 = 36 minutes ∴ The hands will be 18 minutes spaces apart at 3 : 36 pm.
4:
D
Solution: Angle traced by the minute hand in 2 hours 20 minutes,i.e., $$eqalign{
& = 140{ ext{ minutes}} cr
& = {left( {frac{{360}}{{60}} imes 140}
ight)^ circ } cr
& = {840^ circ } cr} $$
& = 140{ ext{ minutes}} cr
& = {left( {frac{{360}}{{60}} imes 140}
ight)^ circ } cr
& = {840^ circ } cr} $$
5:
C
Solution: Angle swept by the minute hand in 35 minutes. $$eqalign{
& = {left( {frac{{360}}{{60}} imes 35}
ight)^ circ } cr
& = {210^ circ } cr} $$ ∴ Required area = Area of a sector of a circle with radius 10 cm and central angle 210° $$eqalign{
& = frac{{pi {r^2} heta }}{{360}} cr
& = left( {frac{{22}}{7} imes 10 imes 10 imes frac{{210}}{{360}}}
ight){ ext{c}}{{ ext{m}}^2} cr
& = frac{{550}}{3}{ ext{c}}{{ ext{m}}^2} cr
& = 183frac{1}{3}{ ext{c}}{{ ext{m}}^2} cr} $$
& = {left( {frac{{360}}{{60}} imes 35}
ight)^ circ } cr
& = {210^ circ } cr} $$ ∴ Required area = Area of a sector of a circle with radius 10 cm and central angle 210° $$eqalign{
& = frac{{pi {r^2} heta }}{{360}} cr
& = left( {frac{{22}}{7} imes 10 imes 10 imes frac{{210}}{{360}}}
ight){ ext{c}}{{ ext{m}}^2} cr
& = frac{{550}}{3}{ ext{c}}{{ ext{m}}^2} cr
& = 183frac{1}{3}{ ext{c}}{{ ext{m}}^2} cr} $$
6:
B
Solution: Let angle between the hands of clock be x When the time is 4 : 25 am Where [M = minutes and H = hours] Required angle $$eqalign{
& = { ext{30}}left( {frac{{ ext{M}}}{5} - { ext{H}}}
ight) - frac{{ ext{M}}}{2} cr
& = 30left( {frac{{25}}{5} - 4}
ight) - frac{{25}}{2} cr
& = 30left( {frac{{25 - 20}}{5}}
ight) - frac{{25}}{2} cr
& = 30left( {frac{5}{5}}
ight) - frac{{25}}{2} cr
& = 30 - frac{{25}}{2} cr
& = frac{{60 - 25}}{2} cr
& = frac{{35}}{2} cr
& = 17{frac{1}{2}^{° }} cr} $$
& = { ext{30}}left( {frac{{ ext{M}}}{5} - { ext{H}}}
ight) - frac{{ ext{M}}}{2} cr
& = 30left( {frac{{25}}{5} - 4}
ight) - frac{{25}}{2} cr
& = 30left( {frac{{25 - 20}}{5}}
ight) - frac{{25}}{2} cr
& = 30left( {frac{5}{5}}
ight) - frac{{25}}{2} cr
& = 30 - frac{{25}}{2} cr
& = frac{{60 - 25}}{2} cr
& = frac{{35}}{2} cr
& = 17{frac{1}{2}^{° }} cr} $$
7:
C
Solution: The watch loses $$frac{1}{2}$$ hour each hour. So, it must have take 8 hours to show 4 pm from 12 noon. Thus, it stopped at 8 pm. So, the correct time is 5 hours ahead of 8 pm, i,e., 1 am.
