Vedanta Excel in Mathematics Book - 8 Final (2078)

Coordinates

Example 2: A point G is on the x-axis 8 units right from the origin. Another point H
is on the y-axis 6 units above the origin. Find the distance between the
points G and H.

Solution: Y

Here, the coordinates of the point G is (8, 0) H
and that of H is (0, 6)

So, x1 = 8, x2 = 0, y1 = 0, y2 = 6 X' O(0, 0) GX
Now, by using distance formula, Y'
GH = (x2 – x1)2 + (y2 – y1)2

= (0 – 8)2 + (6 – 0)2
= 64 + 36
= 100 = 10 units

? The distance between the point G and H is 10 units.

Example 3: The centre of a circle is O (5, 8) and A (12, 9) is any point in its
circumference. Find the diameter of the circle. Does another point
B (9, 4) lie in the circumference of the circle?

Solution:
Here, O (5, 8) is the centre of the circle. A (12, 9) is a point in its circumference.

So, x1 = 5, x2 = 12, y1 = 8 and y2 = 9. A (12, 9)
Now, by using distance formula, O (5, 8)
OA = (x2 – x1)2 + (y2 – y1)2

= (12 – 5)2 + (9 – 8)2
= 72 + 12
= 50 = 25 × 2 = 5 2 units

? The radius of circle (OA) = 5 2 units

? The diameter of the circle = 2(OA) = 2 × 5 2 = 10 2 units.

Again, to find the distance between O (5, 8) and B (9, 4)
OB = (9 – 5)2 + (4 – 8)2 = 42 + (–4)2 = 32 = 16 × 2 = 4 2 units

? OA ≠ OB.

So, OB is not the radius of the given circle. The point B (9, 4) does not lie in the
circumference of the circle.

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Coordinates

Example 4: Show that the points A (5, 3), B (1, 2), C (2, –2), and D (6, –1) are the

vertices of a square.

Solution: D(6, –1) C(2, –2)

Let, A (5, 3), B (1, 2), C (2, –2), and D (6, –1) are the vertices

of a quadrilateral ABCD. A(5, 3) B(1, 2)
Now, by using the distance formula,

AB = (x2 – x1)2 + (y2 – y1)2 = (1 – 5)2 + (2 – 3)2 = (–4)2 + (–1)2 = 17 units

BC = (x2 – x1)2 + (y2 – y1)2 = (2 – 1)2 + (–2 – 2)2 = 12 + (–4)2 = 17 units

CD = (x2 – x1)2 + (y2 – y1)2 = (6 – 2)2 + (–1+ 2)2 = 42 + 12 = 17 units

DA = (x2 – x1)2 + (y2 – y1)2 = (5 – 6)2 + (3+ 1)2 = (–1)2 + 42 = 17 units

Also, diagonal AC = (x2 – x1)2 + (y2 – y1)2 = (2– 5)2 + (–2 – 3)2 = 34 units

And, diagonal BD = (x2 – x1)2 + (y2 – y1)2 = (6– 1)2 + (–1 – 2)2 = 34 units

Thus, in quadrilateral ABCD,
The sides AB = BC = CD = DA = 17 units
Also, the diagonals AC = BD = 34 units

Here, all sides of the quadrilateral ABCD are equal and it's diagonals are also equal.

So, the quadrilateral ABCD is a square and the given points are the vertices of the
square.

Example 5: Show that the points (4, 3), (3, 2), and C (2, 1) are collinear.

Solution: A (4, 3) B (3, 2) C (2, 1)
Let the points be A (4, 3), B (3, 2), and C (2, 1).

Now, by using distance formula,
AB = (x2 – x1)2 + (y2 – y1)2 = (3 – 4)2 + (2 – 3)2 = (–1)2 + (–1)2 = 2 units
BC = (x2 – x1)2 + (y2 – y1)2 = (2 – 3)2 + (1 – 2)2 = (–1)2 + (–1)2 = 2 units
AC = (x2 – x1)2 + (y2 – y1)2 = (2 – 4)2 + (1 – 3)2 = (–2)2 + (–2)2 = 8 = 2 2 units
Here, AB + BC = 2 + 2 = 2 2 = AC

Hence, the given points are collinear.

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Coordinates

Example 6: If the distance between the points (a, 0) and (0, 3) is 5 units, find the
value of a.
Solution: Q (0, 3)

Let P (a, 0) and Q (0, 3) be the two given points. P (a, 0)
Here, PQ = 5 units. Squaring both sides
Now, by using distance formula,
52 = ( a2 + 9)2
PQ = (x2 – x1)2 + (y2 – y1)2 or, 25 = a2 + 9
or, 5 = (0 – a)2 + (3 – 0)2 or, a2 = 25 – 9
or, 5 = (–a)2 + 32 or, a2 = 16
or, 5 = a2 + 9
or, a = 16 = 4

So, the required value of a is 4.

EXERCISE 18.2
General Section

1. Say and write the distance between the given points.
a) (m1, n1) and (m2, n2) = ..........................
b) (0,0) and (3, 4) distance = ..........................
c) (0,0) and (4, 3) distance = ..........................
d) (0,0) and (1, 3) distance = ..........................
e) (0,0) and (2, 3) distance = ..........................

Creative Section - A

2. Find the distance between the following points.

a) (1, 2) and (3, 4) b) (5, 3) and (1, 6) c) (–2, 1) and (4, 3)

d) (4, 5) and (–3, 6) e) (–5, –4) and (–13, 2)

f) (5 + 3, 2 – 3) and (7 + 3, 2 + 3)

3. a) A point P is on the x-axis, 6 units right from the origin and another point Q is
on the y-axis, 8 units above the origin. Find the distance between P and Q.

b) A point A lies on the x-axis 9 units left from the origin and another point B lies
on the y-axis 12 units below the origin. Find the distance between A and B.

c) Find the distance between origin and a point P (–5, 5).

Creative Section - B

4. a) When the map of Nepal is presented in a coordinate plane, the coordinates of
Kathamndu is found to be (5, 3) and that of Pokhara (4, 2 3). Find the map
distance between these places. If 1 unit represents 100 km, find the actual
distance of Pokhara from Kathmandu.

b) Biratnagar lies at the point (1, 5) and Dharan lies at (3, 9) when the map
of Nepal is placed on the graph of coordinate plane. If 1 unit represents
4.5 5 km, find the actual distance between these two places.

