Vedanta Excel in Opt. Maths Book 6 Final (2078)

vEMexdcAaenlTtaiHn AEdMdAitTiIoCnSal

6Book

Authors

Hukum Pd. Dahal Piyush Raj Gosain

Editors

Tara Bdr. Magar P. L. Shah

vedanta

Vedanta Publication (P) Ltd.
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Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
[email protected]
www.vedantapublication.com.np

EvMexdcAaenlTtaiHn AEMddAiTtiIoCnSal

6Book

Authors
Hukum Pd. Dahal and Piyush Raj Gosain

All rights reserved. No part of this publication may
be reproduced, copied or transmitted in any way,
without the prior written permission of the publisher.

¤Vedanta Publication (P) Ltd.

First Edition: B.S. 2077 (2020 A. D.)
Second Edition: B. S. 2078 (2021 A. D.)

Layout and Design
Pradeep Kandel
Price: Rs 95.00
Printed in Nepal

Published by:

Vedanta Publication (P) Ltd.
jb] fGt klAns;] g kf| = ln=

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www.vedantapublication.com.np

Preface

Vedanta Excel in Additional Mathematics for class 6 is completely based on the
contemporary pedagogical teaching learning activities and methodologies. It is
an innovative and unique in the sense that the contents of the book are written
and designed to ful ill the need of integrated teaching learning approaches.

Vedanta Excel in Additional Mathematics has incorporated applied constructivism
the latest trend of learner centered teaching pedagogy. Every lesson of the series
is written and designed in such a manner that makes the classes automatically
constructive and the learner actively participate in the learning process to
construct knowledge themselves, rather than just receiving ready made
information from their instructor. Even teachers will be able to get enough
opportunities to play the role of facilitators and guides shifting themselves from
the traditional methods imposing instractions. The idea of the presentation of
every mathematical item is directly or indirectly re lected from the writer's long
experience, more than two decades, of teaching optional mathematics.

Each unit of Vedanta Excel in Additional Mathematics series is provided with
many more worked out examples, arranged in the hierarchy of the learning
objectives and they are re lective to the corresponding exercises.

Vedanta Excel in Additional Mathematics is an independent kind in its contents
as the Curriculum Development Centre (CDC) does not consider it under its
curriculum. It helps the students of class 6 to lay foundation for class 9 and 10
in compulsory mathematics and optional mathematics. My honest efforts have
been to provide all the essential matters and practice materials to the students.
It is believed that the book serves as a staircase for the students of class 6. The
book contains practice exercises in the form of simple to complex including the
varieties of problems. I have tried to establish relationship between the examples
and the problems set for practice to the maximum extent.

The book aims to give an elementary knowledge of Measurement of Angles,
Trigonometry, Ordered Pair, Coordinate Geometry, Matrices. Special emphasis

has been given to all the chapters as all of them are entirely new to the students.
Questions in each exercise are catagorized into two groups - Short Questions and
Long Questions.

My hearty thanks goes to Mr. Hukum Pd. Dahal, Tara Bahadur Magar and P.L.
Shah, the series editors, for their invaluable efforts in giving proper shape to the
series. I am also thankful to my colleague Mr. Gyanendra Shrestha who helped me
a lot during the preparation of the book.

I am also thankful to my respected parents and my family members for their
valuable support to bring the book out in this form. I would also like to express
my hearty gratitude to all my friends, colleagues and beloved students who
always encouraged me to express my knowledge, skill and experience in the form
of books. I am highly obliged to all my known and unknown teachers who have
laid the foundation of knowledge upon me to be such a person.

Last but not the least, I am hearty thankful to Mr. Pradeep Kandel, the computer
and designing senior of icer of the publication for his skill in designing the series
in such an attractive form.

Efforts have been made to clear the subject matter included in the book. I do hope
that teachers and students will best utilize the series.

Valuable suggestions and comments for its further improvement from the
concerned will be highly appreciated.