8:
B
Solution: Number of rotation $$ = frac{{72}}{{12}} = 6$$
9:
B
Solution: In 1 hour, the hour hand make the angle of 30° The hour hand make the angle of x in $${ ext{8}}frac{{30}}{{60}}$$ hours $$eqalign{
& Rightarrow x = {left( {30 imes 8frac{1}{2}}
ight)^ circ } cr
& Rightarrow x = {left( {30 imes frac{{17}}{2}}
ight)^ circ } cr
& Rightarrow x = {255^ circ } cr} $$ The minute hand make the angle in 1 minute = 6° Minute hand makes the angle in 30 minutes $$eqalign{
& { ext{ = }}{left( {6 imes 30}
ight)^ circ } cr
& = {180^ circ } cr
& { ext{Required angle}} cr
& = {255^0} - {180^0} cr
& = {75^0} cr} $$
& Rightarrow x = {left( {30 imes 8frac{1}{2}}
ight)^ circ } cr
& Rightarrow x = {left( {30 imes frac{{17}}{2}}
ight)^ circ } cr
& Rightarrow x = {255^ circ } cr} $$ The minute hand make the angle in 1 minute = 6° Minute hand makes the angle in 30 minutes $$eqalign{
& { ext{ = }}{left( {6 imes 30}
ight)^ circ } cr
& = {180^ circ } cr
& { ext{Required angle}} cr
& = {255^0} - {180^0} cr
& = {75^0} cr} $$
10:
D
Solution: Time from 12 noon to 9 : 38 A.M. = 12 hours + 9 hours 38 minutes = 21 hours 38 minutes $$eqalign{
& { ext{ = 21}}frac{{38}}{{60}}{ ext{ hours}} cr
& { ext{ = 21}}frac{{19}}{{30}}{ ext{ hours}} cr
& { ext{ = }}frac{{649}}{{30}}{ ext{hours}} cr} $$ Angle traced by the hour hand in 12 hours = 360° Angle traced by the minute hand in $$eqalign{
& Leftrightarrow frac{{649}}{{30}}{ ext{hours}} cr
& = {left( {frac{{360}}{{12}} imes frac{{649}}{{30}}}
ight)^ circ } cr
& = {649^ circ } cr} $$
& { ext{ = 21}}frac{{38}}{{60}}{ ext{ hours}} cr
& { ext{ = 21}}frac{{19}}{{30}}{ ext{ hours}} cr
& { ext{ = }}frac{{649}}{{30}}{ ext{hours}} cr} $$ Angle traced by the hour hand in 12 hours = 360° Angle traced by the minute hand in $$eqalign{
& Leftrightarrow frac{{649}}{{30}}{ ext{hours}} cr
& = {left( {frac{{360}}{{12}} imes frac{{649}}{{30}}}
ight)^ circ } cr
& = {649^ circ } cr} $$
11:
C
Solution: Number of rounds completed by the minute hand in 3 days 5 hours $$eqalign{
& = left( {3 imes 24 + 5}
ight) cr
& = 77 cr} $$ Number of rounds completed by the hour hand in 3 days 5 hours $$eqalign{
& = left( {3 imes 2 + frac{5}{{12}}}
ight) cr
& = 6frac{5}{{12}} cr} $$ ∴ Difference between the distance traversed $${ ext{ = }}left[ {77 imes left( {2 imes frac{{22}}{7} imes 10}
ight) - 6frac{5}{{12}} imes left( {2 imes frac{{22}}{7} imes 7}
ight)}
ight]{ ext{cm}}$$ $$eqalign{
& = left( {4840 - 282.33}
ight){ ext{ cm}} cr
& = 4557.67{ ext{ cm}} cr} $$
& = left( {3 imes 24 + 5}
ight) cr
& = 77 cr} $$ Number of rounds completed by the hour hand in 3 days 5 hours $$eqalign{
& = left( {3 imes 2 + frac{5}{{12}}}
ight) cr
& = 6frac{5}{{12}} cr} $$ ∴ Difference between the distance traversed $${ ext{ = }}left[ {77 imes left( {2 imes frac{{22}}{7} imes 10}
ight) - 6frac{5}{{12}} imes left( {2 imes frac{{22}}{7} imes 7}
ight)}
ight]{ ext{cm}}$$ $$eqalign{
& = left( {4840 - 282.33}
ight){ ext{ cm}} cr
& = 4557.67{ ext{ cm}} cr} $$
12:
A
Solution: One clock show 10 pm, on 21 st January 2010 One clock gains = 2 minutes Other clock loses = 5 minutes Time period between 10 pm and 4 pm = 18 hours ∴ Required difference = (2 × 18 + 5 × 18 ) minutes = 126 minutes
13:
C
Solution: Time gained in 1 hour = 6 minutes Time gained in 6 hours = (6 × 6) minutes = 36 minutes After 6 hours, the correct time is 11 : 00 am and the watch will show 11 : 36 am.
