239 IGCSE Core Maths - fairfieldbooks.com.sg · Preface The Core Mathematics guidebook is written...

Preface

The Core Mathematics guidebook is written for students preparing for the Cambridge IGCSE Mathematics (0580) Core curriculum examination. It is designed to enable students to revise the concepts they have learnt in the classroom, and further strengthen their understanding by practising questions similar to ones in the examination.

The book is structured to enable students to refer to the learning objectives, read succint definitions for concepts, view example questions, and thereafter, practise related questions in order to solidify concepts learnt. In addition, full worked solutions are found at the back of the book to ensure students can check for presentable answers and steps to solve the problems.

Students, may you have a memorable journey with this book and understand and appreciate Mathematics to a deeper level.

Eugene G. Y.http://teach.sg

239_IGCSE_Core_Maths_4thPass.pdf` 30/06/15

Contents

Assessment ....................................................................................................................v

Components For Assessments ..............................................................................v

Instructions To Candidates ................................................................................... vi

1 Number ............................................................................. 1 1.1 Types Of Numbers ..................................................................................................... 5

1.2 Ordering Of Numbers .............................................................................................. 9

1.3 HCF & LCM ..................................................................................................................12

1.4 Approximations & Estimations ..........................................................................15

1.5 Decimals, Fractions & Percentages ..................................................................21

1.6 Ratio, Proportion, Rate & Speed ........................................................................29

1.7 Indices & Standard Form ......................................................................................33

1.8 Money & Finance .....................................................................................................37

1.9 Time ...............................................................................................................................41

2 Algebra & Graphs ............................................................ 45 2.1 Introduction To Algebra........................................................................................48

2.2 Algebraic Manipulation .......................................................................................51

2.3 Simple Equations In 1 Unknown ......................................................................54

2.4 Simultaneous Equations ......................................................................................56

2.5 Number Patterns .....................................................................................................59

2.6 Graphs In Practical Situations ............................................................................60

2.7 Graphs ..........................................................................................................................65

3 Geometry ........................................................................ 71 3.1 Introduction To Geometry ...................................................................................74

3.2 Congruence & Similarity .......................................................................................76

3.3 Geometrical Constructions .................................................................................81


iv

3.4 Angle Bisector & Perpendicular Bisector ........................................................84

3.5 Symmetry....................................................................................................................90

3.6 Angles ...........................................................................................................................93

3.7 Angle Properties Of Polygons .............................................................................98

3.8 Angle Properties Of Circles ................................................................................ 106

3.9 Loci.............................................................................................................................. 112

4 Mensuration .................................................................. 116 4.1 Length, Mass, Capacity, Area, Volume ........................................................ 118

4.2 Perimeter & Area ................................................................................................... 122

4.3 Surface Area & Volume....................................................................................... 134

5 Coordinate Geometry ................................................... 141 5.1 Cartesian Coordinates ....................................................................................... 143

5.2 Gradient .................................................................................................................... 146

5.3 Equation Of A Straight Line ............................................................................. 151

5.4 Parallel Lines ........................................................................................................... 159

6 Trigonometry ................................................................ 161 6.1 Bearings .................................................................................................................... 163

6.2 Pythagoras’ Theorem ......................................................................................... 165

6.3 Trigonometric Ratios Of Acute Angles ........................................................ 168

7 Vectors & Transformations ........................................... 175 7.1 Vectors ....................................................................................................................... 177

7.2 Transformations ................................................................................................... 181

8 Probability .................................................................... 196 8.1 Probability ............................................................................................................... 198

9 Statistics ........................................................................ 202 9.1 Statistical Diagrams ............................................................................................ 204

9.2 Range, Mean, Median & Mode ....................................................................... 220

10 Solutions ....................................................................... 223


v

Assessment

Components For Assessments

Component Time Weighting

Paper 1 (Core)

Short-answer questions based on the Core curriculum.

56 marks.

1 hour 35%

Paper 3 (Core)

Structured questions based on the Core curriculum.

104 marks.