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Y

c) The adjoining figure is the Tikapur
map of Nepal presented in the Bhairahawa
coordinate axes of the graph.
Find the distance between X
Tikapur and Bhairahawa. If
1 unit represents 18 5 km,
find the actual distance of
Tikapur to Bhairahawa.

O

Y

d) From the given graph of Hetauda
map of Nepal, find the Damak
actual distance between
Hetauda and Damak, if 1 unit
represents 4.1 85 km.

OX

5. a) The centre of a circle lies at O (4, 6) and A (–5, 18) is any point in its
circumference.

(i) Find the radius of the circle.

(ii) Find the diameter of the circle.

(iii) Show that B (13, –6) also lies in the circumference of the circle.

b) P (1, 3) is the centre of a circle and a point Q (7, 11) is any point in its
circumference. Find the diameter of the circle. Does another point (–5, 10) lie
in the circumference of the circle?

6. a) Show that the points A (3, 4), B (7, 8), and C (11, 4) are the vertices of an
Isosceles triangle.

b) Show that the points P (–6, 2), Q (1, 7), and R (6, 3) are the vertices of a scalene
triangle.

c) Show that the points A (2, –2), B (8, 4), C (5, 7), and D (–1, 1) are the vertices
of a rectangle ABCD.

d) Show that the points P (1, 1), Q (4, 4), R (4, 8), and S (1, 5) are the vertices of
a parallelogram.

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Coordinates

e) Show that the points A (3, 1), B (8, 1), C (8, 6), and D (3, 6) are the vertices of
a square.

f) Show that the points (4, 6), (–1, 5), (–2, 0), and (3, 1) are the vertices of a
rhombus.

g) Show that the points P (1, 1), Q (–1, –1), and R (– 3, 3) are the vertices of an
equilateral triangle.

7. Show that the following points are collinear.

a) (1, 2), (4, 5), (8, 9) b) (–3, 7), (0, 4), (2, 2)

c) (–1, –1), (2, 3), (8, 11) d) (4, –3), (2, 1), (–1, 7)

8. a) If the distance between the points (a, 3) and (8, 3) is 7 units, find the
value of a.

b) If the distance between the points (0, 5) and (a, 0) is 13 units, find the value

of a. Y
It's your time - Project work!

9. a) Let's draw three straight D A
10. a) lines of length in whole X
number of centimetres (say X' O
3 cm, 4 cm, 5 cm, ... ) in B
a squared graph paper by
using a ruler. Now, write the
coordinates of the joining
points of each straight line
and find the lengths by using
distance formula.

Are the lengths given by C Q
distance formula and the P
lengths measured by a ruler
equal? Y'

Let's draw the following plane shapes in the separate squared graph papers.

(i) An equilateral triangle (ii) An isosceles triangle

(iii) A square (iv) A rectangle

Then write the coordinates of the vertices of these shapes and find the length
of sides of each plane shape by using distance formula. Also find the length of
diagonals in the case of square and rectangle.

b) Let's draw a circle with centre at origin in a squared graph paper by using a
pencil compass. Then find the length of it's any 3 radii and a diameter. Show
that radii of the same circle are equal and diameter is two times the radius of
the circle.

c) Let's draw two right angled triangles in a squared graph paper by using a
ruler. Write the coordinates of the vertices of each triangle. Find the
length of perpendicular, base and hypotenuse by using distance formula.
Show that h2 = p2 + b2 in each of the right-angled triangles.

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Unit Statistics

22

22.1 Review

Statistics is a branch of mathematics in which data of facts and information are collected,
sorted, displayed, and analysed. Statistics are used to make decision and prediction
about the future plans and policies. Statistical information helps to understand the
economic problems and formulation of economic policies.

The word ‘statistics’ comes from the word ‘state’, largely because it was the job of the
state to keep records and make decisions based on census result.

22.2 Collection of data

The marks obtained by 24 students of class 8 in Mathematics out of 20 full marks are
given below.

15, 10, 17, 14, 19, 12, 15, 10, 16, 20
14, 18, 15, 10, 17, 16, 18, 17, 20, 13

Such numerical figures are called data.

Data should be presented in a proper order so that it is easier to get the necessary
information for which they are collected. When the data are arranged either in ascending
or in descending order, they are said to be in proper order. The properly arranged data
are called arrayed data; otherwise, they are said to be raw data.

22.3 Frequency table

The data given below are the hourly wages (in Rs) of 20 workers of a factory:

75, 90, 65, 80, 70, 65, 75, 75, 80, 70, o Raw data
75, 80, 90, 80, 70, 70, 75, 75, 70, 75

65, 65, 70, 70, 70, 70, 70, 75, 75, 75, o Arrayed data
75, 75, 75, 75, 80, 80, 80, 80, 90, 90

Here, Rs 65 is repeated 2 times. So, its frequency is 2.
Rs 70 is repeated 5 times. So, its frequency is 5.
Rs 75 is repeated 7 times. So, its frequency is 7.
Rs 80 is repeated 4 times. So, its frequency is 4.
Rs 90 is repeated 2 times. So, its frequency is 2.

In this way, a frequency is the number of times a value occurs. Data and their frequencies
can be presented in a table called frequency table.

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Statistics
Now, let’s present the above wages of 20 workers in a frequency table.

Wages (in Rs) Tally marks Frequency
65 || 2
70 |||| 5
75 |||| || 7
80 ||| 4
90 || 2
20
Total

Tallying is a system of showing frequencies using diagonal lines grouped in fives. Each
time five is reached, a horizontal line is drawn through the tally marks to make a group
of five. The next line starts a new group. For example,

1 o| 6 o |||| |
2 o || 7 o |||| ||
3 o ||| 8 o |||| |||
4 o |||| 9 o |||| ||||
5 o |||| 10 o |||| ||||

22.4 Grouped and continuous data

Let following are the marks obtained by 20 students in a Mathematics exam of full
marks 50.