14:
B
Solution: Time from 1 pm on Wednesday to 1 pm on Thursday = 48 hours So, the watch gains (1 + 2) minute or 3 minutes in 48 hours. Now, 3 minutes are gained in 48 hours So, 1 minute is gained in $$left( {frac{{48}}{3}}
ight)$$ xa0 = 16 hours. Thus, the watch showed the correct time 16 hours after 1 pm on Tuesday, i.e., 5 am on Wednesday
ight)$$ xa0 = 16 hours. Thus, the watch showed the correct time 16 hours after 1 pm on Tuesday, i.e., 5 am on Wednesday
15:
A
Solution: To be together between 8 am and 9 am, the minute hand has to gain 40 minutes spaces. 55 minutes spaces are gained in 60 minutes. 40 minutes space are gained in $$left( {frac{{60}}{{55}} imes 40}
ight)$$ xa0minutes = $${ ext{43}}frac{7}{{11}}$$ xa0minutes So, Henry started his trip at $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 8 am. Now, to be 180° apart, the hands must be 30 minutes spaces apart. At 2 pm, they are 10 minutes spaces apart. ∴ The minute hand will have to gain (10 + 30) = 40 minutes spaces. As calculate above, 40 minutes spaces are gained in $${ ext{43}}frac{7}{{11}}$$ xa0minutes. So, Henry's trip ended at $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 2 pm ∴ Duration of travel = Duration from $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 8 am to $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 2 pm = 6 hours
ight)$$ xa0minutes = $${ ext{43}}frac{7}{{11}}$$ xa0minutes So, Henry started his trip at $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 8 am. Now, to be 180° apart, the hands must be 30 minutes spaces apart. At 2 pm, they are 10 minutes spaces apart. ∴ The minute hand will have to gain (10 + 30) = 40 minutes spaces. As calculate above, 40 minutes spaces are gained in $${ ext{43}}frac{7}{{11}}$$ xa0minutes. So, Henry's trip ended at $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 2 pm ∴ Duration of travel = Duration from $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 8 am to $${ ext{43}}frac{7}{{11}}$$ xa0minutes past 2 pm = 6 hours
16:
B
Solution: Since the time read by the lady was 57 minutes earlier than the correct time, so the minute hand is (60 - 57) = 3 minutes spaces behind the hour hand. Now, at 5 o'clock, the minute hand is 25 minutes spaces behind the hour hand. To be 3 minutes spaces behind, it must gain (25 - 3) = 22 minutes spaces. 55 minutes spaces are gained in 60 minutes. 22 minutes spaces are gained in $$left( {frac{{60}}{{55}} imes 22}
ight)$$ xa0 = 24 minutes Hence, the correct time was 24 minutes past 5.
ight)$$ xa0 = 24 minutes Hence, the correct time was 24 minutes past 5.
17:
N/A
Solution: Explanation in Detail: To determine how many times the hour hand and minute hand of a clock form a right angle between 1:00 pm and 10:00 pm, we analyze their angular positions over this period. 1. Calculating Angular Movements: - Minute Hand: The minute hand moves 360 degrees in 60 minutes, so in 9 hours (from 1:00 pm to 10:00 pm), it covers: [
360 ext{ degrees/hour} imes 9 ext{ hours} = 3240 ext{ degrees}
] - Hour Hand: The hour hand moves 30 degrees in 60 minutes (or 0.5 degrees per minute), covering: [
30 ext{ degrees/hour} imes 9 ext{ hours} = 270 ext{ degrees}
] 2. Relative Angular Distance: - The difference in their angular positions over 9 hours is: [
3240 ext{ degrees (minute hand)} - 270 ext{ degrees (hour hand)} = 2970 ext{ degrees}
] 3. Calculating Right Angles: - A right angle is formed every 180 degrees. - Therefore, the number of times they form a right angle is: [
frac{2970 ext{ degrees}}{180 ext{ degrees}} = 16.5
] Rounding down, they form a right angle 16 times. 4. Identifying Times of Right Angles: - The first right angle occurs at 1:21.8181... pm. - Subsequent right angles occur approximately every 32.7878... minutes until the last right angle at 9:32.7272... pm. 5. List of Times: - The hour and minute hands form a right angle at the following times: - 1:21.8181 pm - 1:54.5454 pm - 2:27.2727 pm - 3:00 pm - 3:32.7272 pm - 4:05.4545 pm - 4:38.1818 pm - 5:10.9090 pm - 5:43.6363 pm - 6:16.3636 pm - 6:49.0909 pm - 7:21.8181 pm - 7:54.5454 pm - 8:27.2727 pm - 9:00 pm - 9:32.7272 pm Therefore, the hour hand and minute hand of the clock form a right angle 16 times between 1:00 pm and 10:00 pm, as calculated based on their angular movements and the criteria for forming right angles.
