Prime Optional Mathematics 7

iii) For continuous class interval
Rfm Rfx
Mean (x) = N or N

Where,

m = Mid value (mean) of each classes. (Also can be taken x)

f = No. of observations

fm = Product of mid-value and number of observation (fx)

R fm = sum of (fm) or (fx)

N = R f = sum of number of observations.

Worked out Examples

1. Construct a frequency distribution table for the marks obtained by
the students as 20, 18, 14, 20, 16, 16, 14, 14, 16, 18, 12, 14, 12, 16,
18, 18, 20, 14, 16, 16.
Solution:
The marks obtained by the students are :

Marks tally marks frequency
12 || 2
14 |||| 5
16 6
18 |||| | 4
20 |||| 3
|||
N = 20

2. Find the arithmetic mean of the marks obtained by the 8 students

of grade VII in optional mathematics 12, 16, 18, 20, 15, 17, 13, 11.

Solution:

The marks obtained in ascending order are: 11, 12, 13, 15, 16, 17,

18, 20.

No. of observations (N) = 8

Arithmetic mean (x) = ?

We know,
Rx
x = N

= 10 + 12 + 13 + 15 + 16 + 17 + 18 + 20
8

146 PRIME Opt. Maths Book - VII

= 112
8

= 14
` x = 14

3. Find the arithmetic mean of the cost of vegetables in different
shops.

Cost (Rs.) 10 14 18 25 30

No. of shops 4 5 10 4 2
Solution:

Cost (X) No. of Shops (f) f×x

10 4 40

14 5 70

18 10 180

25 4 100

30 2 60

N = 25 ∑fx = 450

Here, / fx

Arithmetic mean (x) =N

= 450
25

= 18

4. Find arithmetic mean from the followings:

Class 0–10 10–20 20–30 30–40 40–50
f 6 8 12 10 4

Solution:

Class f mid-value (x) f×x

0–10 6 0 + 10 =5 30
2

10–20 8 10 + 20 120
2 = 15

20–30 12 20 + 30 = 25 300
2

PRIME Opt. Maths Book - VII 147

30–40 10 30 + 40 = 35 350
2 180
∑fx = 980
40–50 4 40 + 50 = 45
2

N = 40

We have, = / fx
Arithmetic mean (x)
N

= 980
40
= 24.5

5. If mean of the observations is 20, �ind the value of m from the
observations 12, 18, 20, m, and 26.

Solution:

The given observation are:
12, 18, 20, m, and 26.

No. of observations (N) = 5

We have, Rfx
N
Mean(x) =

or, 20 = 12 + 18 + 20 + m + 26
5
76 + m
or, 20 = 5

or, 76 + m = 100

or, m = 100–76

or, m = 24

148 PRIME Opt. Maths Book - VII

Exercise : 7.1

1. Answer the followings.
i. What do you men by primary data?
ii. What is arithmetic mean? Write down its formula for discrete
observations.
iii. What do you mean by secondary data.
iv. Tabulate the observations in frequency distribution table.
20, 24, 24, 20, 32, 26, 32, 26, 36, 40, 36, 40, 20, 26, 26, 24, 32,
32,
v. Tabulate the data in frequency distribution table.
14, 18, 16, 20, 20,24, 24, 22, 22, 20, 16, 14, 18, 18, 20, 22, 22,
18, 22, 24, 24, 20, 22, 20, 20

2. Find arithmetic mean from the followings:
i. ∑X = 200, No. of observations is 10.
ii. ∑fx = 252, ∑f = 18.
iii. 14, 20, 26, 32, 38
iv. 12, 20, 28, 36, 44, 50, 56, 64
v. ∑fx = 400 + 50a, N = 16 +2a

3. Find the arithmetic mean from the followings. 40 50
i. x 10 20 30 3 1

f 268 10
2
ii. Marks 24 15 14 13
55
f 5 8 15 10 2

iii. Age 15 25 35 45 32–40
2
f 4 6 8 10

iv. Class 0–8 8–16 16–24 24–32
f 3483

PRIME Opt. Maths Book - VII 149

v. Marks 0–20 20–40 40–60 60–80 80–100
f 7 9 14 12 8

4. Prime more creative questions.
i. If ∑x = 180 + m, N = 10 and x= 20, find the value of m.
ii. If ∑fx = 200 + 6P, N = 25 and x= 20, find the value of P.
iii. If mean of the observations 14, P, 26, 32 and 38 is 26, find the
value of P.
iv. If mean of the observations is 15, find the value of a.
x 10 13 14 15 24
f 2 10 a 8 5

v. If arithmetic mean of the observations is 35, find the value of
m.