27, 38, 25, 18, 9, 24, 48, 15, 27, 35

23, 45, 32, 16, 26, 39, 20, 33, 40, 37

The above mentioned data are called individual or discrete data. Another way of
organizing data is to present them in a grouped form. For grouping the given data, we
should first see the smallest value and the largest value, then we have to divide the data
into an appropriate class–interval. The numbers of values falling within each class–
interval give the frequency. For example,

Marks Tally marks Frequency
0 – 10 | 1
10 – 20 ||| 3
20 – 30 |||| || 7
30 – 40 |||| | 6
40 – 50 ||| 3
Total 20

In the above series, 9 is the smallest value and 48 is the largest value. So, the data are
grouped into the interval of 0 – 10, 10 – 20, etc., so that the smallest and the largest
values should fall in the lowest and the highest class–interval respectively.

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Statistics

Let’s consider a class–interval 10 – 20.
Here, 10 is called the lower limit and 20 is the upper limit of the class–interval.
The difference between two limit is called the length or height of each class–interval.
For example, in 10 – 20 the length of the class–interval is 10.
Again, let’s take class–intervals, 0 – 10, 10 – 20, 20 – 30, …
Here, the upper limit of a pervious class–interval has repeated as the lower limit of the
consecutive next class–interval. Such an arrangement of data is known as grouped and
continuous data.

22.5 Cumulative frequency table

The word ‘cumulative’ is related to the word ‘accumulated’, which means to ‘pile up’.
The table given below shows the marks obtained by 20 students in a mathematics test
and the corresponding cumulative frequency of each class–interval

Marks Frequency Cumulative o 2 students obtained marks less than 10.
(f) frequency (c.f) o 7 students obtained mark less than 20.
0 – 10 o 15 students obtained marks less than 30.
10 – 20 2 2 o 19 students obtained marks less then 40
20 – 30 5 5+2=7 o 20 students obtained marks less than 50.
30 – 40 8 8 + 7 = 15
40 – 50 4 15 + 4 = 19
1 19 + 1 = 20
Total
20

Thus, cumulative frequency corresponding to a class–interval is the sum of all frequencies
up to and including that class–interval.

22.6 Graphical representation of data

We have already discussed to present data in frequency distribution tables. Alternatively,
we can also present data graphically. Different types of diagrams are used for this
purpose. Here, we shall discuss two types of diagrams: line graph and pie chart.

(i) Line graphs

Data can also be represented by plotting the corresponding frequencies in the graph
paper. The line so obtained by joining the points is called the line graph.

While constructing a line graph, the frequencies of the items are plotted along y–axis.
The line graph given below represents the daily wags of the workers of a company.

Wages in Rs. 40 50 60 70 80 90 100

No. of workers 15 8 22 18 10 8 5

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Statistics

Y

25

20

No. of workers 15

10

5

0 40 50 60 70 80 90 100 X
wages (in Rs)

EXERCISE 22.1

General Section – Classwork

Lets say and write the answers as quickly as possible.

1.a) A branch of mathematics which deals with the collection, presentation, analysis
and interpretation of data is known as ............................................

b) The properly arranged data are called ............................................

c) The number of times that a particular observation occurs in the data
is the ...........................................

d) In a class interval 40 - 50, the lower limit is ....................................... and the
upper limit is ....................................

2. The adjoining line graph shows 120
the average rain fall in mm during
6 months of the year 2020.

(i) In which month was the 100
minimum average rainfall? 80
On which month was it
maximum?

................................................ Rainfall (mm) 60
40
(ii) How much was the average
rainfall recorded in August?

................................................

(iii) Write a paragraph about the 20
common trend of rainfall
during these months in your 0 Apr May June July Aug Sep
exercise book. Months

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Statistics

Creative Section

3. a) From the marks given below obtained by 20 students in Mathematics, construct
a frequency distribution table with tally marks.

15, 18, 12, 16, 18, 10, 15, 16, 15, 12

10, 12, 15, 12, 16, 18, 12, 15, 12, 16
b) The marks obtained by 40 students in mathematics in SEE examination are

given below. Group the data into the class intervals of length 10 and construct a
cumulative frequency distribution table.
42, 68, 80, 45, 92, 36, 8, 17, 49, 30
5, 26, 98, 74, 53, 65, 72, 28, 55, 46
86, 70, 62, 27, 16, 44, 85, 59, 51, 73
66, 78, 38, 81, 97, 77, 69, 45, 33, 67

c) The marks obtained by 50 students of class 8 in Mathematics are shown in the
table given below. Construct a cumulative frequency table to represent the data.

Marks 10 20 30 40 50

No. of students 4 9 20 15 2

d) The hourly wages of 40 workers in a factory are shown in the table given below.
Show their wages in a cumulative frequency table.

Wages (in Rs.) 50 60 70 80 90

No. of workers 7 14 11 5 3

4. a) Class 8 students conducted a survey and recorded the number of children in
primary level in their school in the following table. Construct a line graph to
show the numbers in different classes.

Classes I II III IV V
No. of children 25 20 40 50 45

b) The table given below shows the velocity of a bus at different interval of time.
Draw a line graph to show the velocity–time graph.

Time (in second) 5 10 15 20 25 30
Velocity (in m/s) 5 15 20 30 10 25

It’s your time - Project work!

5. a) Let’s collect and write the marks obtained by your friends in the recently
conducted mathematics exam. Group the data into the class interval of length
10 and show them in a cumulative frequency distribution table.

b) Let’s collect and write the number of students from class 1 to 8 in your school.
Show the data in a line graph.

(ii) Pie chart

A pie chart is a circular statistical graphic which is divided into sectors and the angles
of the sectors represent the frequency.
Constructing pie–charts

Follow these steps to draw a pie chart.

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Statistics

– Add all the frequencies and write each frequency as a fraction of the total frequency.

– Change each fraction into a number of degrees multiplying by 360q. (There are 360q

in a circle) corresponding frequency
Total frequency
i.e. number of degrees of each fraction = × 360q

– Tabulate the angles in ascending or descending order.

– Draw a circle of convenient size. Then draw a radius as a starting point.

– Use a protractor to construct the angles at the centre corresponding to each sector.

Worked-out examples

Example 1: The table given below shows the number of students in classes 1 to 5 of
a school. Draw a pie chart to represent the numbers.