Piyush Raj Gosain

CONTENT

Unit 1 Measurement of Angles ..................................................................7-19

1.0 Review .......................................................................................................... 7
1.1 Different Systems of Measurement of Angles ............................................. 8
1.2 Conversion of degree measure into grade measure ................................. 10
1.3 Conversion of grade measure into degree measure ................................. 10
1.4 Conversion of degree measure into radian measure ................................ 11
1.5 Conversion of grade measure into radian measure .................................. 11

Unit 2 Trigonometry ................................................................................. 20-45

2.0 Introduction ............................................................................................... 20
2.1 Right angled triangle ................................................................................. 20
2.2 Perpendicular (p), base (b) on the basis of reference angle ..................... 21
2.3 Ratios of the sides of a right angled triangle ............................................ 23
2.4 Trigonometric Ratios ................................................................................. 25
2.5 Operations on Trigonometric Ratio .......................................................... 29
2.6 Relation between trigonometric ratios ..................................................... 34
2.7 Trigonometric Identity .............................................................................. 36
2.8 Pythagoras Theorem .................................................................................. 39

Unit 3 Ordered Pair ..................................................................................... 46-50

3.1 Pair ............................................................................................................. 46
3.2 Ordered Pair .............................................................................................. 46
3.3 Equal ordered pairs ................................................................................... 47

Unit 4 Coordinate Geometry ................................................................... 51-63

4.0 Introduction ............................................................................................... 51
4.1 Number lines and coordinate axes ........................................................... 51
4.2 Coordinates of a point in different quadrants .......................................... 53
4.3 Plotting points in different quadrants ...................................................... 54
4.4 Distance between two points .................................................................... 57
4.5 Distance formula ....................................................................................... 59

Unit 5 Matrices ............................................................................................. 64-74

5.1 Matrix - Introduction ................................................................................ 64
5.2 Rows and columns of a matrix ................................................................. 65
5.3 Order of a Matrix ....................................................................................... 66
5.4 Equal Matrices ........................................................................................... 67
5.5 Operations on Matrices.............................................................................. 67

Model Questions ................................................................................................... 75

Measurement of Angles

1Measurement of Angles

1.0 Review

Let us review some basic terms related to an angle.

(a) Ray A
In geometry, a ray can be defined as a part of a line
that has a fixed starting point but no end point. It O B
can extend infinitely in one direction. A

In the figure, OA is a ray.

On its way to infinity, a ray may pass through more than one point.

(b) Vertex ray 2
ray 1
The common end point of two or more rays or line
segments is called vertex.

In the figure, O is the vertex. OA and OB are two O
rays, that cut at a point O.

(c) Angle

An angle is the figure formed by two rays sharing a common end point. The
common end point is the vertex. In the above figure AOB is an angle.

(d) Arms of an angle Q
P
The two rays which form an angle are called arms of

an angle. In the figure OP and OQ are the arms of the

angle POQ. O

Note :

(i) The symbol '‘' is used to represent an angle. Thus, ‘POQ represents the
angle POQ.

(ii) An angle takes three capital letters in such a way that the letter at the middle
is always the vertex of the angle.

vedanta Excel in Additional Mathematics - Book 6 7

Measurement of Angles
For example:

Q

OP

‘POQ = ‘O

(e) Protractor
A protractor is an instrument for measuring angles. It is typically in the form of
a flat semi-circle marked with degrees along the curved edge.

Steps for measuring an angle

The following are the steps to measure an angle. Let the angle to be measured be
POQ
(i) Place the protractor in such a way that its centre point is exactly at O.
(ii) Adjust the protractor so that the 0°-180° line is along the same arm PS.

Q

SO P

(iii) Start counting from the zero which is above OP and read the number of degrees
at the point from where other arm OQ passes.

In the figure, ‘POQ = 50°

1.1 Different Systems of Measurement of Angles

There are different system of measurement of angles. P

(a) Degree Measure (Sexagesimal System)
In this system, 1 right angle is divided into 90 R
equal parts and each part is called 1 degree. The
'Degree' measurement is denoted by the symbol
(°) and written at the top and right side of the
given value of angle.

The figure alongside is a part of a protractor in
which is a right angle is divided into 90 equal

Q

8 vedanta Excel in Additional Mathematics - Book 6

Measurement of Angles

parts. So, 1 right angle = 90° and 2 right angles = 180°. R

In this system of measurement of angles, the
sum of angles of a triangle is 180°.

In the figure, PQR is a triangle. Q P
‘P + ‘Q + ‘R = 180°

i.e., ‘QPR + ‘PQR + ‘PRQ = 180°

(b) Grade Measure (Centesimal System)
In this system, 1 right angle is divided into 100 R
equal parts and each part is called 1 grade. The
'Grade' measurement is denoted by (g) and written
at the top and right side of the given value of
angle.

The adjoining figure is a part of a protractor in

which a right angle is divided into 100 equal

parts. So, 1 right angle = 100g and 2 right angles Q P
= 200g .

In this system of measurement, the sum of angles of a triangle is 200g.

