CONTENTSChapter 1 Numbers1.1 Types of numbers 1

1.2 Sets 6

1.3 Calculating squares, square roots, cubes and cube roots of numbers 18

1.4 Directed numbers 20

1.5 Fractions, decimals and percentages 22

1.6 Ordering 30

1.7 Indices and standard form of numbers 33

1.8 The four rules 37

1.9 Estimation 39

1.10 Upper and lower bounds 42

1.11 Ratio, proportion and rate 44

1.12 Calculations involving percentages 56

1.13 Using a calculator 61

1.14 Time 64

1.15 Money and currency 66

1.16 Simple interest and compound interest 69

1.17 Exponential growth and decay 72

Chapter 2 Algebra2.1 Use of symbols 79

2.2 Algebra (Expansion and factorisation) 86

2.3 Algebraic fractions 101

2.4 Exponent laws 106

2.5 Linear, quadratic and simultaneous equations 111

2.6 Inequalities and linear programming 137

2.7 Sequences and patterns 144

2.8 Variation 150

2.9 Graphs in practical situations 151

2.10 Graphs of different functions 161

2.11 Estimating the gradients of tangents to curves 173

2.12 Functions 176

Chapter 3 Geometry3.1 Fundamental geometric

terminology and concepts 189

3.2 Constructions 194

3.3 Scale drawings 201

3.4 Similar figures and solids 205

3.5 Symmetry 221

3.6 Angle properties (Parallel lines, regular polygons and circles) 226

3.7 Loci 256

Chapter 4 Mensuration4.1 Units of conversion 269

4.2 Perimeter and area 272

4.3 Circles and sectors 276

4.4 Volume and surface area of solids 282

4.5 Compound shapes 286

Chapter 5 Coordinate Geometry5.1 Introduction to co-ordinate

geometry 291

5.2 Gradient (slope) 293

5.3 Distance between two points 297

5.4 Equations of straight lines 301

5.5 Parallel lines 305

5.6 Perpendicular lines 308

Chapter 6 Trigonometry and Pythagoras’ Theorem6.1 Introduction to bearings 313

6.2 Pythagoras’ theorem and trigonometric ratios 315

6.3 Finding the area of a triangle using the sine ratio 330

6.4 Trigonometry problems in three dimensions 348

Chapter 7 Vectors and Matrices7.1 Introduction to vectors 355

7.2 Transformation geometry 367

7.3 Vector geometry 380

7.4 Introduction to matrices 388

7.5 Transformations and matrices 399

Chapter 8 Probability8.1 Introduction to probability 409

8.2 Expectation 412

8.3 Complementary events 414

8.4 Estimating probability 416

8.5 Probability of simple combined events 418

Chapter 9 Statistics9.1 Organising and describing data 429

9.2 Statistical graphs 433

9.3 Measuring the centre of ungrouped data sets 441

9.4 Measuring centre for grouped data 444

9.5 Cumulative frequency diagrams 447

9.6 Correlation 460

Practice Test (Paper 2)

Practice Test (Paper 4)

Answers and markschemes can be downloaded for free at www.ntk.edu.hk

NumbersChapter

1.1 Types of numbers

Rational numbersA rational number is a number which can be expressed as a fraction of the form a

b, where a

and b are both whole numbers. Rational numbers consist of:

1. All natural numbers

e.g. 0, 1, 2, 3, 4, …

2. All integers (Positive, negative and zero)

e.g. …, −3, −2, −1, 0, 1, 2, 3, …

3. All fractions (Proper and improper fractions)

e.g. 12

, 1 37

, 34

, …

4. All recurring decimals

e.g. 0.6, 0.23,� � � …

5. All terminating decimals

e.g. 0.12, 1.278, …

Irrational numbersAn irrational number is a number which cannot be expressed as a fraction. Here are some common examples of irrational numbers:

2, 3, , 7 ,3π

Real numbersThe real numbers are all the rational and irrational numbers.

Square numbersA square number is an integer which is the square of another integer. The first few square numbers are 1, 4, 9 and 16.