Marks 15 25 35 45 55
f 4 6 8m2

Answer

1. Show to your teacher.

2. i. 20 ii. 14 iii. 26 iv. 38.75 v. 25
v. 52
3. i. 27.5 ii. 15 iii. 35 iv. 18.8 v. 10

4. i. 20 ii. 50 iii. 20 iv. 15

150 PRIME Opt. Maths Book - VII

7.2 Median, Mode and Range

1. Median

The numerical value of a statistical data which divides the data into two
equal halves is called median.

Y

50% 50%

OX

Median
The median value divides the data 50% in left side and 50% in right side
as shown in the given diagram.

The value which divides the observation into two qual
halves is called median.

For the calculation of median for individual data.

Observation should be written in ascending or in descending order.
N + 1 jth
Size of ` 2 item should be calculated.
+
Median (Md) = corresponding observation of ` N 2 1 jth item. (for odd

Number of observation.)

For even no. of observations: N + 1
2
Md = sum of two con sec utive observations of size ` jitem.
2

Calculation of median for discrete data:

Finding cumulative frequency column in the table.
N + 1
Finding ` 2 jth item.
Finding +
just greater or equal to ` N 2 1 jth size in c.f. column.

Median (Md) = corresponding observation
N + 1 jth
Where ` 2 lies in c.f.

PRIME Opt. Maths Book - VII 151

2. Mode:

The most repeated observation from a collected data is taken as the
model value of the observations. It gives an idea about the importance of
the observation as compared to the other observations. The model value
is the most valuable as compared with the other values.

The most repeated observations of the data is called
mode.
i.e.
Mode = The most repeated observation

= The observation having highest frequency

Examples: 5 is the model of the collected observations 2, 2, 3, 4, 5, 5, 4,
5, 6, 7, 5, 3.

3. Range:

The highest and lowest observations of the collected data can be taken
from the observations during analyzing the data. The differences of such
observations defines the goodness or badness of the distribution of the
observations in the data. This type of differences gives an idea about the
nature of the collected data called range.

The difference of highest and lowest observations of

the statistical data is called range.

i.e. Range = H – L

and coefficient of range = H–L
H+L

The measurement of difference between highest and lowest observations
of the collected data is called range.

152 PRIME Opt. Maths Book - VII

Where, Range = Highest observation – lowest observation.

i.e. R = H – L

Also,
Coefficient of range is calculated from such values as,
H–L
Coefficient of Range = H+L .

Note : Range measures the measurement of closeness of the observations
from highest and lowest observations. [R = H – L]
• All the observations are not included in it.

Example: Find range and its coefficient of the observations taken in

order of 20, 24, 30, 36, 44, 50, 60.

Here,

Highest observation (H) = 60

Lowest observation (L) = 20

Range (R) = H – L

= 60 – 20

= 40

Again,

Coefficient of Range = H–L
H+L

= 60 – 20
60 + 20

= 40
80

= 0.5

Coefficient of Q.D. = Q3 – Q1
Q3 + Q1

PRIME Opt. Maths Book - VII 153

Worked out Examples 30
6
1. Find the median from the following observations.
i. 12, 15, 18, 21, 24, 27, 30.
ii. 40, 36, 32, 20, 24, 28.
iii. marks 10 15 20 25

f 5 8 12 9

Solution:

i. The give observations in ascending order are:

12, 15, 18, 21, 24, 27, 30

No. of observations (N) = 7

Then, +
2
Median (Md) = Size of ` N 1 th item.

j

= Size of `6 + 1 th item.
2
j

= Size of 4th item.

= 4th observation

= 21.

` Md= 21

ii. The given observations in ascending order are:

20, 24, 28, 32, 36, 40.

No. of observations (N) = 40

then, +
2
Median (Md) = Size of ` N 1 th item.

j

= Size of `6 + 1 th item.
2
j

= Size of 3.5th item.

= ^3rd + 4thh observations
2

= 28 + 32
2

= 30

154 PRIME Opt. Maths Book - VII

iii. marks f c.f

10 5 5=5
15 8 5 + 8 = 13
20 12 13 + 12 = 25
25 9 25 + 9 = 34
30 6 34 + 6 = 40

N = 40

Here, +
2
Median (Md) = ` N 1 th item.

j

= Size of ` 40 + 1 th item.
2
j

= Size of 20.5th item.