In the figure, PQR is a triangle. R

Sum of angle of triangle PQR = 200g

i.e., ‘P + ‘Q + ‘R = 200g

or, ‘QPR + ‘PQR + ‘PRQ = 200g

(c) Radian System (Circular Measure) Q P

An angle subtended at the centre of any circle by an arc whose length is equal

to the radius of the circle is called a radian. B

Let OA = r be the radius of the circle with centre at O. O A
Take an arc AB equal in length to the radius.

i.e. AB = OA = r

Join OB. Then, by definition of radian,

‘AOB = 1c (read as 1 radian)

In circular measure, radian is taken as the standard unit of measurement. It is

symbolized by 1c. In this system of measurement, R

the sum of angles of a triangle is Sc.

In 'PQR,

‘P + ‘Q + ‘R = Sc Q P
or, ‘QPR + ‘PQR + ‘PRQ = Sc

vedanta Excel in Additional Mathematics - Book 6 9

Measurement of Angles

1.2 Conversion of degree measure into grade measure

In degree measurement, 1 right angle = 90°

In grade measurement, 1 right angle = 100g

Thus, 90° and 100g both of them represent a right angle.

90° = 100g

1° = 100 g
90

= 10 g
9

x° = 10 × x g
9

where x is the given angle in degree and to be converted into grade.

For example: Convert 45° into grade.

we have,

1° = 10 g
9

45° = 10 × 45 g
9

? 45° = 50g

1.3 Conversion of grade measure into degree measure

In this case, 100g = 90°

1g = 90 °
100

= 9°
10

xg = 9 × x °
10

where x is the given angle in grade and asked to convert into degree.

For example: Convert 60g into degree.

we have,

1g = 9°
10
9 °
60g = 10 × 60

? 60g = 54°

10 vedanta Excel in Additional Mathematics - Book 6

Measurement of Angles

1.4 Conversion of degree measure into radian measure

Since, sum of angles of a triangle = 180°

Also, sum of angles of a triangle = Sc

? 180° = Sc

i.e. 1° = Sc
180

For example : Convert 60° into radian.

We have 1° = Sc
180
Sc
Now, 60° = 180 × 60

= Sc
3

1.5 Conversion of grade measure into radian measure

Since, sum of angles of a triangle = 200g

Also, sum of angles of a triangle = Sc

? 200g = Sc

i.e. 1g = Sc
200

For example: Convert 150g into radian.

We have, 1g = Sc
200
Sc
Now, 150g = 200 × 150g

= 3Sc
4

Worked out Examples

Example 1. Convert 72° into grade measurement.
Solution:
We know that

90° = 100g

1° = 100 g= 10 g
90 9

72° = 10 × 72 g
9

= 80g

vedanta Excel in Additional Mathematics - Book 6 11

Measurement of Angles

Example 2. Convert 80g into degree measurement .
Solution:
We know that,

100g = 90°

1g = 90 °= 9°
100 10

80g = 9 × 80 °
10

= 72°

Example 3. Convert 45° into radian.
Solution:
We have, 1° = Sc
Example 4. 180
Solution: Sc
Now, 45° = 180 × 45
Example 5.
Solution: = Sc
4

Convert 120g into radian.

We have, 1g = Sc
200
Sc
Now, 120g = 200 × 120

= 3Sc
5
Sc
Convert 4 into

(a) degree measure (b) grade measure

(a) Since 1c = 180°
S
Sc 180° S
Now, 4 = S × 4

= 45°

(b) Since 1c = 200g
S
Sc 200g S
Now, 4 = S × 4

= 50g

Example 6. Find the unknown angles in the following figures.

(a) in degrees (b) in grades
R
P

x° Q

20° xg 80g M
S O P QN

12 vedanta Excel in Additional Mathematics - Book 6

Measurement of Angles

Solution: (a) Here, x° + 20° = 90° (complementary angle)
or, x° = 90° – 20° (straight angle = 200g)
? x° = 70°

(b) Here, xg + 80g = 200g
or, xg = 200g – 80g
? xg = 120g

Example 7. If the degree measurement of one of the complementary angles is
Solution: 63°, find the other angle in grade measurement.

We know that,

90° = 100g

1° = 100 g= 10 g
90 9
10
63° = 9 × 63 g

= 70g

Let the other complementary angle be xg.

Now, x + 70g = 100g

or, x = 100g – 70g

? x = 30g

The required angle is 30g.

Example 8. If the grade measurement of one of the supplementary angles is 75g,
Solution: find the other angle in degree measurement.

We know that

100g = 90°

1g = 90 ° = 9°
100 10
9 °
75g = 10 × 75 = 67.5°

Let the other supplementary angle be x.