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Cambridge IGCSE Mathematics Study Guide

Solution

3, 3.14, , 4 , 1037

, 73πa

Solution

a 4 (as 4 2)=3.14, 4 , 103

7(3.14 is rational as 3.14 314

100)=

c 3, , 73π

Prime numbersA prime number is a positive integer greater than 1 that is not divisible by any integer except

1 and itself. This definition means the number 1 is not counted as a prime.

Below is a list of all the prime numbers between 1 to 50:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

The prime factors of a number are the prime numbers which divide that number exactly.

For example, 2 and 3 are the prime factors of 6.

a

Solution

a

Prime numbersUntil recently, research into prime numbers was considered ‘pure’ mathematics because it had no practical application. But in the 1970s computer programmers discovered they could use prime numbers to make their programs more secure.

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Chapter 1 Numbers

A quick way to find the prime factors of a number is by short division. Example 1.4 shows

you how to do this.

a

Solution

a 2 12

2 6

3

2 84

2 42

3 21

7

2 124

2 62

31

Highest common factor (HCF)The HCF of two integers a and b is the largest integer that divides both a and b without a

remainder.

For example: HCF 8, 12 4( ) = , HCF 15, 30 15( ) =

Lowest common multiple (LCM)The LCM of two integers a and b is the smallest integer that is a multiple of both a and b.

For example: LCM 8, 12 24( ) = , LCM 3, 5 15( ) = .

The HCF and LCM can be found by short division. Example 1.5 shows you how to do this.

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Cambridge IGCSE Mathematics Study Guide

c

Solution

a 2 12 18

3 6 9

2 3

3 45 75

5 15 25

3 5

c 2 24 36 60

2 12 18 30

3 6 9 15

2 3 5

Exercise 1.1

1 A collection of ten numbers is

7, , 2, 34

, 111, 2.4, 3, 25, 0, 5 12

π −

From this collection write down all the:

a integers, b rational numbers,

c irrational numbers, d square numbers.

2 A collection of eight numbers is

6.2, 7 , 4 , 89

, 3, 71, 81, 3.13 − −

From this collection write down all the:

a positive integers, b negative integers,

c rational numbers, d square numbers.

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Chapter 1 Numbers

3 Identify and write down all the prime numbers in each of the following sets of numbers:

a 1, 3, 7, 10, 13 b 2, 6, 15, 17, 20

c 12, 16, 23, 29, 31 d 5, 9, 14, 37, 43

e 38, 41, 43, 47, 51 f 53, 56, 58, 61, 79

4 From the list of numbers 2, 2.1, 1, 3, 5, 4.62, 43

, 25− − write down all the:

a natural numbers, b irrational numbers,

c real numbers, d prime numbers.

5 Identify and write down all the multiples of 7 in the following collection of numbers:

13, 56, 70, 30, 14, 63, 50, 1, 7

6 Write down all the positive factors of the following numbers:

a 6 b 12

c 25 d 13

e 30 f 64

7 By writing down all the positive factors of the numbers, find the HCF of:

a 8 and 12 b 16 and 40

8 By writing down the first 6 multiples of each number, find the LCM of:

a 8 and 6 b 12 and 10

9 Using short division, express the following numbers as products of prime factors:

a 45 b 72

c 84 d 120

e 126 f 135

g 210 h 315

10 Find the highest common factor (HCF) and the lowest common multiple (LCM) of the

following sets of numbers using short division:

a 78, 120 b 135, 450

c 105, 675 d 132, 216, 360

e 108, 288, 324 f 180, 270, 360

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Cambridge IGCSE Mathematics Study Guide

1.2 Sets

A set is a group of objects. The objects inside a set are called the elements of the set. We

usually list the elements of a set inside curly brackets e.g. { }3, 6,12 .

Set notationThe following table shows all the notation you are required to know for the IGCSE exam:

Notation Meaning

ℰ The universal set, which contains all the possible elements for the problem.

∅ The empty set, which contains no elements.

n A( ) The number of elements in set A.

A′The complement of set A, which includes all elements in the universal set

which are not in set A.

∈ Means ‘… is an element of…’. For example, a ∈A means a is an element of

the set A.

∉Means ‘… is not an element of …’ For example, a ∉A means a is not an

element of the set A.

A ⊆ B A is a subset of B, which means all elements of set A are also elements of set B.

A ⊂ B A is a proper subset of B, which means A is a subset of B, but also A ≠ B.

