Ace Education Mathematics O'level (Ace Education Book Series)

341 Pages·2021·14.792 MB·English

by  Nswana Ching'ambu

ACE EDUCATION
MATHEMATICS O’LEVEL

Grade 10 − 12

GCSE, GCE

NSWANA CHING’AMBU

The Authorship and Career Network
[email protected]
[phone], [phone]

i
The Authorship and Career Network
‘Impossibility Our Possibility’

The Authorship and Career Network is an organisation with a sole
objective of promoting excellence in research, literacy, scholarship, education
and skill development by supporting authors in Africa and beyond
to publish their works.

Published for Africa by
The Authorship and Career Network
Indeco House, Cairo Road, Lusaka,
Zambia.

Copyright © 2021 by Nswana Ching’ambu

All rights reserved. No part of this publication may be reproduced,
stored or transmitted in any form or by any means, electronic, mechanical,
photocopying, recording, scanning, or otherwise without written
permission from the publisher. It is illegal to copy this book,
post it to a website, or distribute it by any other means
without permission.

Nswana Ching’ambu
asserts the moral right as an author

First edition

ISBN: 978-9982-913-90-4

ii
To the students

Ace Education Book Series aims at giving detailed material in the simplest way to help students
understand and recall information easily. The books also highlight the importance and application of each
topic in real life so that students can understand why they are learning the material, how the material relates to
or can be used in real life.

Ace Education Mathematics O’level consists of 33 units. Each unit begins with the introduction and
overview of the unit, and ends with the review questions and solutions, except few introductory units. To
fully benefit, students are advised to cover everything in each unit considered. Students can also learn more
from our social media platforms. On these platforms, students can find additional information, ask questions
or participate in helping other students.

With full confidence, this book will help a number of students, not just by boosting their scores, but also to
understand Mathematics Ordinary Level.

Ace Education

About the author

Nswana Ching’ambu has a bachelor’s degree in Human Biology from the University of Zambia and Teaching
Methodology from Gideon Robert University. He has been lecturing Clinical Medicine courses at Gideon
Robert University for many years. He lectures Anatomy and Physiology, Medical Biochemistry and Nutrition,
and previously used to lecture Cellular Pathology and Basic Microbiology. He is the founder of Ace
Education and the author of Ace Education Book Series for O’Levels: Biology, Chemistry, Physics and
Mathematics.
[phone]
[email protected]

[email protected]
[email protected]

iii
TABLE OF CONTENTS

UNIT 1: NUMBERS………..……………………………………………………………………………...…..1
UNIT 2: SETS……………………………………..…………………………………………...………..…....12
UNIT 3: LAWS OF ARITHMETIC…………………...……………………………………………….…...21
UNIT 4: FRACTIONS………………………………………………………………………………….…….24
UNIT 5: PERCENTAGES…………………………………………………………...………………....…….29
UNIT 6: RATIO AND PROPORTION……………….………………………………………….....………34
UNIT 7: INDICES………………...………………………………………………………………………….40
UNIT 8: SEQUENCES AND SERIES………..………………………………………………….…………47
UNIT 9: ALGEBRA………………..……………………………………………………………….………..56
UNIT 10: LINEAR EQUATIONS………………...…………………………………………….…………..71
UNIT 11: COORDINATE GEOMETRY..……………………………………………………….………….82
UNIT 12: QUADRATIC EQUATIONS……………………………….……………………….……………92
UNIT 13: LINEAR INEQUALITIES……………………………….………………………..…………….108
UNIT 14: RELATIONS AND FUNCTIONS………………………...……………………………..……..123
UNIT 15: LINEAR FUNCTIONS…………….……………………………………………………………136
UNIT 16: GRAPHS OF LINEAR FUNCTIONS…………………………………………………………142
UNIT 17: GRAPHS OF CUBIC FUNCTIONS……………..……………………………………………156
UNIT 18: MOTION GRAPHS…………..………………………………………………………..………..166
UNIT 19: MATRICES…………..………………………………………………………………..…………174
UNIT 20: MENSURATION………………………..……………………………………………...………..190
UNIT 21: ANGLES…………………………………..……………………………………………..………204
UNIT 22: CIRCLE THEOREMS……………………………………………………………………..……211
UNIT 23: POLYGONS………………………….……………………………………………………...…...221
UNIT 24: TRIANGLES……………..…………………………………………………………………...….225
UNIT 25: SYMMETRY…………………………………………………………………………………….230
UNIT 26: CONSTRUCTION…………………………………………………………………….…………234
UNIT 27: EARTH GEOMETRY……………………...…………………………………………………...241
UNIT 28: PYTHAGORAS’ THEOREM……………………………………………………………...….. 257
UNIT 29: TRIGONOMETRY………..……………………………………………………….…………….260
UNIT 30: VECTORS…………………………………………………………………………………….….277
UNIT 31: TRANSFORMATIONS..………………………………………………………………………...297
UNIT 32: STATISTICS…………...………………………………………………………………….……..309
UNIT 33: PROBABILITY…..………………………………………………………………………………329