Now, x + 67.5° = 180°

or, x = 180° – 67.5°

? x = 112.5°

So, the required angle is 112.5°.

vedanta Excel in Additional Mathematics - Book 6 13

Measurement of Angles

Example 9. Find the unknown angles in the following triangle in degree:

(a) P (b) A

x
30°

Q 30° 70° R

B xC

Solution: (a) We have,
the sum of angles of a triangle = 180°
i.e. x + 30° + 70° = 180°
or, x + 100° = 180°
or, x = 180° – 100°
? x = 80°

(b) Here 'ABC is a right angled triangle.
So, ‘B = 90°
The sum of angles of a triangle = 180°
or, 90° + x + 30° = 180°
or, 120° + x = 180°
or, x = 180° – 120°
? x = 60°

Example 10. Find the unknown angles in grade in the following triangles.

(a) P (b) C
90g 70g

60g x A x
Q R B

Solution: (a) We have the sum of angles of a triangle PQR is 200g

‘Q + ‘P + ‘R = 200g

i.e. 60g + 90g + x = 200g

or, 150g + x = 200g

14 vedanta Excel in Additional Mathematics - Book 6

Measurement of Angles

or, x = 200g – 150g
? x = 50g
(b) Here, 'ABC is a right angled triangle
i.e., ‘A = 100g
The sum of angles of triangle is 200g
i.e. ‘A + ‘B + ‘C = 200g
or, 100g + x + 70g = 200g
or, 170g + x = 200g
or, x = 200g – 170g
? x = 30g

Example 11. Find the each of angles of given triangles in radian:

(a) P (b) A

2x 3x

3x xR x x
Q B C

Solution: (a) Here, the sum of angles of triangle in radian is Sc

In triangle PQR,

‘P + ‘Q + ‘R = Sc

or, 2x + 3x + x = Sc

or, 6x = Sc

? x = Sc
6
Sc Sc
Now, ‘P = 2x = 2 × 6 = 3

‘Q = 3x = 3 × Sc = Sc
6 2
Sc
‘R = x = 6

(b) Here, the sum of angles of a triangle in radian is Sc.

In 'ABC,

‘A + ‘B + ‘C = Sc

or, 3x + x + x = Sc

or, 5x = Sc

vedanta Excel in Additional Mathematics - Book 6 15

Measurement of Angles

? x = Sc
5
Sc 3Sc
Now, A = 3x = 3 × 5 = 5

and, ‘B = ‘C = x = Sc
5

Exercise 1

Short Questions : In degree In grade In radian
1. Complete the table.
Sc
SN Given angle 2
(a) Right angle

(b) Two right angle 200g

or

Sum of angles of a triangle

(c) Half of right angle 1 × 90° = 45°
2

(d) 2 × right angle 2 × 90° = 36°
5 5

2. Convert the following degree into grade:

(a) 45° (b) 36° (c) 54°

(d) 9° (e) 72° (f) 81°

3. Convert the following grade measure into degree:

(a) 70g (b) 20g (c) 50g

(d) 40g (e) 60g (f) 30g

4. Convert the following degree measure into radian:

(a) 10° (b) 20° (c) 30°

(d) 45° (e) 90° (f) 36°

5. Convert the following grade measure into radian:

(a) 20g (b) 40g (c) 50g

(d) 100g (e) 75g (f) 150g

16 vedanta Excel in Additional Mathematics - Book 6

Measurement of Angles

6. Convert the following radian measure into degree:

(a) Sc (b) Sc (c) Sc
2 3 5
Sc Sc 4Sc
(d) 6 (e) 8 (f) 5

7. Convert the following radian measure into grade:

(a) Sc (b) 2Sc (c) Sc
5 5 4
3Sc Sc 4Sc
(d) 5 (e) 2 (f) 5

8. Find the unknown angles in degree from the following figures:

(a) (b)
x° 45°

40° x°

(c) (d)

x° 70° x°
60°

9. Find the unknown angles in grade from the following grade:

(a) (b)
xg
60g 72g
xg

(c) (d)
150g xg xg 108g

10. (a) If the degree measurement of one of the complementary angles is 27°, find
the other angle in grade measurement.

(b) If the grade measurement of one of the complementary angles is 50g, find
the other angle in degree measurement.

(c) If the degree measurement of one of the supplementary angles is 81°, find
the other angle in grade measurement.