360 ext{ degrees/hour} imes 9 ext{ hours} = 3240 ext{ degrees}
] - Hour Hand: The hour hand moves 30 degrees in 60 minutes (or 0.5 degrees per minute), covering: [
30 ext{ degrees/hour} imes 9 ext{ hours} = 270 ext{ degrees}
] 2. Relative Angular Distance: - The difference in their angular positions over 9 hours is: [
3240 ext{ degrees (minute hand)} - 270 ext{ degrees (hour hand)} = 2970 ext{ degrees}
] 3. Calculating Right Angles: - A right angle is formed every 180 degrees. - Therefore, the number of times they form a right angle is: [
frac{2970 ext{ degrees}}{180 ext{ degrees}} = 16.5
] Rounding down, they form a right angle 16 times. 4. Identifying Times of Right Angles: - The first right angle occurs at 1:21.8181... pm. - Subsequent right angles occur approximately every 32.7878... minutes until the last right angle at 9:32.7272... pm. 5. List of Times: - The hour and minute hands form a right angle at the following times: - 1:21.8181 pm - 1:54.5454 pm - 2:27.2727 pm - 3:00 pm - 3:32.7272 pm - 4:05.4545 pm - 4:38.1818 pm - 5:10.9090 pm - 5:43.6363 pm - 6:16.3636 pm - 6:49.0909 pm - 7:21.8181 pm - 7:54.5454 pm - 8:27.2727 pm - 9:00 pm - 9:32.7272 pm Therefore, the hour hand and minute hand of the clock form a right angle 16 times between 1:00 pm and 10:00 pm, as calculated based on their angular movements and the criteria for forming right angles.
18:
B
Solution: After 12 days, i.e., after 12 × 24 hours clock A will gain 48 minutes and will show 12 : 48 noon. After 12 days, i.e., after 12 × 24 hours clock B will loose 16 minutes and will show 11 : 44 am The two clocks will show the same time after time after 135 days. The time difference has to be 12 hours between then = 720 minutes. A will gain 540 minutes in 135 days. B will loose 180 minutes in 135 days, total 720 minutes. Further if we consider only time then the problem becomes simpler Total difference of minutes between the times shown by the clocks after 36 hours ⇒ $$frac{{16}}{3}$$ xa0 minutes difference in 1 day ⇒ 12 × 60 minutes difference in $$frac{3}{{16}}$$ × 12 × 60 = 135 days ∴ 12 noon, after 135 days
19:
C
Solution: Time interval from 9 am on Monday to 8 : 30 pm on Wednesday. $$eqalign{
& { ext{ = }}left( {24 imes 2.5}
ight) - { ext{0:30 hours }} cr
& { ext{ = 60}} - { ext{0}}{ ext{:30 hours}} cr
& { ext{ = 59 hours 30 minutes}} cr
& = 59frac{{30}}{{60}} cr
& = 59frac{1}{2} cr
& = frac{{119}}{2}{ ext{ hours}} cr
& { ext{Also 24 hours}} - { ext{12 minutes}} cr
& = { ext{23 hours 48 minutes}} cr
& = 23 + frac{{48}}{{60}} cr
& = 23frac{4}{5} cr
& = frac{{119}}{5}{ ext{ hours}} cr
& herefore frac{{119}}{2}{ ext{ hours of this clock}} cr
& = frac{{24 imes 5}}{{119}} imes frac{{119}}{2} cr
& = 60{ ext{ hours}} cr
& left( {60 - frac{{119}}{2}}
ight){ ext{ hours}} cr
& { ext{ = }}frac{{120 - 119}}{2}{ ext{ hours}} cr
& { ext{ = }}frac{1}{2}{ ext{ hours}} cr
& = 30{ ext{ minutes}} cr} $$ Hence, the correct time is 30 minutes after 8:30 pm i.e., 9 pm