Class I II III IV V

No. of students 40 50 30 45 15

Solution:

Here, total number of students = 40 + 50 + 30 + 45 + 15 = 180

Number of degrees of the sector of class I = 40 × 360° = 80° Class II
180 100°
Class I
Number of degrees of the sector of class II = 50 × 360° = 100° Class III 80°
180

Number of degrees of the sector of class III = 30 × 360° = 60° 60° 30° Class V
180 90°

Number of degrees of the sector of class IV = 45 × 360° = 90° Class IV
180

Number of degrees of the sector of class V = 15 × 360° = 30°
180

Example 2: The table given below shows the monthly income of a family from
different sources. Represent the data in a pie chart.

Source House rent Business Agriculture Salary Others

Income (in Rs) 1500 1700 1200 2500 300

Solution:

Here, total monthly income = Rs 7200

Number of degrees of the sector of house rent = 1500 × 360° = 75° Salary
Number of degrees of the sector of business 7200 125°
Business
Number of degrees of the sector of agriculture = 1700 × 360° = 85° 85° Othe1r5°
7200
Agriculture
House rent 60°

75°

Number of degrees of the sector of salary = 1200 × 360° = 60°
7200
2500
Number of degrees of the sector of others = 7200 × 360° = 125° = 300 × 360° = 15°
7200

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Now, tabulating the angles in descending order: Statistics

Source Salary Business House rent Agriculture Others
60q 15q
Angle 125q 85q 75q

Example 3: The monthly budget of a family is shown in Education Food
the given pie chart. If the total expenditure is 95° 120°
Rs 10800, calculate the expenditure on each item
and show in a table. Rent Miscellaneous
75° 30°
Solution:
Here, total expenditure = Rs 10800 Fuel
40°

Expenditure on food = Rs 10800 × 120 Whole circle represents the total
360 expenditure.
i.e. 360q represents Rs 10800
= Rs 3600
10800
Expenditure on education = Rs 10800 × 95 1q represents Rs 360
360
120q represents Rs 10800 × 120
= Rs 2850 360

Expenditure on rent = Rs 10800 × 75 = Rs 2250
360

Expenditure on fuel = Rs 10800 × 40 = Rs 1200
360

Expenditure on miscellaneous = Rs 10800 × 30 = Rs 900
360

Now, the table given below shows the actual expenditure.

Items Food Education Rent Fuel Miscellaneous
Expenditure (Rs) 3600 2850 2250 1200 900

EXERCISE 22.2 Farming
General Section – Classwork 180°

1. The given pie-chart shows the income of a family from three Business Services
different sources. Answer the following questions. 60°

120°

a) How many degrees represent the total income of the family? ..............................

b) From which source does the family have maximum income? ..............................

c) From which source does the family have minimum income? ..............................

d) If the annual income of the family is Rs 4,50,000, how much is the income from
farming? ..............................

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Statistics

Creative Section

2. a) The table given below shows the number of students of a school from class 4 to 8.
Represent the data in a pie chart.

Class IV V VI VII VIII

No. of students 50 45 40 35 10

b) The monthly budget of a family is given below.

Food – Rs 2100 Clothing – Rs 3300

Miscellaneous – Rs 2250 Saving – Rs 3150

Represent the above data in a pie chart.

c) Mr. Limbu’s spends Rs 10,800 in a month. His monthly expenditure on different

headings are given below:

Food – Rs 2,400 Education – Rs 3,600

Rent – Rs 4,200 Miscellaneous – Rs 600

Show the above data in a pie chart.

d) Represent the following data in a pie chart. Monthly expenditure of a family.

Item Education House rent Food Others

Amount (Rs) 8000 4000 10000 14000

e) World consumption of energy is given below:

Natural gas – 20 % Nuclear – 5 %

Coal – 25 % Hydroelectric – 10 %

Oil – 40 %

Represent the above data in a pie chart.

3. a) Mrs. Rai’s monthly expenditure is Rs 10,800. The Food Rent
diagram on the right is a pie chart showing her 150° 70°
expenditure on different headings. Work out how
much was spent under each heading. 30° Miscellaneous

60° 50°
Study Transportation

b) The diagram on the right is a pie chart showing the expenses Wages Fuel
of a small manufacturing firm. The total expenses were 150° 40°
Rs 1,44,000 in a month. Calculate the expenditure on
each heading. Raw Rent
materials 50°

120°

c) The pie chart given alongside shows the votes secured by YX
three candidates A, B and C in an election. If A secured 120°
5,760 votes,
i) how many votes did C secure? Z
140°
ii) who secured the least number of votes? How many votes
did he secure?

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Statistics

d) The given pie chart shows the composition of different Cotton
materials in a type of cloth in percent. 90° Nylon
54°
i) Calculate the percentage of each material found in the
cloth. 144° 72°
Polyester Others
ii) Calculate the weight of each material contained by a
bundle of 50 kg of cloth.

It’s your time - Project work!

4. a) Make a group of your friend. Take the number of students from classes 1 to 5

separately and present the data you obtained in pie-chart.

b) Collect the number of students of your class who secured A+, A, B+, B, etc.
grades in Mathematics exam and show the data in pie-chart.

22.7 Measures of central tendency

The measure of central tendency gives a single central value that represents the
characteristics of entire data. A single central value is the best representative of the
given data towards which the values of all other data are approaching.

Average of the given data is the measure of central tendency. There are three types of
averages which are commonly used as the measure of central tendency. They are: mean,
median, and mode.

22.8 Arithmetic mean

Arithmetic mean is the most common type of average. It is the number obtained by
dividing the sum of all the items by the number of items.

i.e. mean = sum of all the items
the number of items

(i) Mean of non–repeated data

If x represents all the items and n be the number of items, then mean (x) = ∑x
n

(ii) Computation of combined mean

We can compute a single mean from the means of different sets of data. Such mean is

called a combined mean. and x1 be its mean. Also,
its mean. If the combined
Lneutm, nb1erbeofthiteemnusminbethr eofseitceomnsd in the first set of data nm2 ebaenthbee
x , then, set of data and x2 be

x= n1 x1 + n2 x2
n1 + n2

Worked-out examples

Example 1: Calculate the average of the following marks obtained by 10 students
of a class in mathematics.
25, 18, 35, 24, 15, 20, 33, 28, 22, 30

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Statistics

Solution:

Here, 6x = 25 + 18 + 35 + 24 + 15 + 20 + 33 + 28 + 22 + 30 = 250

n = 10 ∑x 250
n 10
Now, mean (x) = = = 25

Example 2: The average weight of 90 students from class I to V of a school is 20 kg

and the average weight of 60 students from class VI to X is 35 kg. Find

average weight of the students of the school.