(d) If the grade measurement of one of the supplementary angles is 32g, find
the other angle in degree measurement.

vedanta Excel in Additional Mathematics - Book 6 17

Measurement of Angles

Long Questions :
11. Find the unknown angles in degree from the following triangle:

(a) (b)
x° 30°

55° 70° 110° x°

(c) 70° 35° (d)
50°
x°
x°

12. Find the unknown angles in grade from the following figure:

(a) 70g (b)
x° 80g x°

120g

20g

(c) 40g (d)
x° 70g

x°

13. Find the measure of each angle in radian from the following figure:

(a) A (b) E
5x 2x

B 4x 3x

C F xG

(c) X (d) P
x x

Y xZ QR
vedanta Excel in Additional Mathematics - Book 6
18

Measurement of Angles

14. (a) If 3x°, 5x°, and 2x° are angles of a triangle, find the each angle in degree.
(b) If 4x°, 6x°, and 5x° are the angles of a triangle, find the each angle is degree.
(c) If 5xg, 2xg, and 3xg are the angles of a triangle, find the each angle in grade.
(d) If 3xg, 4xg, and 3xg are the angles of a triangle, find the each angles in grade.
(e) If 2xc, 3xc, and 4xc are the angles of a triangle, find each angle in radian.
(f) If 4xc, 6xc, and 10xc are the angles of a triangle, find each angle in radian.

2. (a) 50g (b) 40g (c) 60g (d) 10g

(e) 80g (f) 90g

3. (a) 63° (b) 18° (c) 45° (d) 36°

(e) 54° (f) 27°

4. (a) Sc (b) Sc (c) Sc (d) Sc
18 9 6 4
Sc Sc
(e) 2 (f) 5

5. (a) Sc (b) Sc (c) Sc (d) Sc
10 5 4 2
3Sc 3Sc
(e) 8 (f) 4

6. (a) 90° (b) 60° (c) 36° (d) 30°

(e) 22.5° (f) 144°

7. (a) 40g (b) 80g (c) 50g (d) 120g

(e) 100g (f) 160g

8. (a) 50° (b) 45° (c) 120° (d) 110°

9. (a) 40g (b) 28g (c) 50g (d) 92g

10. (a) 70g (b) 45g (c) 110g (d) 151.2°

11. (a) 55° (b) 40° (c) 75° (d) 40°

12. (a) 50g (b) 60g (c) 60g (d) 30g

13. (a) 5Sc , Sc , Sc (b) Sc , Sc , Sc (c) S4c, Sc , Sc (d) Sc , Sc , Sc
12 3 4 3 2 6 2 4 3 3 3
14. (a) 54°, 90°, 36° (b) 48°, 72°, 60° (c) 100g, 40g, 60g (d) 60g, 80g, 60g

(e) 29Sc, Sc , 4Sc (f) Sc , 3Sc , Sc
3 9 5 10 2

vedanta Excel in Additional Mathematics - Book 6 19

Matrices

5Matrices

5.1 Matrix - Introduction

In a class test, Hary secured 48 marks in Mathematics, 45 marks in Science and 44
marks in English out 50 full marks.

Now, let’s present these marks in a table:

Subjects Mathematics Science English
Marks obtained 48 45 44

The marks obtained by Hary presented on the above table can be arranged as:
[48 45 44]

Here, the marks are arranged in a row enclosing by the brackets [ ].
Again, let’s present these marks in table by another ways.

Subjects Marks
Mathematics 48
Science 45
English 44

The marks obtained by Hary presented on the above table can also be arranged as :

48
45
44
Here, the marks are arranged in a column enclosed by the brackets [ ].

The arrangement of numbers in rows and column enclosing by brackets [ ] is called
matrix. Matrices is the plural form of matrix.

Usually, matrices are represented by capital letters like A, B, C, D, ....., X, Y, Z.

For example:

A = [48 45 44], B = 2 4 8 and so on.
3 ,C= 7 5

4

64 vedanta Excel in Additional Mathematics - Book 6

Matrices

Furthermore, the table given below shows the marks obtained by Jomes, Jerry, and
Kristina in a test examination of 50 full marks of each subject.

Subjects James Name of Students Kristina
44 Jerry 36
Mathematics 43 38 42
Science 40 42 45
English 40

Now, the marks can be arranged in the matrix form marks:

44 38 36
A = 43 42 42

40 40 45

Thus, the marks are arranged in 3 rows and in 3 column in the given matrix.

Again, let’s present the above marks obtained by three students in the table by
another way.

Name of Mathematics Subjects English
Students 44 Science 40
38 40
James 36 43 45
Jerry 42
Kristina 42

In this case, the marks can be arranged in the matrix form.