& { ext{ = }}left( {24 imes 2.5}
ight) - { ext{0:30 hours }} cr
& { ext{ = 60}} - { ext{0}}{ ext{:30 hours}} cr
& { ext{ = 59 hours 30 minutes}} cr
& = 59frac{{30}}{{60}} cr
& = 59frac{1}{2} cr
& = frac{{119}}{2}{ ext{ hours}} cr
& { ext{Also 24 hours}} - { ext{12 minutes}} cr
& = { ext{23 hours 48 minutes}} cr
& = 23 + frac{{48}}{{60}} cr
& = 23frac{4}{5} cr
& = frac{{119}}{5}{ ext{ hours}} cr
& herefore frac{{119}}{2}{ ext{ hours of this clock}} cr
& = frac{{24 imes 5}}{{119}} imes frac{{119}}{2} cr
& = 60{ ext{ hours}} cr
& left( {60 - frac{{119}}{2}}
ight){ ext{ hours}} cr
& { ext{ = }}frac{{120 - 119}}{2}{ ext{ hours}} cr
& { ext{ = }}frac{1}{2}{ ext{ hours}} cr
& = 30{ ext{ minutes}} cr} $$ Hence, the correct time is 30 minutes after 8:30 pm i.e., 9 pm
20:
D
Solution: There are 8 intervals in 9 tinging 10 intervals in 11 tinging. Time duration of 8 intervals = 9 seconds ∴ Required time = Duration of 10 intervals $$eqalign{
& = left( {frac{9}{8} imes 10}
ight){ ext{ seconds}} cr
& = 11.25{ ext{ seconds}} cr} $$
& = left( {frac{9}{8} imes 10}
ight){ ext{ seconds}} cr
& = 11.25{ ext{ seconds}} cr} $$
21:
B
Solution: $$eqalign{
& { ext{Time lost in }}6frac{1}{2}{ ext{ hours}} cr
& = { ext{ }}left( {6frac{1}{2} imes 4}
ight)sec cr
& = 26sec cr} $$ Correct time after $${ ext{6}}frac{1}{2}$$ hours = 7 hours 40 minutes 6 seconds + 6 hours 30 minutes = 14 hours 10 minutes 6 seconds Time show by the clock = 14 hours 10 minutes 6 seconds - 26 sec = 14 hours 9 minutes 40 seconds
& { ext{Time lost in }}6frac{1}{2}{ ext{ hours}} cr
& = { ext{ }}left( {6frac{1}{2} imes 4}
ight)sec cr
& = 26sec cr} $$ Correct time after $${ ext{6}}frac{1}{2}$$ hours = 7 hours 40 minutes 6 seconds + 6 hours 30 minutes = 14 hours 10 minutes 6 seconds Time show by the clock = 14 hours 10 minutes 6 seconds - 26 sec = 14 hours 9 minutes 40 seconds
22:
C
Solution: In 12 hours, they are at right angles 22 times. ∴ In 24 hours, they are at right angles 44 times.
23:
B
Solution: $$eqalign{
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{17}}{2},{ ext{hrs}} cr
& { ext{ = }},{left( {frac{{360}}{{12}} imes frac{{17}}{2}}
ight)^ circ } cr
& = 255^ circ cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},{ ext{30}},{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 30}
ight)^ circ } cr
& = 180^ circ cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {255 - 180}
ight)^ circ } = {75^ circ } cr} $$
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{17}}{2},{ ext{hrs}} cr
& { ext{ = }},{left( {frac{{360}}{{12}} imes frac{{17}}{2}}
ight)^ circ } cr
& = 255^ circ cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},{ ext{30}},{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 30}
ight)^ circ } cr
& = 180^ circ cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {255 - 180}
ight)^ circ } = {75^ circ } cr} $$
24:
B
Solution: The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o'clcok only). So, in a day, the hands point in the opposite directions 22 times.