2 hours 65%

Weightage For Assessment Objectives

Assessment objectivePaper 1 (marks)

Paper 3 (marks)

Core assessment

AO1: Mathematical techniques 42–48 78–88 75–85%

AO2: Applying mathematical techniques to solve problems

8–14 16–26 15–25%

Weightage For Topics

Components Number AlgebraSpace and

shapeStatistics and

probability

Core (Papers 1 & 3)

30–35% 20–25% 30–35% 10–15%


Chapter

Number1

Number1

1.1 Types Of Numbers ............................................................. 5 1.1.1 Natural numbers ....................................................................................................... 5

1.1.2 Whole numbers .......................................................................................................... 5

1.1.3 Integers .......................................................................................................................... 5

1.1.4 Prime numbers ........................................................................................................... 5

1.1.5 Square/cube numbers ............................................................................................ 6

1.1.6 Rational/irrational numbers ................................................................................ 6

1.1.7 Real numbers .............................................................................................................. 7

1.1.8 Negative numbers in practical situations ....................................................... 7

1.2 Ordering Of Numbers ....................................................... 9 1.2.1 BEDMAS ......................................................................................................................... 9

1.2.2 Inequalities .................................................................................................................10

1.2.3 Number line ...............................................................................................................10

1.2.4 Ordering numbers ...................................................................................................11

1.2.5 Using a calculator ...................................................................................................11

1.3 HCF & LCM ....................................................................... 12 1.3.1 Factors ..........................................................................................................................12

1.3.2 Multiples ......................................................................................................................13

1.3.3 Highest Common Factor (HCF) .........................................................................13

1.3.4 Lowest Common Multiple (LCM) ......................................................................14

1.4 Approximations & Estimations ....................................... 15 1.4.1 Rounding answers to whole numbers ...........................................................15

1.4.2 Rounding answers to decimal places .............................................................16

1.4.3 Significant figures ....................................................................................................17

1.4.4 Rounding answers to significant figures .......................................................17

1.4.5 Estimating numbers, quantities and lengths ..............................................18

1.4.6 Lower & upper bounds ..........................................................................................20


IGCSE Core Maths2

1.5 Decimals, Fractions & Percentages ................................. 21 1.5.1 Decimals – add, subtract, multiply, divide ....................................................21

1.5.2 Fractions – add, subtract, multiply, divide ....................................................23

1.5.3 Calculating fractions & percentages ...............................................................25

1.5.4 Converting between decimals, fractions & percentages ........................26

1.5.5 Further percentages ...............................................................................................27

1.6 Ratio, Proportion, Rate & Speed ..................................... 29 1.6.1 Ratio ..............................................................................................................................29

1.6.2 Direct proportion .....................................................................................................30

1.6.3 Inverse proportion ...................................................................................................32

1.7 Indices & Standard Form ................................................. 33 1.7.1 Indices ..........................................................................................................................33

1.7.2 Laws of indices .........................................................................................................34

1.7.3 Standard form ..........................................................................................................35

1.8 Money & Finance ............................................................. 37 1.8.1 Profit & loss .................................................................................................................37

1.8.2 Percentage profit & loss .........................................................................................38

1.8.3 Money exchange .....................................................................................................38

1.8.4 Simple interest ..........................................................................................................39

1.8.5 Compound interest ................................................................................................41

1.9 Time ................................................................................. 41 1.9.1 Converting between 12-hour and 24-hour clock ......................................41

1.9.2 Reading clocks & timetables ...............................................................................43


Number | Syllabus3

Syllabus

C1.1

Identify and use natural numbers, integers (positive, negative and zero), prime numbers, square numbers, common factors and common multiples, rational and irrational numbers (e.g. p, 2 ), real numbers.

C1.2

Extended curriculum only.

C1.3

Calculate squares, square roots, cubes and cube roots of numbers.

C1.4

Use directed numbers in practical situations.

C1.5

Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts.

Recognise equivalence and convert between these forms.

C1.6

Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, ≥, ≤

C1.7

Understand the meaning and rules of indices.