44 43 40
A = 38 42 40

36 42 45

The data written in the form of a matrix are called the members or elements of the
matrix.

Definition: A rectangular arrangements of elements in rows and columns enclosed
by a pair of round or square brackets is called a matrix.

5.2 Rows and Columns of a Matrix

In a matrix, numbers are arranged horizontally and vertically. The horizontal
arrangements are called rows and the vertical arrangements are called columns.

For example :

A = [4 5 6] It has only one row.

vedanta Excel in Additional Mathematics - Book 6 65

Matrices

15 It has only one column.
B = 12 It has two rows and two columns.

13

C= 6 7
8 9

P= 8 9 12 It has two rows and three columns.
11 14 12

In a matrix, pq r First row (R1)
A= s tu Second row (R2)
vwx Third row (R3)

First Second Third
column column column
(C1) (C2) (C3)

The members of the first row are p, q, r; the second row has s, t, u; and the third row
has v, w, x.

Also, the members of the first column are p, s, v; the second column has q, t, w, and
the third column has r, u, x.

5.3 Order of a Matrix

Let us consider a matrix. A = 1 2 3
4 5 6
Here, number of rows of matrix A = 2

number of columns of matrix A = 3.

It means that the matrix A has 2 rows and 3 columns.

This matrix is called order of 2 × 3 (read as 2 by 3).

Definition: The number of "rows × columns" of a matrix is known as the order of
the matrix.

Example: Let P = 1 2
3 4
It has 2 rows and 2 columns.

Thus, the order of matrix P is 2 × 2. It is also written as P2 × 2 to say that the matrix
P is of order 2 × 2.

66 vedanta Excel in Additional Mathematics - Book 6

Matrices

5.4 Equal Matrices

Two matrices are said to be equal if their orders are equal and the corresponding
elements are the same.

For example:

If A = 4 7 and B= 4 7 , then matrix A = matrix B
6 8 6 8

Also, if x 8 = 5 w , then x = 5, w = 8, z = 9 and y = 4.
9 y z 4

5.5 Operations on Matrices

Let us study the following two matrices of sales of newspapers in two shops in two
days.

Shop M

Day Annapurna Newspapers Naya Patrika
40 Kantipur 60
Sunday 45 50 65
Monday 60

Shop N

Day Annapurna Newspapers Naya Patrika
44 Kantipur 62
Sunday 48 60 68
Monday 65

Then, we can find the total number of newspapers sold in two days. To find the total
number of newspapers sold in two days, we can add the corresponding elements:

i.e., M + N = 40 + 44 50 + 60 60 + 62
45 + 48 60 + 65 65 + 68

= 84 110 122
93 125 133

Addition, subtraction, and multiplication are operations on matrices. Here, we
study only about addition and subtraction of matrices.

(a) Addition of Matrices

Let P = 1 2 and Q = 6 2 be two matrices. They are of the same order.
4 5 1 3
Then, the sum or addition of matrices P and Q is denoted by P + Q. Then, their

corresponding elements are added to find P + Q.

vedanta Excel in Additional Mathematics - Book 6 67

Matrices

i.e., P + Q = 1 2 + 6 2
4 5 1 3

= 1+6 2+2
4+1 5+3

= 7 4
5 8

Two or more matrices of the same orders can be added.

(b) Subtraction of Matrices

Let M = 2 4 and N = 1 2 be two matrices.
8 2 6 0
Then, the difference of two matrices M and N is denoted by M – N and it is

calculated by subtracting their corresponding elements.

i.e., M – N = 2 4 – 1 2
8 2 6 0

= 2–1 4–2
8–6 2–0

= 1 2
2 2

Two matrices of same orders can be subtracted.

Worked out Examples

Example 1. State the orders of the following matrices:
Solution:
(a) M= 1 2 (b) N= 2 46
4 5 3 21

12 2 4]

(c) P = 3 4 (d) Q = [1

56

a

(e) R = b

c

(a) Here, M = 1 2
4 5

In matrix M,

number of rows = 2

number of columns = 2

Thus, M is a matrix of order 2 × 2.

68 vedanta Excel in Additional Mathematics - Book 6

Matrices

(b) Here, N = 2 4 6
3 2 1

In matrix N,

number of rows = 2

number of columns = 3

Thus, N is a matrix of order 2 × 3.

12
(c) Here, P = 3 4

56

In matrix P,

number of rows = 3

number of columns = 2

Thus, order of matrix P is 3 × 2.