Solution:
Here, n1 = 90 and x1= 20 kg , n2 = 60 and x2 = 35 kg

? Combined mean x = n1 x1 + n2 x2
n1 + n2

90 × 20 + 60 × 35 = 1800 + 2100 = 26 kg
= 90 + 60 150

Hence, the average weight of the students of the school is 26 kg.

Example 3: If the average of the following wages received by 5 workers is Rs 35,

find the value of p.
30, 36, p, 40, 44.

Solution:
Here, 6x = 30 + 36 + p + 40 + 44 = 150 + p

n =5 ∑x
n
Now, average =

or, 35 = 150 + p
5

or, 150 + p = 175
or, p = 25

So, the required value of p is Rs 25.

(ii) Mean of individual repeated data (Mean of a frequency distribution)

In the case of repeated data, follow the steps given below to calculate the mean.

– Draw a table with 3 columns

– Write down the items (x) in ascending or descending order in the first column and
the corresponding frequencies in the second column.

– Find the product of each item and its frequency (fx) and write in the third column.

– Find the total of f column and fx column.

– Divide the sum of fx by the sum of f (total number of items), the quotient is the
required mean.

Example 4: From the table given below, calculate the mean mark.

Age (in years) 6 7 8 9 10 11

No. of students 3 5 4 9 7 2

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Solution:
Calculation of average age:

Marks (x) No. of students (f) fx
6 3 18
7 5 35
8 4 32
9 9 81
10 7 70
11 2 22
6fx =258
Total N = 30

Now, mean mark (x) = ∑fx = 258 = 8.6 years
N 30

So, the required average age is 8.6 years.

Example 5: If the mean of the data given below be 17, find the value of m.

x5 10 15 20 25 30
f2 5 10 m 4 2

Solution: f fx
x 2 10
5 5 50
10 10 150
15 m 20m
20 4 100
25 2 60
30 N = 23 + m 6fx = 370 + 20m

Total

Now, mean (x) = ∑fx
N

or, 17 = 370 + 20 m
23 + m

or, 391 + 17m = 370 + 20 m

or, 3m = 21

or, m = 7

So, the required value of m is 7.

(iii) Mean of grouped and continuous data

In the case of grouped and continuous data, we should find the mid–values (m) of
each class interval and it is written in the second column. The mid–value of each class

interval is obtained as: Mid–value = lower limit + upper limit
2

Then, each mid–value is multiplied by the corresponding frequency and the product
fm is written in the fourth column. The process of calculation of mean is similar to the
above mentioned process.

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Example 6: Calculate the mean from the table given below.

Solution: Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
No. of students 3 8 12 7 2

Calculation of mean

Marks (x) mid–value (m) No. of students (f) fm

0 – 10 5 3 15
10 – 20 15 8 120
20 – 30 25 12 300
30 – 40 35 7 245
40 – 50 45 2 90

Total N = 32 6fm = 770

Now, mean marks (x) = ∑ fx = 770 = 24.06
N 32

EXERCISE 22.3
General Section – Classwork

1. Let’s say and write the answers as quickly as possible.

a) Average of 4 and 6 is ........................ b) Average of 2, 5 and 9 is .......................

c) Average of 4, 6, 8, 10 is ........................ d) Average of 3 and x is 5, x = ..................

e) Average of p and 4 is 3, p = ...............................

2. a) 6fx = 50 , n = 5, x = .............. b) 6fx = 60, n = 10, x = ..............

c) 6fx = 80, n = 8, x = .............. d) 6fx = 150, n = 25, x = ..............

e) 6fx = 40, x = 5, n = ............... f) 6fx = 70, x = 14, n = ..............

Creative Section A

3. a) The ages of Ram, Hari, Shyam, Krishna, and Gopal are 12, 18, 13, 16, and 6 years
respectively. Find their average age.

b) Find the mean value of the following:
5, 11, 14, 10, 8, 6

c) Find the mean from the data given below.
57, 74, 83, 76, 60

4. a) If mean (x) = 9, 6x = 80 + p and N = 10, find the value of p.
b) In an individual series, if 6x = 60 + a, N = a – 4 and mean (x) = 5, find the
value of a.
c) If 6x = 400 – m, N = 18 + m and x = 10, find the value of m.

5. a) The average age of 5 students is 9 years. Out of them the ages of 4 students are
5, 7, 8 and 15 years. What is the age of the remaining student?

b) If 7 is the mean of 3, 6, a, 9, and 10, find the value of a.

c) Find x if the mean of 2, 3, 4, 6, x, and 8 is 5.

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Statistics

6. a) The average rainfall in Kathmandu valley in the first 6 months of the year 2077
was 75 mm and the average rain fall of the last 6 months was 15 mm. Find the
average rainfall in Kathmandu in the whole year.

b) The average weight of 30 girls in class 8 is 42 kg and that of 20 boys is 45 kg.
Find the average weight of the students of the class.

c) The average height of x number of girls and 15 boys is 123 cm. If the average
height of boys is 125 cm and that of girls is 120 cm, find the number of girls.

7. a) If 6fx = 40 + a, N = 4 + a and x = 5, find the value of a.

b) If the mean of a series having 6fx = 100 – k and N = k – 4 is 15, find the value
of k.

Creative Section - B

8. a) Find the mean from the given table

Marks obtained 15 25 35 45 55
No. of students 7 8 12 7 6

b) The ages of the students of a school are given below. Find the average age.

Age (in years) 5 8 10 12 14 16

No. of students 20 16 24 18 25 15

c) Compute the arithmetic mean from the following frequency distribution table.

Height (in cm) 58 60 62 64 66 68

No. of plants 12 14 20 13 8 5

9. a) Find the mean of the following frequency distribution.

Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60

Frequency 75 6 12 8 2

b) Find mean.

Wages (Rs.) 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100

No. of workers 3 6 12 7 8 4

c) Find the mean from the following data.