Use the standard form A × 10n where n is a positive or negative integer, and 1 ≤ A < 10.

C1.8

Use the four rules for calculations with whole numbers, decimals and vulgar (and mixed) fractions, including correct ordering of operations and use of brackets.

C1.9

Make estimates of numbers, quantities and lengths, give approximations to specified numbers of significant figures and decimal places and round off answers to reasonable accuracy in the context of a given problem.

C1.10

Give appropriate upper and lower bounds for data given to a specified accuracy.

C1.11

Demonstrate an understanding of ratio and proportion.

Use common measures of rate.

Calculate average speed.

C1.12

Calculate a given percentage of a quantity.

Express one quantity as a percentage of another.

Calculate percentage increase or decrease.

C1.13

Use a calculator efficiently.

Apply appropriate checks of accuracy.


IGCSE Core Maths4

C1.14

Calculate times in terms of the 24-hour and 12-hour clock.

Read clocks, dials and timetables.

C1.15

Calculate using money and convert from one currency to another.

C1.16

Use given data to solve problems on personal and household finance involving earnings, simple interest and compound interest.

Extract data from tables and charts.

C1.17

Extended curriculum only.


Number | 1.1 Types Of Numbers5

1.1 Types Of Numbers

1.1.1 Natural numbers

כ Definition 1.1.1 — Natural numbers.Natural numbers are counting numbers, whole numbers starting from 1. e.g. 1, 2, 3, …

׀׀ Some books define 0 as a natural number as well.

1.1.2 Whole numbers

כ Definition 1.1.2 — Whole numbers.Whole numbers are counting numbers, starting from 0.

e.g. 0, 1, 2, 3, …

1.1.3 Integers

כ Definition 1.1.3 — Integers.Integers are positive and negative whole numbers, including 0.

e.g…, −3, −2, −1, 0, 1, 2, 3, 4, …

1.1.4 Prime numbers

כ Definition 1.1.4 — Prime numbers.Prime numbers are integers that are divisible by 1 and itself only. The smallest prime number is 2.

e.g. 2, 3, 5, 7, 11, 13, 17, …

Ѣ׀ Example 1.1.1 — Prime numbers.

Express 21 as a product of prime factors.

Solution:

21 = 3 × 7

Ѣ׀ Example 1.1.2 — Prime numbers.

Express 18 as a product of prime factors.

Solution:

18 = 2 × 3 × 3

Exercise 1.1 — Prime numbers.

Express the following as a product of prime factors.

(a) 10 (b) 30 (c) 72 (d) 210


IGCSE Core Maths6

1.1.5 Square/cube numbers

כ Definition 1.1.5 — Square numbers.Square numbers are integers that are derived from multiplying a number by itself.

e.g. 1, 4, 9, 16, 25, 36, …

כ Definition 1.1.6 — Cube numbers.Cube numbers are integers that are derived from multiplying a number by itself twice.

e.g. 1, 8, 27, 64, 125 …

Exercise 1.2 — Square numbers, cube numbers.

Evaluate the following.

(a) 52 (b) 0.82

(c) 33 (d) 6.93

(e) 81 (f) 1 96.

(g) −3433 (h) 8

156253

1.1.6 Rational/irrational numbers

כ Definition 1.1.7 — Rational numbers.Rational numbers are numbers that can be expressed as fractions that have numerators and denominators as integers.

e.g. 47

318

0 3 2 65 92 16, , . , . , ,−

׀׀ Decimals which are recurring are always rational.

e.g. 0.2222… = 29

, 0.67777… = 6190

, 0.323232… = 3299

כ Definition 1.1.8 — Irrational numbers.Irrational numbers are numbers that cannot be expressed as fractions that have numerators and denominators as integers.

e.g. p, 2 , e

׀׀ Decimals which are non-recurring are therefore irrational.

e.g. p = 3.141592654…, 2 = 1.414213562…, e = 2.71828182846…


Number | 1.1 Types Of Numbers7

Exercise 1.3 — Rational numbers, irrational numbers.