iv
1.0 NUMBERS

Introduction

We always use mathematics in our everyday lives. For example, you go to a store to buy exercise books. How
many will be purchased and for how much? Math. You wake up and want to plan your day. What time will
you do that and for how long? Math. You are cooking. How much water, salt and seasoners to add? Math.
You are building a house. How wide and high will it be? It is math again. In short, we knowingly or
unknowingly apply mathematics in most of our everyday life activities. Undoubtedly, without mathematics
life can be chaotic. We cannot solve even the simplest problem nor can we communicate with each other
clearly.

Mathematics is a language of science. It deals with numbers, quantities, shapes and how they relate to each
other. Any language has the alphabet, words, sentences, punctuation marks and rules of grammar.
Mathematics being a language also has these. The alphabet of mathematics are numbers. Words of
mathematics are expressions, such as 1 cup of water, 3 teaspoons of sugar, etc. Sentences of mathematics are
equations such as moving at 30km/hr we will cover 90km in 3 hours. The punctuation marks of mathematics
are signs and symbols such as +, −, × and ÷. Mathematics also has sets of rules which govern how numbers,
expressions, equations and signs have to be used.

What are numbers? They are a way of expressing the quantity of something. This unit covers different sets of
numbers. It also covers multiples, factors, squares and reciprocal of a number and many more.

Specific outcomes

By the end of the unit, you will be able to:
❖ Describe the sets of numbers: integers, natural numbers, whole numbers, rational numbers, irrational

numbers and real numbers
❖ Find the lowest common multiple (LCM) for the given numbers
❖ Find the highest common factor (HCF) for the given numbers
❖ Find the prime factors of a numbers
❖ Find a square number and cube number of a numb
❖ Find the square root and cube root
❖ Find the reciprocal of a number
❖ Work out significant figures
❖ Standard form:
• Convert standard notation to standard form
• Convert standard form to standard notation
• Add and subtract standard form
• Multiply and divide standard form

Numbers 1
NUMBERS
1.1 SETS OF NUMBERS
1. INTEGERS
• Integers are numbers that are expressed without a fraction.
• They are from negative infinity to positive infinity.
• The symbol for integers: ℤ
• Examples of integers: …, −3, −2, −1, 0, 1, 2, 3, …

2. POSITIVE INTEGERS
• Positive integers begin from one to infinity.
• The symbol for positive integers: ℤ+
• Examples of positive integers: 1, 2, 3, 4, 5, …

3. NEGATIVE INTEGERS
• Negative integers begin from infinity to negative one.
• The symbol for negative integers: ℤ−
• Examples: −1, −2, −3, −4, −5, …

4. NATURAL NUMBERS
• Natural numbers are positive integers from one to infinity.
• They are used for counting.
• The symbol for natural numbers: ℕ
• Examples: 1, 2, 3, 4, 5, …