25:
D
Solution: At 4 o'clock, the hands of the watch are 20 min. spaces apart. To be in opposite directions, they must be 30 min. spaces apart. ∴ Minute hand will have to gain 50 min. spaces. 55 min. spaces are gained in 60 min. 50 min. spaces are gained in $$eqalign{
& left( {frac{{60}}{{55}} imes 50}
ight){ ext{min}}{ ext{.}},{ ext{or}},54frac{6}{{11}},{ ext{min}}. cr
& herefore { ext{Required}},{ ext{time}} cr
& = 54frac{6}{{11}}{ ext{min}}{ ext{.}},{ ext{past}},4 cr} $$
& left( {frac{{60}}{{55}} imes 50}
ight){ ext{min}}{ ext{.}},{ ext{or}},54frac{6}{{11}},{ ext{min}}. cr
& herefore { ext{Required}},{ ext{time}} cr
& = 54frac{6}{{11}}{ ext{min}}{ ext{.}},{ ext{past}},4 cr} $$
26:
C
Solution: To be together between 9 and 10 o'clock, the minute hand has to gain 45 min. spaces. 55 min. spaces gained in 60 min. 45 min. spaces are gained in $$left( {frac{{60}}{{55}} imes 45}
ight){ ext{min}}{ ext{.}}{kern 1pt} ,{ ext{or}}{kern 1pt} ,49frac{1}{{11}}{kern 1pt} { ext{min}}.$$ ∴ The hands are together at $$49frac{1}{{11}}$$ min. past 9
ight){ ext{min}}{ ext{.}}{kern 1pt} ,{ ext{or}}{kern 1pt} ,49frac{1}{{11}}{kern 1pt} { ext{min}}.$$ ∴ The hands are together at $$49frac{1}{{11}}$$ min. past 9
27:
D
Solution: At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand. To be coincident, it must gain 15 min. spaces. 55 min. are gained in 60 min. 15 min. are gained in $$left( {frac{{60}}{{55}} imes 15}
ight)$$ xa0 min. = $$16frac{4}{{11}}$$ min. ∴ The hands are coincident at $$16frac{4}{{11}}$$ min. past 3
ight)$$ xa0 min. = $$16frac{4}{{11}}$$ min. ∴ The hands are coincident at $$16frac{4}{{11}}$$ min. past 3
28:
C
Solution: The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock). AM 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55 PM 12:00 1:05 2:11 3:16 4:22 5:27 6:33 7:38 8:44 9:49 10:55 The hands overlap about every 65 minutes, not every 60 minutes. ∴ The hands coincide 22 times in a day.
29:
C
Solution: In 12 hours, the hands coincide or are in opposite direction 22 times. ∴ In 24 hours, the hands coincide or are in opposite direction 44 times a day.
30:
B
Solution: Time from 12 pm on Monday to 2 pm on the following Monday
= 7 days 2 hours = 170 hours ∴ The watch gains [left( {2 + 4frac{4}{5}}
ight)] xa0 minutes or [frac{{34}}{5}] xa0 minutes in 170 hours. Now, $$frac{{34}}{5}$$ minutes are gained in 170 hours ∴ 2 minutes are gained in [left( {170 imes frac{5}{{34}} imes 2}
ight)] xa0 hours = 50 hours ∴ Watch is correct 2 days 2 hours after 12 pm on Monday
i.e., it will be correct at 2 pm on Wednesday.
= 7 days 2 hours = 170 hours ∴ The watch gains [left( {2 + 4frac{4}{5}}
ight)] xa0 minutes or [frac{{34}}{5}] xa0 minutes in 170 hours. Now, $$frac{{34}}{5}$$ minutes are gained in 170 hours ∴ 2 minutes are gained in [left( {170 imes frac{5}{{34}} imes 2}
ight)] xa0 hours = 50 hours ∴ Watch is correct 2 days 2 hours after 12 pm on Monday
i.e., it will be correct at 2 pm on Wednesday.
31:
D
Solution: Number of minutes in a day = (24 × 60) = 1440 ∴ Required percentage $$eqalign{
& = left( {frac{5}{{1440}} imes 100}
ight)\% cr
& = frac{{50}}{{144}}\% cr} $$
& = left( {frac{5}{{1440}} imes 100}
ight)\% cr
& = frac{{50}}{{144}}\% cr} $$
32:
D
Solution: $$eqalign{
& { ext{Angle}},{ ext{traced}},{ ext{be}},{ ext{the}},{ ext{hour}},{ ext{hand}},{ ext{in}},{ ext{6}},{ ext{hours}} cr
& = {left( {frac{{360}}{{12}} imes 6}
ight)^ circ } = {180^ circ } cr} $$
& { ext{Angle}},{ ext{traced}},{ ext{be}},{ ext{the}},{ ext{hour}},{ ext{hand}},{ ext{in}},{ ext{6}},{ ext{hours}} cr
& = {left( {frac{{360}}{{12}} imes 6}
ight)^ circ } = {180^ circ } cr} $$
33:
D
Solution: Angle traced by hour hand in $$frac{{125}}{{12}}$$ hrs $$eqalign{
& = {left( {frac{{360}}{{12}} imes frac{{125}}{{12}}}