= 25 is c.f. just greater then 20.5

= 20 is the corresponding marks.

= 20

` Md = 20

2. Find mode from the observations 15, 12, 14, 15, 14, 15, 15, 16, 17,
15, 16, 18.
Solution:
The given observations taken in order are :12, 14, 14, 15, 15, 15,
15, 15, 16, 16, 17, 18.
Here,
15 is the most repeated obsertion (5 times).
` Mode = the most repeated observation.
= 15
` Mode = 15

3. Find the range and its coefficient of the observations 8, 12, 16, 18,
24, 28, 32.
Solution:
The given observations are: 8, 12, 16, 18, 24, 28, 32.
Here,
Highest observation (H) = 32
lowest observation (L) = 8

PRIME Opt. Maths Book - VII 155

` Range = H– L
= 32 – 8
= 24

4. If median of the observations taken in order 12, 16, 20+m, 30, 36
and 40 is 27, find the value of m.

Solution:

The observations taken in order are: 12, 16, 20+m, 30, 36, 40.

No. of observations (N) = 6

Median (Md) = 27
We have,
+
Median (Md) = Size of ` N 2 1 th item.

j

or, 27 = Size of ` 6 + 1 th item.
2
j

or, 27 = Size of 3.5th item.

^3 + 4hth
or, 27 = 2 observations.

or, 27 = 20 + m + 30
2

or, 54 = 50 + m

or 54–50 = m

or, 4 = m

` m=4

5. If 12 is the lowest observation of a data whose range is 16, find the
highest observation. Also find the coefficient of range.
Solution:
In a data,
Lowest observation (L) = 12
Range = 16
Highest observation (H) = ?
We have,
Range = H – L
or, = H – 12
or, 16 –12= H
or, 28 = H
` H = 28

156 PRIME Opt. Maths Book - VII

Also,

Coefficient of Rang = H–L
H+L

= 28–12
28 + 12

= 16
40

` = 0.4

Exercise : 7.2

1. Find the median from the followings:

i. 14, 18, 22, 26, 30.

ii. 20, 28, 36, 42, 48, 56.

iii. 8, 14, 20, 26, 50, 44, 38, 32.

iv. x 16 20 24 28 32
9 6
f5 8 12
v. Marks 10 20 30 40 50 60 70
10 5 2
f 4 6 9 14

2. Find mode from the followings.

i. 8, 10, 8, 12, 12, 14,16, 12, 14, 8, 16, 14, 12, 12

ii. 20, 20, 24, 24, 24, 26, 24, 26, 28, 30, 26, 24.

iii. 50, 52, 54, 52, 54, 50, 60, 58, 56, 54, 56, 54.

iv. x 12 18 24 30 40
4
f 7 8 12 5
60
v. Marks 20 30 40 50 7

f 9 11 15 8

3. Find range and its coefficient from the followings.
i. 14, 18, 20, 24, 26.

ii. 20, 25, 30, 35, 40, 45, 50, 55, 60.

iii. 10, 15, 20, 25, 30, 35, 40.

PRIME Opt. Maths Book - VII 157

iv. x 10 20 30 40 50 60 70
f 4 6 8 12 5 3 2

v. Marks 8 14 20 26 32
f 5
9 10 7 4

4. Prime more creative questions.

i. If median of the observations taken in order 12, 16, 20, 20 + P,
28, 32, and 36 is 24, find the value of P.

ii. If median of the observations taken in order. 32, 38, 44, 45 +m,
51 + m, 62, 68 and 74 is 53, find the value of m.

iii. If lowest observation and range of a data are 10 and 15
respectively. Find the highest observation. Also find the
coefficient of range.

iv. If highest observation and range of a data are 60 and 24
respectively, find the lowest observation. Also find the
coefficient of range.

v. If median of the observations taken in order 20, 30, 6P – 15,
60, and 70 is 45, find the value of P.

1. i. 22 ii. 39 Answer v. 40

iii. 29 iv. 24

2. i. 12 ii. 24 iii. 54 iv. 24 v. 40

3. i. 12, 0.3 ii. 40, 0.5 iii. 30, 0.6 iv. 60, 0.75 v. 24, 0.6

4. i. 4 ii. 5 iii. 25, 0.25 iv. 36, 0.25 v. 10

158 PRIME Opt. Maths Book - VII

Statistics

Unit Test - 1
Time : 30 minutes

[1 × 1 + 3 × 2 + 2 × 4 + 1 × 5 = 20]

Attempt all the questions:
1. Write down the calculating formula of coefficient of range.
2. a. Find the mean of the observations 10, 14, 18, 22 and 26.

b. If median of the observations taken in order 20, 24, 20 + m, 32
and 36 is 28, find the value of m.

c. Find the range of the observations 8, 12, 16, 20, 24 and 28.