(d) Here, Q = [1 2 4]

In matrix Q,

number of rows = 1

number of columns = 3

Thus, order of matrix Q is 1 × 3.

a
(e) Here, R = b

c

In matrix R,

number of rows = 3

number of columns = 1

Thus, order of matrix R is 3 × 1.

Example 2. Find the values of x, y, z, and w of x y = 4 5 .
Solution: z w 6 7

Here, x y = 4 5
z w 6 6

Equating the corresponding elements of the equal matrices

we get, x = 4, y = 5, z = 6 and w = 7.

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Matrices

Example 3. Add two matrices P and Q if P = 2 3 and Q = 6 7 .
8 4 3 2

Solution: Here, P= 2 3 and Q = 6 7
8 4 3 2

The given matrices are of same order we can find P + Q.

Now, P + Q = 2 3 + 6 7
8 4 3 2

= 2+6 3+7
8+3 4+2

= 8 10
11 6

Example 4. If A = 1 2 and B = 3 4 , show that A + B = B + A.
5 3 7 4

Solution: Here, A = 1 2 and B = 3 4
5 3 7 4

Now, A + B = 1 2 + 3 4
5 3 7 4

= 1+3 2+4
5+7 3+4

= 4 6
12 7

Again, B + A = 3 4 + 1 2
7 4 5 3

= 3+1 4+2
7+5 4+3

= 4 6
12 7

? A + B = B + A Proved.

Note :

In matrix addition, A + B = B + A.

It is called commutative property of matrix addition.

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Example 5. If A = 10 3 and B = 4 5 , find A – B and B – A.
7 4 2 3

Solution: Here, A = 10 3 and B = 4 5
7 4 2 3

Now, A – B = 10 3 – 4 5
7 4 2 3

= 10 – 4 3–5
7–2 4–3

= 6 –2
5 1

Again, B – A = 4 5 – 10 3
2 3 7 4

= 4 – 10 5–3
2–7 3–4

= –6 2
–5 –1

Here, A – B and B – A are not equal.

Example 6. Mr. Hanuman secured marks of 45 in English, 40 in Mathematics
and 38 in Science in a class test of 50 marks. Write the marks in a
table as well as in matrix form. (use 1 × 3 matrix form)

Solution: The marks secured by Mr. Hanuman can be written in a table as below:

English Mathematics Science
45 40 38

In matrix form [45 40 38].

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Matrices

Exercise 5

Short Questions :

1. State the orders of the following matrices:

(a) A = [1 2 3] (b) B= 4 8
2 3

(c) C= 2 (d) D= 1 2 3
7 4 5 6

24 123

(e) E = 5 6 (f) P = 4 5 6

79 789

2. Find the values of a , b, c and d in the following equal matrices.

(a) a 8 = 5 c (b) 8 4a = 4c 2
7 b d 10 3b 15 9 d

3. (a) Find the values of x and y if x–5 8 = 2 8 .
9 y+6 9 8

(b) Find the values of p and q if p–4 12 = 2–p 12 .
20 2q + 1 20 q+8

4. Add the following matrices:

(a) A = [1 2 3] and B = [2 7 5]

(b) P = [6 2 4] and Q = [3 4 5]

(c) E= 2 3 and F = 6 2
4 5 7 4

(d) T= –2 4 and U = 6 2
5 6 4 8

(e) M= 0 2 and N = 2 3
4 6 4 5

(f) G= 2 4 and H = –2 –4
6 7 –6 –7

(g) T= 1 2 3 and P = 6 2 7
4 5 6 8 4 8

5. Find P – Q in the following cases.

(a) P= 2 4 and Q = 1 2
6 7 3 5

72 vedanta Excel in Additional Mathematics - Book 6

Matrices

(b) P= 6 7 and Q = 2 3
8 2 3 4

(c) P= 2 4 and Q = 2 8
6 14 3 4

(d) P = [4 5 6] and Q = [2 4 6]

47

(e) P = 5 and Q = 8

69

(f) P= 2 4 6 and Q = 3 2 7
3 2 4 9 2 4

Long Questions :

6. (a) The marks obtained by Romeo a class 6 student in a class test are given as:
Mathematics 47, Science 44, English 46. Present the given marks in a table.
Then, arrange the marks in a matrix form. (use 1 × 3 matrix form)

(b) Shushma obtained 43 marks in mathematics and 42 marks in science in an
exam. In the same exam, her friend Kajal obtained 48 marks in mathematics
and 42 marks in science. Present their marks in a table. Then, arrange the
marks in a matrix form.