5. WHOLE NUMBERS
• Whole numbers are positive integers including zero.
• The symbol for whole numbers: 𝕎
• Examples of whole numbers: 0, 1, 2, 3, 4, 5, …

6. RATIONAL NUMBERS
𝑎
• Rational numbers are numbers that can be expressed as , where a and b are integers,
𝑏
and b ≠ 0
• The symbol for rational numbers: ℚ
1 5 7
• Examples of rational numbers: , , − , …
2 3 8

7. IRRATIONAL NUMBERS
𝑎
• Irrational numbers are numbers that cannot be expressed as , where a and b are integers,
𝑏
and b ≠ 0
• The symbol for irrational numbers: 𝕀
• Examples of irrational numbers: √2, √3,√5, π, e

8. REAL NUMBERS
• Real numbers are a set of rational numbers, irrational numbers, and integers.
• The symbol for real numbers: ℝ
5
• Examples of real numbers: −2, 0, 4, , √2, …
13

2 Numbers
❖ WORKED EXAMPLES
1. Identify the following numbers as rational, irrational or integer
2
A.
3
B. −10
C. √15
D. 2.5
E. 10000
F. √7
G. 0
H. −0.2

❖ SOLUTIONS
1. A. Rational number
B. Integer
C. Irrational number
D. Rational number
E. Integer
F. Irrational number
G. Integer
H. Rational number

1.2 MULTIPLES
• Multiples are numbers that are a product of a number by another integer.
• Examples:
i. Multiples of 2 are 2, 4, 6, 8, 10, …; 2 is a product of 2 and 1. 4 is a product of 2 and 2 and so on.
ii. Multiples of 3 are 3, 6, 9, 12, 15, …; 6 is a product of 3 and 2. 12 is a product of 3 and 4.
iii. Multiples of 4 are 4, 8, 12, 16, 20, …; 12 is a product 4 and 3, 20 is a product of 4 and 5.

❖ LOWEST COMMON MULTIPLE (LCM)
• The lowest common multiple is the smallest number which is a multiple of both given numbers.
• Examples:
i. Common multiples of 2 and 3 are 6, 12, 18, 24, … The lowest common multiple of 2 and 3
is 6.
ii. Common multiples of 4 and 5 are 20, 40, 60, … The lowest common multiple of 4 and 5
is 20.
iii. Common multiples of 3 and 6 are 6, 12, 18, 24, … The lowest common multiple of 3 and 6
is 6.

1.3 FACTORS
• A factor is a number that divides into another number without leaving a remainder.
• Examples:
i. Factors of 12 are 1, 2, 3, 4, 6 and 12.
ii. Factors of 18 are 1, 2, 3, 6, 9 and 18.

❖ HIGHEST COMMON FACTOR
• The highest common factor is the largest number which is a factor of both given numbers.
• Examples:
i. Common factors of 12 and 18 are 1, 2, 3 and 6. The highest common factor of 12 and 18 is
6.
ii. Common factors of 20 and 30 are 1, 2, 3, 5 and 10. The highest common factor of 20 and
30 is 10.
iii. Common factors of 9 and 18 are 1, 3 and 9. The highest common factor of 9 and 18 is 9.
Numbers 3
1.4 PRIME FACTORISATION
❖ PRIME NUMBER
• A prime number is a number that can be divided only by one and itself.
• Examples of prime numbers: 2, 3, 5, 7, 13, 17, 19, 23, 31, …

❖ PRIME FACTOR
• A prime factor is a factor that is also a prime number.
• Prime factors of 15 are 3 and 5; 3 and 5 are factors of 15 and they are also prime numbers.
• Prime factors of 30 are 2, 3 and 5.