3. a. Find the median value of the observations.

x 10 15 20 25 30 35

f 3 4 6 12 9 5
b. If range of observations having lowest term 20 is 30, find the

highest observation.

4. Find arithmetic mean of the data.

Class 0-10 10-20 20-30 30-40 40-50
f 8 12 15 9 6

PRIME Opt. Maths Book - VII 159

Proposed Syllabus with Grid for
First Terminal Examination

S.N. Contains Topics K-1 U-2 A-4 HA-5 TQ TM Periods

1 Algebra i. Order Pairs 111 – 37 5

ii. Cartesian Product

2 Matrices All 111 – 37 5

3 Co-ordinate i. Distance Formula 1 1 1 1 4 12 4
Geometry

4 Trigonometry Exercise 4.1 1 3 3 1 8 24 12
Exercise 4.2
Exercise 4.3 4 6 6 2 18
Exercise 4.4
4 12 24 10 50 26
Total Questions

Total Marks

K = Knowledge, U = Understanding, A = Application, HA = Higher ability

Model Question Set for First Terminal Examination

Group ‘A’ [4 × 1 = 4]

1. a. Write down antecedent and consequence from the ordered pair

(2, 4).

b. 2 –1 , find the matrix AT.
If A = <3 –2F

2. a. Convert 90° into grades.
b. Write down the co-ordinate of a point an x-axis and on y-axis.

Group ‘B’ [6 × 2 = 12]

3. a. If A = {3, 2}, B = {6, 7, 8}, find Cartesian product A × B.

b. Find the distance between the points A(2, –3) and B(8, 5).

2 3 1 2 , find the value of 2A – B.
4. a. If A = < F and B = < F
1 4 3
–1

b. Find the ratio of the angles 80g and 108°.

5. a. Factories : Sin4A – Cos4A.

b. Find the trigonometric ratio Sinq from the adjoining diagram.

A

8cm

C q B
6cm

160 PRIME Opt. Maths Book - VII

Group ‘C’ [6 × 4 = 24]
6. If P = {a, b, c}, Q = {x, y}, find P × Q and show in arrow diagram.

7. 2 41 2 , find the matrix A.
If A + B = <6 3F and B = <3 –1F

8. Prove that the points A(4, –3) and B(–3, –4) are equidistant from the origin.

9. Prove that: Sin²A.Cos²B – Cos²A.Sin²B = Sin²A – Sin²B.
10. If 3TanA = 4, find the value of Sin²A + Cos²A.
11. Prove that : Sec4A – Tan4A = 2Sec²A – 1.

Group ‘D’ [2 × 5 = 10]
12. The one angle of right angled triangle is 40g, find the third angle in degrees.
13. Prove that the points A(3, 2), B(3, –3) and C(8, –3) are the vertices of

an isosceles triangle.

Proposed Syllabus with Grid for
Second Terminal Examination

S.N. Contains Topics K-1 U-2 A-4 HA-5 TQ TM Periods

1 Algebra i. Polynomial 111 – 37 6
ii. Surd

2 Matrices – 111 – 37 –

3 Co-ordinate Remaining All –11 – 26 4
Geometry

4 Trigonometry Remaining All 12 3 – 6 17 12
6 Transformation Reflection 1– – 1 26 6

7 Statistics Arithmetic Mean –1 – 1 27 6

First Term Review 6

Total Questions 4 6 6 2 18

Total Marks 4 12 24 10 50 40

K = Knowledge, U = Understanding, A = Application, HA = Higher ability

Model Question Set for Second Terminal Examination
Group ‘A’ [4 × 1 = 4]

1. a. What types of polynomial is x² – 3x + 4 ?

b. What do you mean by square matrix ?

2. a. Write down the relation of SinA in terms of CosA.
b. Write down the image of a point A (3, –2) under reflection about x-axis.

Group ‘B’ [6 × 2 = 12]

3. a. If A × B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}, find the sets A
and B. Also find n(A × B).

PRIME Opt. Maths Book - VII 161

b. 3 –42F, find the transpose of the matrix 2A.
If A = <1

4. a. Find the mid-point of line joining the points A(3, –2) and B(5, –4).
b. Find the median of the observations 12, 16, 20, 24, 28, 32.