Marks obtained 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80

No. of students 4 5 2 4 3 2

d) Compute the mean from the table given below.

Age (years) 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No. of people 2 5 7 6 3 2

e) The table below gives the daily earnings of 110 workers in a textile mill.

Daily earning 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700

No. of workers 40 20 15 25 10

Find the average weekly earning.

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Statistics

f) The ages of workers in a factory are as follows

Age in years 18 – 24 24 – 30 30 – 36 36 – 42 42 – 48 48 – 54

No. of workers 6 8 12 8 4 2

Calculate the average age of the groups.
10. a) Find the mean by constructing a frequency table of class interval of 10 from the

data given below.

7, 47, 36, 39, 31, 19, 41, 49, 9, 51, 29, 22,

59, 17, 49, 21, 24, 12, 31, 8, 36, 18, 32, 16, 23
b) Construct a frequency table of class interval of 10 from the given data and find

the mean.
23, 5, 17, 28, 39, 52, 16, 22, 69, 75

41, 33, 9, 49, 34, 59, 72, 46, 65, 58

60, 48, 64, 32, 50, 73, 57, 51, 63, 36

It’s your time - Project work!

11. a) Let’s ask your parents and write the ages of your family members. Then, calculate
the average age of your family members.

b) Let’s collect the marks obtained by your any 10 friends in the recently conducted
mathematics exam and calculate the mean mark.

c) Let’s collect and write the number of students in each class from class 1 to 8
in your school. Then find the average number of students in each class in your
school.

22.9 Median

Look at the following series. 5, 9, 13, 17 21, 25, 29
3 items Middle item 3 items

In the above series, the numbers are arranged in ascending order. Here, the fourth item
17 has three items before it and three items after it. So, 17 is the middle item in the
series. 17 is called the median of the series.

Thus, median is the value of the middle–most observation, when the data are arranged
in ascending or descending order of magnitude.

(i) Median of ungroupped data

To find the median of an ungroupped data, arrange them in ascending or descending
order. Let the total number of observation be n.

– If n is odd, the median is the value of the n+1 th
2 observation.

– If n is even, the median is the average of the n th n +1 th
2 and 2 observation.

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Worked-out examples

Example 1: The weights in kg of 7 students are given below. Find the median
weight.

35, 45, 43, 30, 52, 40, 37
Solution:
Arranging the weights in ascending order, we have,

30, 35, 37, 40, 43, 45, 52

Here, n = 7 n+1 th
Now, the position of median = 2 item

= 7+1 th
2 item = 4th item
i.e., 4th item is the median.
? Median = 40 kg.

Example 2: The marks obtained by 6 students are given below. Calculate the
median mark.
15, 25, 10, 30, 20, 35

Solution:
Arranging the marks in ascending order, we have,

10, 15, 20, 25, 30, 35

Here, n = 6 th
item
Now, the position of median = n+1
2

= 6+1 th
2 item = 3.5th item

3.5th item is the average of 3rd and 4th items.

? Median = 20 + 25= 24 = 22.5
2 2

(ii) Median of Discrete series

To compute the median of a discrete series of frequency distribution, we should display

the data in ascending or descending order in a cumulative frequency table. Then, the

median is obtained by using the formula:
th
Median = value of N+1 item
2

Example 3: Compute the median from the table given below.

x4 8 12 16 20 24
f3 54762

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Statistics

Solution:
Cumulative frequency table

xf c.f.

4 3 3
6 5 8
12 4 12
16 7 19
20 6 25
24 2 27
Total N = 27

Now, position of median = N+1 th 27 + 1 th = 14th item
2 2
item = item

In c.f. column, the c.f. just greater than 14 is 19 and its corresponding values is 16.
? Median = 16.

22.10 Quartiles

Quartiles are the values that divide the data arranged in ascending or descending order
into four equal parts. A distribution is divided into four equal parts by three quartiles.

– athbeovfierswt hoirchlo7w5er%qoufatrhtieleit(eQm1)s is the point below which 25 % of the items lie and
lie.

– The second quartile (Q2) is the point below which 50 % of the items lie and above
which 50 % of the items lie. Of course, the second quartile is the median.

– The third or upper quartile (Q3) is the point below which 75 % of the items lie and
above which 25 % of the items lie.

If N be the number of items in ascending (or descending) order of a distribution, then in

the case of discrete data = N+1 th
the position of the first quartile (Q1) 4
item

the position of the second quartile (Q2) = 2(N + 1) th N+1 th
4 2
= item

the position of the third quartile (Q3) = 3(N + 1) th
4
item

Similarly, in the case of grouped data, the positions of quartiles are obtained in the

following ways. th

The first quartile (Q1) = value of N item
4

The second quartile (Q2) = 2(4N)thitem = value of N th
The third quartile (Q3) = value of 3(4N)thitem 2
item

After finding the positions, the process of computing the quartiles is as similar as the

process of computing median.

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Statistics

Example 4: Find the first quartile (Q1) and the third quartile (Q3) from the data
given below.

7, 17, 10, 20, 13, 28, 24

Solution:

Arranging the data in ascending order,

7, 10, 13, 17, 20, 24, 28

Here, n = 7 n+1 th = 7+1 th
The position of the first quartile (Q1) = 4 4
The value of 2nd item is 10. item item = 2nd item

? The first quartile (Q1) = 10. n+1 th
4
Again, the position of the third quartile (Q3) = 3 item = 6th item
The value of 6th item is 24.

? The third quartile (Q3) = 24.

Example 5: The marks obtained by 10 students of class 8 in Mathematics are given

below. Compute Q1 and Q3.
18, 14, 16, 10, 15, 12, 8, 5, 11, 20

Solution:

Arranging the marks in ascending order,

5, 8, 10, 11, 12, 14, 15 16, 18, 20

The position of the Q1 = n + 1 th = 10 + 1 th = 2.75th item
Here, the 2nd item is 14 and 34rd 4
item item

item is 16.

? Q1 = 14 + (16 – 14) × 75% = 14 + 2 × 75 = 15.5
100 3 × 2.75th
aonf dQ93 t=h it3emnis+4118. th

Again, the position item = item = 8.25th item
Here, 8th item is 16

? Q3 = 16 + (18 – 16) × 25% = 16 + 2 × 25 = 16.5
100
Example 6: Compute the first and the third quartiles from the table given below.