State if the following are rational or irrational.

(a) 6 (b) − 92

(c) 16 (d) − 3

(e) −5.3 (f) 4.35

(g) 53 (h) 643

1.1.7 Real numbers

כ Definition 1.1.9 — Real numbers.Real numbers are all rational and irrational numbers, which includes integers, decimals, fractions and so on. Some numbers are not included like ∞, −1.

e.g. 6, −0.0023, 73

, sin 55°, 413 , −ln 4

1.1.8 Negative numbers in practical situations

Ѣ׀ Example 1.1.3 — Using numbers in practical situations.

The table shows the temperature reading of Chicago for the first 6 months.

Jan Feb Mar Apr May June

−9°C −8°C 0°C 9°C 16°C 22°C

Find the difference in temperature between

(a) January and February

(b) January and March

(c) January and May

(d) February and June

Solution:

(a) −8 − (−9) = 1°C

(b) 0 − (−9) = 9°C

(c) 16 − (−9) = 25°C

(d) 22 − (−8) = 30°C

׀׀ Take the highest temperature minus the lowest temperature. A difference must be a positive value.


IGCSE Core Maths8

Exercise 1.4 — Using numbers in practical situations.

1. The table shows the temperature reading of Siberia for the first 6 months.

Jan Feb Mar Apr May June

−16.2°C −14.7°C −7.2°C 3.2°C 11.6°C 18.2°C

Find the difference in temperature between

(a) January and March (b) February and April

(c) May and June (d) March and June

2. A business owner started off with $1000 at the start of the month. Calculate his balance as a positive or negative number after the following transactions took place in order:

(a) −$140 for products

(b) +$300 from sales

(c) −$1280 for rent

(d) +$450 from sales

(e) −$200 for salary

3. A man standing at a cliff threw a ball vertically upwards until it was 33 m above the cliff, before it fell for 195 m all the way down into the sea.

Calculate the height of the cliff from sea level.

4. The table shows several important dates and countries in the discover of zero in Mathematics.

Year Events

1740 BC Egyptians used 0 as a base line for drawings.

300 BC Babylonians used a placeholder to represent 0.

650 AD Brahmagupta used 0 for addition and subtraction.

1637 AD Rene Descartes defined origin of co-ordinate geometry as (0, 0).

Using the values above,

(a) Calculate the number of years from the time of the Egyptians to Brahmagupta.

(b) Calculate the number of years from the time of the Babylonians to Rene Descartes.

(c) Calculate the number of years from the time of the Egyptians to Rene Descartes.


Number | 1.2 Ordering Of Numbers9

1.2 Ordering Of Numbers

1.2.1 BEDMAS

כ Definition 1.2.1 — BEDMAS.BEDMAS stands for

Brackets

↓Exponents

↓Divide and Multiply

↓Add and Subtract

Whenever we have calculations, we carry them out in that order.

׀׀ Exponents are also known as indices or powers, you will learn them later in this chapter.

׀׀ Divide and multiply or add and subtract, just calculate from left to right.

Ѣ׀ Example 1.2.1— BEDMAS.

Evaluate 4 + (5 − 2)2 ÷ 3 × 2 − 7.

Solution:

4 + 32 ÷ 3 × 2 − 7 (brackets first)

= 4 + 9 ÷ 3 × 2 − 7 (exponents next)

= 4 + 6 − 7 (divide and multiply next)

= 3 (add and subtract last)

Exercise 1.5 — BEDMAS.

Evaluate the following.

(a) 12 − 4 × 2 (b) 8 + 6 ÷ 3

(c) 81 − (4 − 3) (d) 8 × (7 − 2)

(e) 5 − 22 (f) 9 − 2 × 4 + 32

(g) 3 − (5 + 23) ÷ 4 × 22 (h) 4 + 32 × 5 − 18 ÷ 9

(i) 8 − (2 − 3) × 6 + 12 ÷ 4


239 IGCSE Core Maths - fairfieldbooks.com.sg · Preface The Core Mathematics guidebook is written...

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