ight)^ circ } cr
& = 312{frac{1}{2}^ circ } cr} $$ Angle traced by minute hand in 25 min $$eqalign{
& = {left( {frac{{360}}{{60}} imes 25}
ight)^ circ } cr
& = {150^ circ } cr} $$ $$eqalign{
& herefore { ext{Reflex angle}} cr
& = {360^ circ } - {left( {312frac{1}{2} - 150}
ight)^ circ } cr
& = {360^ circ } - 162{frac{1}{2}^ circ } cr
& = 197{frac{1}{2}^ circ } cr} $$
& = {left( {frac{{360}}{{12}} imes frac{{125}}{{12}}}
ight)^ circ } cr
& = 312{frac{1}{2}^ circ } cr} $$ Angle traced by minute hand in 25 min $$eqalign{
& = {left( {frac{{360}}{{60}} imes 25}
ight)^ circ } cr
& = {150^ circ } cr} $$ $$eqalign{
& herefore { ext{Reflex angle}} cr
& = {360^ circ } - {left( {312frac{1}{2} - 150}
ight)^ circ } cr
& = {360^ circ } - 162{frac{1}{2}^ circ } cr
& = 197{frac{1}{2}^ circ } cr} $$
34:
C
Solution: Angle traced by hour hand in 12 hrs = $${360^ circ }$$ Angle traced by hour hand in 5 hrs 10 min. i.e., $$eqalign{
& frac{{31}}{6}{ ext{hrs}} = {left( {frac{{360}}{{12}} imes frac{{31}}{6}}
ight)^ circ } cr
& ,,,,,,,,,,,,,,,,,, = {155^ circ } cr} $$
& frac{{31}}{6}{ ext{hrs}} = {left( {frac{{360}}{{12}} imes frac{{31}}{6}}
ight)^ circ } cr
& ,,,,,,,,,,,,,,,,,, = {155^ circ } cr} $$
35:
A
Solution: $$eqalign{
& 55,min .,{ ext{spaces}},{ ext{are}},{ ext{covered}},{ ext{in}},60,min cr
& 60,min .,{ ext{spaces}},{ ext{are}},{ ext{covered}},{ ext{in}} cr
& = left( {frac{{60}}{{55}} imes 60}
ight),min . cr
& = 65frac{5}{{11}},min . cr
& { ext{Loss}},{ ext{in}},64,min . cr
& = {65frac{5}{{11}} - 64} = frac{{16}}{{11}},min . cr
& { ext{Loss}},{ ext{in}},24,hrs. cr
& = left( {frac{{16}}{{11}} imes frac{1}{{64}} imes 24 imes 60}
ight),min. cr
& = 32frac{8}{{11}},min. cr} $$
& 55,min .,{ ext{spaces}},{ ext{are}},{ ext{covered}},{ ext{in}},60,min cr
& 60,min .,{ ext{spaces}},{ ext{are}},{ ext{covered}},{ ext{in}} cr
& = left( {frac{{60}}{{55}} imes 60}
ight),min . cr
& = 65frac{5}{{11}},min . cr
& { ext{Loss}},{ ext{in}},64,min . cr
& = {65frac{5}{{11}} - 64} = frac{{16}}{{11}},min . cr
& { ext{Loss}},{ ext{in}},24,hrs. cr
& = left( {frac{{16}}{{11}} imes frac{1}{{64}} imes 24 imes 60}
ight),min. cr
& = 32frac{8}{{11}},min. cr} $$
36:
D
Solution: When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart. At 7 o'clock, they are 25 min. spaces apart. ∴ Minute hand will have to gain only 5 min. spaces. 55 min. spaces are gained in 60 min. 5 min. spaces are gained in $$eqalign{
& = left( {frac{{60}}{{55}} imes 5}
ight){kern 1pt} min . cr
& = 5frac{5}{{11}}{kern 1pt} min . cr
& herefore { ext{Required time}} = 5frac{5}{{11}}{kern 1pt} min .{kern 1pt} ,{ ext{past}}{kern 1pt} 7 cr} $$
& = left( {frac{{60}}{{55}} imes 5}
ight){kern 1pt} min . cr
& = 5frac{5}{{11}}{kern 1pt} min . cr
& herefore { ext{Required time}} = 5frac{5}{{11}}{kern 1pt} min .{kern 1pt} ,{ ext{past}}{kern 1pt} 7 cr} $$
37:
B
Solution: At 5 o'clock, the hands are 25 min. spaces apart. To be at right angles and that too between 5.30 and 6, the minute hand has to gain (25 + 15) = 40 min. spaces. 55 min. spaces are gained in 60 min. 40 min. spaces are gained in $$eqalign{
& = left( {frac{{60}}{{55}} imes 40}
ight){kern 1pt} {kern 1pt} min . cr
& = 43frac{7}{{11}}{kern 1pt} {kern 1pt} min . cr
& herefore { ext{Required time}} = 43frac{7}{{11}}{kern 1pt} {kern 1pt} min .,{ ext{past}},5 cr} $$
& = left( {frac{{60}}{{55}} imes 40}
ight){kern 1pt} {kern 1pt} min . cr
& = 43frac{7}{{11}}{kern 1pt} {kern 1pt} min . cr
& herefore { ext{Required time}} = 43frac{7}{{11}}{kern 1pt} {kern 1pt} min .,{ ext{past}},5 cr} $$
38:
B
Solution: $$eqalign{
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{13}}{3},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{13}}{3}}