Marks 30 40 50 60 70 80

Solution: No. of students 46 10 12 5 2

Cumulative frequency distribution table

Marks (x) No. of students (f) c.f.
30 4 4
40 6 10
50 10 20
60 12 32
70 5 37
80 2 39

Total N = 39

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Statistics

Now, the position of the first quartile (Q1) = N+1 th 39 + 1 th
4 4
item = item = 10th item

In c.f. column, the corresponding value of the c.f. 10 is 40.

? The first quartile (Q1) = 40 +4its1cotrhirteesmpo=nd3i0ntgh

Again, the position of the third quartile (Q3) =3 N item is 60.
In c.f. column, the c.f. just greater than 30 is 32 and value

? The third quartile (Q3) = 60

22.11 Mode

The mode of a set of data is the value with the highest frequency. A distribution that has
two modes is called bimodal. The mode of a set of data is denoted by Mo.

(i) Mode of discrete data

In the case of discrete data, mode can be found just by inspection, i.e. just by taking an
item with highest frequency.

Example 7: Find the mode for the following distribution.
25, 18, 20, 18, 22, 18, 20, 18, 20

Solution:
Arranging the data in ascending order.

18, 18, 18, 18, 20, 20, 20, 22, 25
Here, 18 has the highest frequency.
? Mode = 18.

22.12 Range

The difference between the largest and the smallest score is called range.
? Range = Largest score – Smallest score

Example 8: The marks obtained by 10 students of class 8 in Mathematics are given
below. Find the range.

78 36 27 95 43
15 69 84 72 51
Here, the highest marks = 95
The lowest marks = 15

? Range = highest score – lowest score

= 95 – 15 = 80

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Statistics

EXERCISE 22.4
General Section – Classwork

1. Let’s say and write the correct answers as quickly as possible.

a) Median of 4, 6, 9 is .....................

b) Median of 5, 8, 10, is .....................
c) Median of 6, 2, 5, 4, 8 is .....................

d) The first quartile of 2, 5, 6, 8, 10, 13, 15, is .....................
e) The third quartile of 3, 4, 7, 10, 12, 15, 17, is .....................

f) The observation which occurs maximum number of times in a data is
called .....................

g) The mode of 4, 5, 7, 5, 5, 4, 9, 7, 4, 5 is .....................

h) The difference between the largest and score the smallest score is called ............
g) The range of 10, 30, 20, 80, 40, 60, 50, 70 is .....................

Creative Section - A

2. a) Find the medians of the following sets of data.
i) 21, 18, 35, 46, 40
ii) 15, 30, 35, 25, 20, 45, 40
iii) 22, 16, 14, 26, 32, 30

b) The weights of five students are as follows. Find their median weight.
48 kg, 59 kg, 43 kg, 63 kg, 52 kg

c) Find the median age of a group of 8 people whose ages in years are as follows:
47, 61, 13, 34, 56, 22, 30, 20

3. a) If the following numbers are in ascending order and median is 4, find the value
of x.
x, x + 1, x + 2, x + 3, x + 4

b) The numbers x – 2, x – 1, x + 2, x + 3, x + 5 are in ascending order. If the
median is 10, find the value of x.

4. a) Find the first quartiles (Q1) of the following sets of data.
(i) 14, 12, 17, 23, 20, 16, 10
(ii) 16, 25, 10, 30, 35, 8, 12
(iii) 40, 20, 30, 10, 16, 12, 18, 24, 28

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Statistics

b) Find the third quartiles (Q3) of the following sets of data.
(i) 15, 9, 21, 33, 27, 39, 45

(ii) 18, 26, 14, 22, 30, 38, 34

(iii) 30, 20, 50, 80, 40, 60, 70, 90
5. a) Find the modes of the following distributions:

(i) 7, 9, 5, 7, 10, 9, 7, 12

(ii) 15 kg, 21 kg, 17 kg, 21 kg, 28 kg, 21 kg, 15 kg, 21 kg

b) In a class, there are 15 students of 16 years, 14 students of 17 years, and 16
students of 18 years. What will be the modal age of the class?

c) In a factory, number of labourers and their remuneration are as follows. Find the
modal class.

Remuneration (in Rs) 1500 – 2000 2000 – 2500 2500 – 3000

No. of labourers 220 215 120

d) Find the modal size of the shoes from the data given below:

Size of shoe 56789 10
No. of men 10 15 30 25 18 12

e) Find the mode from the following data.

Daily wages (in Rs) 70 90 110 130 150 170
No. of workers 4 12 15 18 20 12

Creative Section -B

6. Find the median marks from the data given below:

a) Marks 24 36 50 65 78

No. of students 2 4 12 11 6

b) Marks 10 15 20 25 30 35
No. of students 3 7 15 12 7 3

c) Wage in Rs 45 55 65 75 85 95
No. of workers 20 25 24 18 15 7

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Statistics

7. a) Calculate the first quartile (Q1) from the data given below.

Marks obtained 32 36 40 44 48 52

No. of students 259632

b) Compute the third quartile (Q3) from the following table.

Wages in Rs 50 60 70 80 90 100
No. of workers
6 10 15 13 8 3

c) Find the first quartile (Q1) and the third quartile (Q3) from the following
distribution

Ages in years 22 27 32 37 42
No. of people 35 42 40 30 24

8. a) The heights of 10 students of class 8 are given below.

95 cm, 110 cm, 120 cm 90 cm, 100 cm,
116 cm
105 cm, 98 cm, 115 cm, 112 cm,

(i) What is the height of the tallest student?

(ii) What is the height of the shortest student?

(iii) Find the range of the height.

b) Find the range of the following data.

25, 20, 38, 50, 45, 27, 36, 18

c) The marks obtained by 20 students in Mathematics are given below. Find the
range.

Marks 4 6 8 10 12 14

No. of students 2 3 5 7 2 1

It’s your time - Project work!

