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Mathematics [5 marks sum]

Graphical Representations · ICSE STD 10 (ICSE - English Medium). 18 questions.

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[5 marks sum]

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18 questions · timed · auto-graded

Question 15 Marks
Draw an ogive for the following :
Marks (More than) 0 10 20 30 40 50 60 70 80 90 100
Cumulative Frequency 100 87 65 55 42 36 31 21 18 7 0
Answer
Steps:
1. Start with lower limits of class intervals and from cumulative frequency, subtract the frequency of each class to obtain c.f distribution .
2. Mark lower class limits along $x$-axis $.1 \mathrm{~cm}=5$ units
3. Mark cumulative frequencies along y-axis. $1 \mathrm{~cm}=5$ units
4. Plot points $(x, f)$ where $x$ is the lower limit of one class and $f$ is the corresponding c.f. $(0,100),(10,87)$, $(20,65),(30,55),(40,42),(50,36),(60,31),(70,21),(80,18),(90,7),(100,0)$
5. Join the points to get the ogive.
Marks more than Cumulative Frequency
0 100
10 87
20 65
30 55
40 42
50 36
60 31
70 21
80 18
90 7
100 0
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Question 25 Marks
Draw an ogive for the following :
Marks obtained More than 10 More than 20 More than 30 More than 40 More than 50
No. of students 8 25 38 50 67
Answer
Steps:
1. Start with lower limits of class intervals and from cumulative frequency, subtract the frequency of each class to obtain c.f distribution.
2. Mark lower class limits along $x$-axis. $1 \mathrm{~cm}=5$ units
3. Mark cumulative frequencies along y-axis. $1 \mathrm{~cm}=5$ units
4. Plot points $(x, f)$ where $x$ is the lower limit of one class and $f$ is the corresponding c.f. $(10,188)$, $(20,180),(30,155),(40,117),(50,67)$
5. Join the points to get the ogive.
Marks more than Frequency Cumulative Frequency
10 8 188
20 25 180
30 38 155
40 50 117
50 67 67
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Question 35 Marks
Draw an ogive for the following :
Age in years Less than 10 Less than 20 Less than 30 Less than 40 Less than 50
No. of people 0 17 42 67 100
Answer
Steps :
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot age .
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequecies . In this case $(10,0),(20,17),(30,42),(40,67),(50,100)$.
4. Join the points plotted by a smooth curve.
Age less than Cumulative Frequency
10 0
20 17
30 42
40 67
50 100
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Question 45 Marks
Draw an ogive for the following :
Marks obtained Less than 10 Less than 20 Less than 30 Less than 40 Less than 50
No. of students 8 22 48 60 75
Answer
Steps :
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot marks.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies . In this case $(10,8),(20,22),(30,48),(40,60),(50,75)$.
4. Join the points plotted by a smooth curve.
Marks less than Cumulative Frequency
10 8
20 22
30 48
40 60
50 75
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Question 55 Marks
Draw an ogive for the following :
Class Interval 10-19 20-29 30-39 40-49 50-59
Frequency 28 23 15 20 14
Answer
Steps:
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies . In this case $(19,28),(29,51),(39,66),(49,86),(59,100)$.
4. Join the points plotted by a smooth curve.
less than Cumulative Frequency
19 28
29 51
39 66
49 86
59 100
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Question 65 Marks
Draw an ogive for the following :
Class Interval 100-150 150-200 200-250 250-300 300-350 350-400
Frequency 10 13 17 12 10 8
Answer
Steps :
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies . In this case $(150,10),(200,23),(250,40)(300,52)(350,62),(400,70)$.
4. Join the points plotted by a smooth curve.
less than Cumulative Frequency
150 10
200 23
250 40
300 52
350 62
400 70
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Question 75 Marks
Draw an ogive for the following :
Class Interval 0-10 10-20 20-30 30-40 40-50
Frequency 8 12 10 14 6
Answer
Steps:
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies . In this case $(10,8),(20,20),(30,30),(40,44),(50,50)$.
4. Join the points plotted by a smooth curve .
less than cumulative frequency
10 8
20 20
30 30
40 44
50 50
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Question 85 Marks
Construct a frequency polygon without using a histogram for the following frequency distribution :
Class Interval 1-10 11-20 21-30 31-40 41-50
Frequency 8 12 10 16 6
Answer
Steps:
1. Make the class intervals continuous by subtracting 0.5 from the lower limit of each class and add 0.5 to the upper limit of each class.
2. Find class mark by calculating the average of the class interval.
3. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
4. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
5. Plot the points on the graph. $(5.5,8),(15.5,12),(25.5,10),(35.5,16),(45.5,6)$.
6. Mark two more midpoints of zero frequency on $x$-axis at the start and at the end.
7. Now connect the points using straight lines.
Class Interval Class Mark Frequency
0.5-10.5 $
=\frac{0.5+10.5}{2}=5.5
$		 8
10.5-20.5 $
=\frac{10.5+20.5}{2}=15.5
$		 12
20.5-30.5 $
=\frac{20.5+30.5}{2}=25.5
$		 10
30.5-40.5 $
=\frac{30.5+40.5}{2}=35.5
$		 16
40.5-50.5 $
=\frac{40.5+50.5}{2}=45.5
$		 6
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Question 95 Marks
Construct a frequency polygon without using a histogram for the following frequency distribution :
Class Interval 10-20 20-40 40-60 60-80 80-100
Frequency 9 17 15 20 14
Answer
Steps:
1. Find class mark by calculating the average of the class interval.
2. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
3. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
4. plot the points on the graph. $(15,9),(30,17),(50,15),(70,20),(90,14)$.
5. Mark two more midpoints of zero frequency on $x$-axis at the start and at the end .
6. Now connect the points using staright lines.
Class Interval Class mark Frequency
10-20 $
=\frac{10+20}{2}=15
$		 9
20-40 $
=\frac{20+40}{2}=30
$		 17
40-60 $
=\frac{40+60}{2}=50
$		 15
60-80 $