5. a. Convert 20° 25' 28'' into seconds.
b. Find the value of : Sin²30° + Cos²60° + Tan²45°.

Group ‘C’ [6 × 4 = 24]

6. If A = {3, 4, 5}, B ={1, 2}, find A × B and B × A. Show A × B in arrow
diagram.

7. If =x + 2 y 3 G = 3 3 are equal matrices, find the value of x and y.
1 – 3 < F
1
2

8. Find the co-ordinate of centroid of a triangle having vertices
A(1, –2), B(3, –4) and C(2, 3), find in the diagram.
1 – Cos4 i
9. Prove that : Sin4 i = 1 + 2Cot²q

10. One angle of a triangle is 40° and the second angle is 60g, find the

third angle in degrees.

11. Length of a ladder as shown in diagram taken against a wall is

20m. Find the height of the wall where the ladder makes 30°

angle with the ground.

20m

30°

Group ‘D’ [2 × 5 = 10]
12. Find the co-ordinate of image a triangle having vertices A (3, 2),

B (7, 5) and C (5, 7) under reflection about y = 0. Also plot the
object and image in graph.
13. Find the arithmetic mean of :

Class 0 - 8 8 - 16 16 - 24 24 - 32 32 - 40
f 34751

162 PRIME Opt. Maths Book - VII

Specification Grid for
Final Examination referred by CDC Nepal

S.N. Contains Topics K-1 U-2 A-4 HA-5 TQ TM Periods

1 Algebra i. Order Pairs 111 – 37 16

ii. Cartesian Product

iii. Surd

iv. Polynomial

2 Matrices i. Introduction 1 1 1 – 3 7 10
ii. Addition
iii. Transpose

3 Co-ordinate i. Distance Formula – 1 1 – 2 6 12
Geometry ii. Mid-Point Formula

4 Trigonometry i. Measurement of 11 2 – 4 11 22
Angles

ii. Trigonometric
Ratios

iii. Conversion of TR
iv. Standard Angles

5 Vector i. Introduction 1– 1 – 25 6
–1 – 1 27 8
ii. Addition

6 Transformation i. Reflection

ii. Translation

7 Statistics i. Central Tendency –1 – 1 27 6
ii. Range

First Term Review 4

Second Term Review 4

Total Questions 466 2 16

Total Marks 4 12 24 10 50 80

K = Knowledge, U = Understanding, A = Application, HA = Higher ability

Model Question Set for Final Terminal Examination
Group ‘A’ [4 × 1 = 4]

1. a. Convert the surd 23 3 into pure surd.
b. What do you mean by rectangular matrix ?

2. a. Find the value of Sin30° + Cos60°.
b. If A(2, 4) and B(3, 7) are any two points, Find the vector AB .

Group ‘B’ [6 × 2 = 12]
3. a. If (x + 2, 4) and (5, y – 1) are equal ordered pairs, Find the value

of x and y.

PRIME Opt. Maths Book - VII 163

b. If A = 2 –1 and B= 1 –3 , find the value of A
<3 –2F <2 4F
a
2A + B. 2 cm

4. a. Find the trigonometric ratio Sinθ from the C q B
adjoining diagram. acm

b. Prove that : (Sinθ + Cosθ)² = 1 + 2Sinθ.Cosθ.

5. a. Find the co-ordinate of a point A(3, –2) under a translation

2
T = <F .
3

b. If highest observation and range of a data are 60 and 40
respectively, find the lowest observation of the data.

Group ‘C’ [6 × 4 = 24]
3– 1 4
6. Simplify : 6– 3 3+ – 6– 2
2

3 –4 1 2 , find the matrix A and B.
7. If A + B = < 2 F and A – B = < F
5 1
–6

8. Prove that : Cosi + Cosi = 2Secq.
1 – Sini 1 + Sini
9. A boy is flying a kite where the kite is at a
height of 100m and the string makes an angle ? 100m

of 30° with the ground as shown in diagram. 30°
Find the length of the string.

10. If 5 is the magnitude of a vector a = 4 n , find the value of m.
dm

11. If (2, 3) is the mid- point of line joining the points (a, –3) and
(4, b), find the value of a and b.

Group ‘D’ [2 × 5 = 10]

12. Find the image of a triangle having vertices P(3, –2), Q(–4, 3) and

4
R(0, 5) under translation about T = <F . Also plot the object and
image in graph. 3

13. If mean of the observations given below is 17, find the value of

'P'.

x 5 10 15 20 25 30

f 25P742

164 PRIME Opt. Maths Book - VII