(c) In an examination Sambhu got 45 marks in computer, 42 marks in Nepali,
and 43 marks in social studies. In the same exam, his friend Parvati got 46
marks, 40 marks, and 45 marks respectively in the three above subjects.
Tabulate the marks obtained by these two students. Then, arrange their
marks in a matrix form. (use 2 × 3 matrix form)

(d) A shopkeeper sold 22 kg of Sugar on Sunday, 25 kg on Monday, and 30 kg
on Tuesday. Similarly, she sold 28 kg of rice on Sunday, 30 kg on Monday,
and 34 kg on Tuesday. She also sold 21 kg of Wheat flour on Sunday, 28 kg
on Monday, and 33 kg on Tuesday. Present the sales of these different items
on different days in a table. Then, arrange the data in a matrix form.

7. (a) If A = 2 4 and B = 7 4 , show that A + B = B + A.
6 10 3 2

(b) If P = 4 5 and Q = 2 8 , show that P + Q = Q + P.
6 7 7 4

(c) If G = –4 –3 and H = 14 6 , show that G + H = H + G.
2 –1 4 8

8. (a) If A = 2 4 and B = 7 5 , find A – B and B – A.
6 7 8 10

(b) If M = 2 4 and N = 10 6 , find M – N and N – M.
5 6 8 2

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Matrices

(c) If T = 2 4 and U = –2 8 , then T – U and U – T.
–6 7 4 9

1. (a) 1 × 3 (b) 2 × 2 (c) 2 × 1 (d) 2 × 3

(e) 3 × 2 (f) 3 × 3

2. (a) a = 5, b = 10, c = 8, d = 7 (b) a = 1 , b = 3, c = 2, d = 15
3. (a) x = 7, y = 2 2
(b) p = 3, q = 7

4. (a) [3 9 8] (b) [9 6 9] (c) 8 5 (d) 4 6
11 9 9 14
2 5 0 0 7 4 10
(e) 8 11 (f) 0 0 (g) 12 9 14

5. (a) 1 2 (b) 4 4 (c) 0 –4 (d) [2 1 0]
3 2 5 –2 3 10

–3 (f) –1 2 –1
(e) – 3 –6 0 0

–3

6. (a) [47 44 46] (b) 43 42 (c) 45 42 43
48 42 46 40 45
Sugar Rice Wheat

Sunday 22 28 21

(d) Monday 25 30 28

Tuesday 30 34 33

8. (a) –5 – 1 , 5 1 (b) –8 – 2 , 8 2
–2 – 3 2 3 –3 – 4 3 4
4 – 4 4 4
(c) – 10 – 2 , 10 2

74 vedanta Excel in Additional Mathematics - Book 6

Class : 6 Model Questions Matrices
Full marks : 50
Annual Examination
Time : 1.5 hours

Attempt all the questions.

Group A [9×2 = 18]

1. Find the values of 'x' and 'y' if (4x + 3, y + 2) = (11, 4).

2. If P = 1 2 and Q = 6 3 , find P + Q.
3 4 1 4

3. If A = 2 4 and B = 4 6 , find A – B.
6 7 2 7

4. Convert 81° into grade and radian measure.

5. In a right angled triangle, if one of acute angle is 27°, find the other angle in
grade.

6. From the given right angled triangle, find the unknown side.
C

5 cm

AB
3 cm

7. Simplify : 4sinA – 3cosA + sinA + 8cosA
8. Prove that : sinA + cosA . tanA = 2sinA
9. Find the distance between A(6, 7) and B(3, 3).

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Matrices

Group B [8×4 = 32]

10. If 3x°, 5x° and 2x° are angles of a triangle, find each angle in degree.

11. Convert the following degree in radian and grade.

(a) 45° (b) 72°

12. From the given right angled triangle, find six trigonometric ratios of angle T.

C

2
3

AT B
1

13. Prove the following :

(a) sin2A + cos2A = 1 (b) (1 – cos2T) . sec2T = tan2T

14. Plot the given points on a graph paper A(3, 3), B(9, 3), and C(9, 11).

Join AB, BC, and CA show that it is a right angled triangle.
15. If A(6, 7), B(9, 11), C(7, 2) and D(4, 6) are four points, then show that AB = CD.

16. If P = 2 4 and Q = –6 2 , prove that P + Q = Q + P.
6 7 5 –4

17. If M = 3 6 and N = 2 7 , find M – N and N – M, are they equal?
7 8 –3 –4

76 vedanta Excel in Additional Mathematics - Book 6