❖ PRIME FACTORISATION
• Prime factorisation is the process of expressing a number into products of prime numbers.
• Examples:
i. Prime factorisation of 30 = 2 × 3 × 5

ii. Prime factorisation of 48 = 2 × 2 × 2 × 2 × 3

iii. Prime factorisation = 280

4 Numbers
1.5 SQUARE AND CUBE NUMBERS
1. SQUARE NUMBERS
• A square number is the product of a number multiplied by itself.
• Examples of square numbers:
i. 9 is a square number; a product of 3 × 3.
ii. 36 is a square number; a product of 6 × 6.
iii. 16 is a square number; a product of 4 × 4.
iv. 144 is a square number; a product of 12 × 12.
• Squares are written as 𝑎2 where a is a real number.
i. 32 = 9
ii. 62 = 36
iii. 42 = 16
iv. 122 = 144

2. CUBE NUMBERS
• A cube number is the product of a number multiplied by itself three times.
• Examples of cube numbers:
i. 27 is a cube number; a product of 3 × 3 × 3.
ii. 8 is a cube number; a product of 2 × 2 × 2.
iii. 64 is a cube number; a product of 4 × 4 × 4.
iv. 125 is a cube number; a product of 5 × 5 × 5.
• Cube numbers are written as 𝑎3 where a is a real number.
i. 23 = 8
ii. 43 = 64
iii. 53 = 125

1.6 SQUARE AND CUBE ROOTS
1. SQUARE ROOTS
• A square root is a number which when multiplied by itself twice gives a number.
• Examples of square roots:
i. 4 is a square root of 16; 4 × 4 gives 16.
ii. 3 is a square root of 9; 3 × 3 gives 9.
iii. 5 is a square root of 25; 5 × 5 gives 25.
iv. 7 is a square root of 49; 7 × 7 gives 49.
• Square roots are written as √𝑎
i. √16 = 4
ii. √9 = 3
iii. √25 = 5
iv. √49 = 7

2. CUBE ROOTS
• A cube root is a number which when multiplied by itself three times gives a number.
• Examples of cube roots:
i. 2 is a cube root of 8; 2× 2 × 2 gives 8.
ii. 3 is a cube root of 27; 3 × 3 × 3 gives 27.
iii. 4 is a cube root of 64; 4 × 4 × 4 gives 64.
iv. 5 is a cube root of 125; 5 × 5 × 5 gives 125.
• Cube roots are written as 3√𝑎
i. 3√8 = 8
ii. 3√27 = 3
iii. 3√64 = 4
iv. 3√125 = 5
Numbers 5
1.7 RECIPROCAL
1
• A reciprocal is a number which when multiplied by a fraction gives one, that is, a × = a.
𝑎
• The reciprocal of a number or a fraction is found by flipping over a number or a fraction.
𝑎 𝑏 1
• The reciprocal of = , reciprocal of b = .
𝑏 𝑎 𝑏

❖ WORKED EXAMPLES
1. Find the reciprocal of the following
2
A.
3
50
B.
7
C. 10
D. √2

❖ SOLUTIONS
3
1. A.
2
7
B.
50
1
C.
10
1
D.
√2

1.8 SIGNIFICANT FIGURES
1. RULES OF SIGNIFICANT FIGURES
1. All non-zero digits are significant. For example, 157 has three significant figures.
2. Zeros between non-zero digits are significant. For example, 2001 has four significant figures.
3. Trailing zeros are not significant, but they are if there is a decimal point, for example:
i. 200 has one significant figure while 200. has three significant figures.
ii. 1500 has two significant figures while 5.100 has four significant figures.
4. Terminal zeros to the right of the decimal point are significant, for example:
i. 0.100 has three significant figures.
ii. 2.100 has four significant figures.
iii. 1.00 has three significant figures.
5. Leading zeros are not significant, for example:
i. 05 has one significant figure.
ii. 0.0013 has two significant figures.

❖ WORKED EXAMPLES
1. How many significant figures are in each of the following?
A. 1.0030
B. 28.20
C. 500720
D. 0.00021
E. 720000

❖ SOLUTIONS
1. A. 1.0030 has five significant figures.
B. 28.20 has four significant figures.
C. 500720 has five significant figures.
D. 0.00021 has two significant figures.
E. 720000 has two significant figures.

6 Numbers