9. a) Let’s collect the marks obtained by any of your 9 friends and find:

(i) the median marks (ii) the first and third quartiles marks.

b) Let’s conduct a survey to find the number of students from class 4 to class 8
in your school. Show the number of students of different ages in a frequency
distribution table and compute the following:

(i) median age (ii) the first quartile age (iii) the third quartile age

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Lower Secondary Level Final Examination

Specification Grid - 2069

Subject: Mathematics

S. Areas Level of Knowledge Skills Problem

N. Objectives and Solving

Understanding

Types of Questions Very Short Short Long Long
Short
Total Marks
Topics No. of ques. Time Allocations
Marks Remarks
No. of ques.
Marks
No. of ques.
Marks
No. of ques.
Marks
No. of ques.
Marks

1. Line & Angles 1 1 12

2. Triangle, Parallel- 12 28 40 Minutes

1G ogram and Polygon 24 10 22
3. Similarity and
2 C-
G Congruency 11 12 2 3 5 Min-
4. Circle 11 12
3M 5. Solid Shapes 12
4T 6. Co-ordinates
5.
6. A 7. Perimeter, Area 12 14 26 utes
11
7S
8A and Volume Min-

- 8. Transformation 14 utes
9. Bearing and Scale 1 9 Min-
1 25 utes
Drawing 1 1 12 14 37
10. Sets 1 13 min
11. Whole Number

12. Integer 1 1 214 37
13. Rational Number
14. Real Number 12 38 Minutes
15. Ratio, Proportion

& Percentage 14 4 14
16. Profit and Loss
17. Unitary Method 14 3 7 13 min
18. Simple Interest 51
19. Statistics 14
20. Algebraic
1 1 1 2 14
1
11 2 3 12

Expression Min-
12 29 utes
21. Indices 11 24
2 4 14
22. Equation, In- 1 1

equality & Graphs

180

Total 10 10 3 6 1 4 28 10 40 4 16 41 100 Min-

utes

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Class: VIII BLE Model Set
Compulsory Mathematics
Maximum Marks: 100
Time: 3 hrs.

Attempt all the questions: P X Q
M B
Group-A (5 × (1 + 1) = 10]
N
1. (a) In the given figure, write the corresponding angle of ‘PMN. A Y

(b) If the diameter of a circular pond is x meter, what is Q Y
the measure of its circumference? B

2. (a) From the given graph, find the solution of the x – y = –3 x+y=1
equations: x + y = 1, x – y = –3

(b) Sketch to show the bearing of 1000. X' A OP X

3. (a) Write down the set notation to represent the shaded region Y'
in the given Venn-diagram. U

P
Q

(b) Find the mode of the data given below. x – 3x – 10°
3cm, 4cm, 2cm, 3cm, 4cm, 5cm, 2cm, 3cm, 5cm, 3cm, 2cm 10°

4. (a) Factorize: 1 – 9x2
(b) If x ≠ 0, what is the value of (8x)°?

5. (a) If m = –3 and n = 2, what is the value of mn.
(b) If 4y – 1 = 19, find the value of y.

Group-B (17 × 2 = 34)

6. (a) Find the value of x from the given figure.

(b) Write two distinct properties of rectangle and parallelogram.

A

(c) In given figure, M is the mid-point of side BC and AM A BC,
by what axiom ∆ABM and ∆ACM are congruent. If AB = 5 cm,
what is the length of AC?
BMC

7. (a) In the given figure, AB // CD, show that ∆AOB a ∆COD. AB
O

CD

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Model Questions

(b) If the area of a circular meadow is 154 m2, what is its radius? (S = 22 )
7

(c) Draw the net of a cylinder.

8. (a) If the distance between the points (0, 6) and Q (x, 0) is 5 units, find the value
of x.

(b) If the length and breadth of rectangular meadow is 16m and 12m, what is the
length of its diagonal?

(c) If M = {natural numbers less than 6} and N = {first five even numbers}, find
M ª N and show it in a Venn-diagram.

9. (a) Change 1100112 into denary number system.

(b) There are 80 students in a class. If 4 of them are absent, find the percentage of
present students.

(c) Find the median of the data given below:

30, 10, 40, 20, 60, 50, 80, 70, 90

10. (a) Show a2 – b2 = (a + b) (a – b) in a diagram.

(b) Simplify: xa+b-c × xc-a-b × 1a+b+c
x0

(c) Prove that: 3p+1 – 3p = 1
2 × 3p

11. (a) Solve the inequality and show it in a number line: 2x – 3 ≤ 7.

(b) Solve: x2 – 1 = 5
3

Group-C (14 × 4 = 56)

12. Construct a rectangle PQRS, in which PQ = 4cm and diagonal PR = 6.2 cm.

13. Verify experimentally that the base angles of an isosceles triangle are equal. (Two
figures of different measurement are necessary)

14. The vertices of ∆PQR are P (3, 2), Q (2, 5), and R (-4, 2). Draw the ∆PQR on a graph
paper and reflect it about x – axis, find the coordinates of its image and draw it on
the same graph paper.

15. In an examination, 10% students secured A+ grade in Mathematics, 15% secured
A+ grade in Science, and 5% secured A+ grade in both the subjects.

(i) Sketch a Venn-diagram to show the above information.

(ii) What percent of students secured grades other than A+ in both the subjects?

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Model Questions

16. The length of a hall is three times and width is two times of its height. If the room
contains 384 cubic metres of air, find the area of its floor.

17. The marked price of a mobile set is Rs 11,000. If it is sold at 20% discount and
levying 13% value added tax, find the selling price of the mobile set after value
added tax.

18. 20 men can complete a piece of work in 24 days, how many men should be left the
work so that it would complete in 32 days?

19. Mr. Suresh took a loan from an Agricultural Development Bank to start his fish
farming. If he paid Rs. 5454 and cleared his debt in 2 years 6 months at the rate of
8% per annum, how much loan was taken by him?

20. Simplify: 6 – 16 + 2 50 – 3 8
2 32

21. Find the arithmetic mean from the data given below.

Marks obtained 50 60 70 80 90

No. of students 7 4 14 11 4

22. If a + 1 = 5 , prove that: (i) a2 + 1 = 23 (ii) a3 + 1 = 110
a a2 a3

23. Find the highest common divisor of: x2 – 4, x3 + x2 – 6x and x4 – 8

24. Simplify: x 2 1 + 2x – x2 + 3
+ x–1 x2 – 1

25. Solve the equations 2x – y = 5 and x – 1 = y by graphical method.

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