ight)^ circ } = {130^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},{ ext{20}},{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 20}
ight)^ circ } = {120^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {130 - 120}
ight)^ circ } cr
& = {10^ circ } cr} $$
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{13}}{3},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{13}}{3}}
ight)^ circ } = {130^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},{ ext{20}},{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 20}
ight)^ circ } = {120^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {130 - 120}
ight)^ circ } cr
& = {10^ circ } cr} $$
39:
C
Solution: $$eqalign{
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{21}}{4},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{21}}{4}}
ight)^ circ } = 157{frac{1}{2}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},15,{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 15}
ight)^ circ } = {90^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {157frac{1}{2}}
ight)^ circ } - {90^ circ } cr
& = 67{frac{1}{2}^ circ } cr} $$
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},frac{{21}}{4},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{21}}{4}}
ight)^ circ } = 157{frac{1}{2}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},15,{ ext{min}} cr
& = {left( {frac{{360}}{{60}} imes 15}
ight)^ circ } = {90^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {157frac{1}{2}}
ight)^ circ } - {90^ circ } cr
& = 67{frac{1}{2}^ circ } cr} $$
40:
C
Solution: $$eqalign{
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},12,{ ext{hrs}},{ ext{ = }},{ ext{36}}{{ ext{0}}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{it}},{ ext{in}},frac{{11}}{3},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{11}}{3}}
ight)^ circ } = {110^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},60,{ ext{min}},{ ext{ = }},{ ext{36}}{{ ext{0}}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{it}},{ ext{in}},{ ext{40}},{ ext{min}}. cr
& = {left( {frac{{360}}{{60}} imes 40}
ight)^ circ } = {240^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {240 - 110}
ight)^ circ } = {130^ circ } cr} $$
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{hour}},{ ext{hand}},{ ext{in}},12,{ ext{hrs}},{ ext{ = }},{ ext{36}}{{ ext{0}}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{it}},{ ext{in}},frac{{11}}{3},{ ext{hrs}} cr
& = {left( {frac{{360}}{{12}} imes frac{{11}}{3}}
ight)^ circ } = {110^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{min}}{ ext{.}},{ ext{hand}},{ ext{in}},60,{ ext{min}},{ ext{ = }},{ ext{36}}{{ ext{0}}^ circ } cr
& { ext{Angle}},{ ext{traced}},{ ext{by}},{ ext{it}},{ ext{in}},{ ext{40}},{ ext{min}}. cr
& = {left( {frac{{360}}{{60}} imes 40}
ight)^ circ } = {240^ circ } cr
& herefore { ext{Required}},{ ext{angle}} cr
& = {left( {240 - 110}
ight)^ circ } = {130^ circ } cr} $$
41:
B
Solution: Time from 7 a.m. to 4:15 p.m. = 9 hrs 15 min. = $$frac{{37}}{4}$$ hrs 3 min. 5 sec. of this c
ocl = 3 min. of the correct clock ⇒ $$frac{{37}}{{720}}$$ hrs. of this clock = $$frac{1}{{20}}$$ hrs of the correct clock ⇒ $$frac{{37}}{4}$$ hrs. of this clock = $$left( {frac{1}{{20}} imes frac{{720}}{{37}} imes frac{{37}}{4}}
ight)$$ xa0 xa0 hrs. of the correct clock = 9 hrs. of the correct clock ∴ The correct time is 9 hrs. after 7 a.m. i.e., 4 p.m.
ocl = 3 min. of the correct clock ⇒ $$frac{{37}}{{720}}$$ hrs. of this clock = $$frac{1}{{20}}$$ hrs of the correct clock ⇒ $$frac{{37}}{4}$$ hrs. of this clock = $$left( {frac{1}{{20}} imes frac{{720}}{{37}} imes frac{{37}}{4}}
ight)$$ xa0 xa0 hrs. of the correct clock = 9 hrs. of the correct clock ∴ The correct time is 9 hrs. after 7 a.m. i.e., 4 p.m.