A ⊄ B A is not a proper subset of B.

A ∪ BMeans the union of set A and set B, which includes all elements of set A or set

B, or both.

A ∩ BMeans the intersection of set A and set B, which includes all elements

common to both set A and set B.

Set builder notationInstead of writing e.g. all integers less than 10{ }, mathematicians often use set builder (or

predicate) notation. This set would be written as { }x x: is an integer less than 10 , and we would pronounce it as ‘the set of x such that x is an integer less than 10’. The expression after

the colon (‘x is an integer less than 10’) tells us the condition that x has to satisfy to qualify

for set membership.

It’s easy to confuse O and ∅. O is a number but ∅ is a set. If you think of a set as a container then ∅ is just an empty container, (and you wouldn’t mistake a number for a container!)

COMMON MISTAKE

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Chapter 1 Numbers

A prime numbers between 30 and 40{ }=

Solution

A 31, 37{ }=

A multiples of 3 between 10 and 20=n A

Solution

A 12, 15, 18{ }=

( ) =n A 3

1, 2, 3, 4, 5, 6, 7= A 1, 3, 5, 7{ }=b n A( )′

Solution

a A 2, 4, 6{ }′ = b n A 3( )′ =

A 3, 5, 8, 10= B 5, 7, 10, 20=

c

Solution

c A B 3, 5, 7, 8, 10, 20{ }∪ = A B 5, 10{ }∩ =

5, 7, 10

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Cambridge IGCSE Mathematics Study Guide

Exercise 1.2A

1 Given that A prime numbers between 10 and 20{ }= , list all the members of set A.

2 Given the following:

ℰ { }= whole numbers less than 20

{ }=P multiples of 4

{ }=Q multiples of 3

a List all the members of set P.

b List all the members of set Q.

3 Given that ℰ { }= odd numbers less than 15 , A 5, 7, 11{ }= , A ∩ B = ∅ and A ∪ B = ℰ ,

write down all the members of B.

4 Let A 12, 25, 53{ }= and B 12, 17, 25, 35, 37, 53, 55{ }= .

a Explain why A ⊂ B.

b Is it true that 26 ∉B? Give a reason for your answer.

5 Let A 2, 3, 7, 8, 9{ }= and B 3, 5, 7, 9, 11{ }= . Find:

a A ∩ B

b A ∪ B

6 If ℰ π{ }= 2, 7 , 1.3, 79

, , 43 2 , { }=A rational numbers and { }=B integers , find:

a n A( ) b n B( )′ c n A B B( )∩ ∩ ′ d all subsets of B.

7 It is given that ℰ { }= even numbers less than 16 , A multiples of 4{ }= and

B multiples of 3{ }= .

a Explain why 9 ∈B.

b List the members of set A ∩ B.

c Write down n A B( )∪ ′ .

8 Consider { }=A prime numbers less than 8 .

a List all the members of set A.

b Find n A( ). c Is 6.5 ∉A true? Give a reason for your answer.

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Chapter 1 Numbers

9 Let ℰ { }= ≤ <−x x x: is an integer and 8 8 , A 8, 3, 0, 1, 5, 7{ }= − − and

B 5, 3, 2, 5, 8{ }= − − . Find:

a A′ b A ∩ B

c A ∪ B

10 Let ℰ { }= ≤ <−x x x: is an integer and 8 4 , A 5, 3, 1, 3{ }= − − and B 4, 3, 1, 2, 3{ }= − − .

Find:

a B ′ b A′ ∩ B

c A ∪ B ′

Venn diagramsA Venn diagram consists of a universal set ℰ represented by a rectangle and sets within the

universal set represented by circles or ovals.

Complement sets

A′ is the complement of set A, and includes all elements in the universal set that are not in

set A. In the following diagram, the shaded region represents the set A′.

ℰ

A′

A

Subsets

A ⊆ B means ‘A is a subset of B’. All elements of set A are also elements of set B. In the

diagram, the shaded region represents the situation that A ⊆ B. Every element of the shaded

set A is automatically an element of set B.

Venn diagramsVenn diagrams were invented in 1880 by John Venn, although at the time he called them ‘Eulerian Circles’. He named them after the mathematician Leonhard Euler, who had invented a similar system earlier.

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