=\frac{60+80}{2}=70
$		 20
80-100 $
=\frac{80+100}{2}=90
$		 14
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Question 105 Marks
Construct a frequency polygon without using a histogram for the following frequency distribution :
Class Mark 10 15 20 25 30 35 40
Frequency 4 20 40 45 30 25 5
Answer
Steps:
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Mark the given data on the graph. $(10,4),(15,20),(20,40),(25,45),(30,30),(35,25),(40,5)$
4. Mark two more midpoints of zero frequency on $x$-axis at the start and at the end.
5. Now connect the points using staright lines.
Class Mark Frequency
10 4
15 20
20 40
25 45
30 30
35 25
40 5
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Question 115 Marks
Construct a histogram and then a frequency polygon for the following frequency distribution.
Class Interval 100-150 150-200 200-250 250-300 300-350
Frequency 10 15 17 12 10
Answer
Steps :
1. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency .
3. Draw rectangles of histogram as per given data .
4. For each rectangle, mark the midpoint of its length at the top part. In this case, points are $(125.10),(175,15),(225,17),(275,12),(325,10)$.
5. Mark two more midpoints of Zero frequency on $x$-axis at the start and at th end.
6. Now connect the points using straight lines.
Class Interval Frequency
100-150 10
150-200 15
200-250 17
250-300 12
300-350 10
Image
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Question 125 Marks
Construct a histogram and then a frequency polygon for the following frequency distribution.
Class Interval 11-20 21-30 31-40 41-50 51-60 61-70
Frequency 15 28 50 35 20 12
Answer
1. Organize the data into equal intervals by subtracting 0.5 from lower limit of each class and add 0.5 to the upper limit of each class .
2. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval .
3. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
4. Draw rectangles of histogram as per given data.
5. For each rectangle, mark the midpoint of its length at the top part . In this case, points are $(15.5,15),(25.5,28),(35.5,50),(45.5,35),(55.5,20),(65.5,12)$.
6. Mark two more midpoints of zero frequency on $x$-axis at the start and at the end .
7. Now connect the points using straight lines.
Class Interval Frequency
10.5-20.5 15
20.5-30.5 28
30.5-40.5 50
40.5-50.5 35
50.5-60.5 20
60.5-70.5 12
Image
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Question 135 Marks
Construct a histogram and then a frequency polygon for the following frequency distribution.
Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 8 13 27 35 14 3
Answer
Steps:
1. On the $x$-axis , take $1 \mathrm{~cm}$ as 5 units and plot class interval.
2. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
3. Draw rectangle of histogram as per given data.
4. For each rectangle, mark the midpoint of its length at the top part . In this case, points are $(5,8)$, $(15,13),(25,27),(35,35),(45,14),(55,3)$.
5. Mark two more midpoints of zero frequency on $x$-axis at the start and at the end.
6. Now connect the points using straight lines.
Class Interval Frequency
0-10 8
10-20 13
20-30 27
30-40 35
40-50 14
50-60 3
Image
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Question 145 Marks
Construct histograms for following frequency distribution:
Class Mark 6 12 18 24 30 36
Frequency 8 12 15 18 25 7
Answer
Steps :
a. Make the class intervals continuous by taking class mark as midpoints.
b. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
c. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
d. Draw rectangles of histogram as per given data.
Class Interval Frequency
3-9 8
9-15 12
15-21 15
21-27 18
27-33 25
33-39 7
Image
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Question 155 Marks
Construct histograms for following frequency distribution:
Class Interval 110-119 120-129 130-139 140-149 150-159
Frequency 15 23 30 20 16
Answer
Steps:
a. Make the class intervals continuous by subtracting 0.5 from the lower limit of each class and add 0.5 to the upper limit of each class.
b. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class intervals.
c. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
d. Draw rectangles of histogram as per given data.
Class Interval Frequency
109.5-119.5 15
119.5-129.5 23
129.5-139.5 30
139.5-149.5 20
149.5-159.5 16
Image
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Question 165 Marks
Construct histograms for following frequency distribution:
Class Interval 1-10 11-20 21-30 31-40 41-50
Frequency 11 23 30 20 16
Answer
Steps :
a. Make the class intervals continuous by subtracting 0.5 from the lower limit of each class and add 0.5 to the upper limit of each class .
b. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
c. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
d. Draw rectangles of histogram as per given data.
Class Interval Frequency
0.5-10.5 11
10.5-20.5 23
20.5-30.5 30
30.5-40.5 20
40.5-50.5 16
Image
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Question 175 Marks
Construct histograms for following frequency distribution:
Class Interval 130-140 140-150 150-160 160-170 170-180
Frequency 24 16 29 20 11
Answer
Steps :
a. On the $x$-axis , take $1 \mathrm{~cm}$ as 5 units and plot class interval.
b. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency.
c. Draw rectangles of histogram as per given data .
Class Interval Frequency
130-140 24
140-150 16
150-160 29
160-170 20
170-180 11
Image
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Question 185 Marks
Construct histograms for following frequency distribution:
Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 8 20 34 22 10 6
Answer
Steps:
a. On the $x$-axis, take $1 \mathrm{~cm}$ as 5 units and plot class interval.
b. On the $y$-axis, take $1 \mathrm{~cm}$ as 5 units and plot frequency .
c. Draw rectangle of histogram as per given data.
Class Interval Frequency
0-10 8
10-20 20
20-30 34
30-40 22
40-50 10
50-60 6
Image
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[5 marks sum] - Mathematics STD 10 Questions - Vidyadip