Cambridge IGCSE® and O Level Additional Mathematics Coursebook

471 Pages·2018·52.396 MB·English

by  Sue Pemberton

Sue Pemberton
Cambridge IGGSE®and O Level
Additional
Mathematics
Coursebook
Second edition
il Cambridge
UNIVERSITY PRESS
Cambridge
UNIVERSITY PRESS
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Second published 2019
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IGCSE® is a registered trademark.
Past exam paper questions throughout are reproduced by permission of Cambridge
Assessment International Education.
Cambridge Assessment International Education bears no responsibility for the example
answers to questions taken from its past question papers which are contained in
this publication.
All exam-style questions and sample answers in this title were written by the authors.
In examinations, the way marks are awarded may be different.
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M
4. ,4 A'/'gj, >*.
Contents
Acknowledgements
VI
Introduction vii
How to use this book viii
1 Functions 1
1.1 Mappings
2
1.2 Definition of a function 3
1.3 Composite functions
5
1.4 Modulus functions 7
1.5 Graphs of )) = |f(x|) w here f(x) is linear
10
1.6 Inverse functions 12
1.7 The graph of a function and its inverse
15
Summary
18
Examination questions
19
Simnltaneous equations and qnadratics
23
2.1 Simultaneous equations (one linear and one non-linear)
25
2.2 Maximum and minimum values of a quadratic function
28
2.3 Graphs of y = |f(x)| where f(x) is quadratic
34
2.4 Quadratic inequalities
37
2.5 Roots of quadratic equations
39
2.6 Intersection of a line and a curve 42
Summary
44
Examination questions
46
Indices and snrds 49
3.1 Simplifying expressions involving indices
50
3.2 Solving equations involving indices
51
3.3 Surds 55
3.4 Multiplication, division and simplification of surds
57
3.5 Rationalising the denominator of a fraction
60
3.6 Solving equations involving surds
63
Summary
67
Examination questions
67
Factors and polynomials
70
4.1 Adding, subtracting and multiplying polynomials
71
4.2 Division of polynomials
73
4.3 The factor theorem 75
4.4 Cubic expressions and equations
78
4.5 The remainder theorem 82
Summary
86
Examination questions
87
Equations, inequalities and graphs
89
5.1 Solving equations of the type \ax - b\ = \cx - d\
90
5.2 Solving modulus inequalities
94
5.3 Sketching graphs of cubic polynomials and their moduli
98
5.4 Solving cubic inequalities graphically
102
5.5 Solving more complex quadratic equations
103
Summary
105
Examination questions
107
Logarithmic and exponential functions
111
6.1 Logarithms to base 10
112
6.2 Logarithms to base a
115
6.3 The laws of logarithms
118
Cambridge IGCSE and 0 Level Additional Mathematics
6.4 Solving logarithmic equations 120
6.5 Solving exponential equations 122
6.6 Change of base of logarithms 124
6.7 Natural logarithms 126
6.8 Practical applications of exponential equations 128
6.9 The graphs of simple logarithmic and exponential functions 129
6.10 The graphs oiy = ke"'' + a and y=k\n {ax+ b) where n, k, a
and b are integers 130
6.11 The inverse of logarithmic and exponential functions 133
Summary 134
Examination questions 135
7 S traight-line graphs 138
7.1 Problems involving length of a line and midpoint 140
7.2 Parallel and perpendicular lines 143
7.3 Equations of straight lines 145
7.4 Areas of rectilinear figures 148
7.5 Converting from a non-linear equation to linear form 151
7.6 Converting from linear form to a non-linear equation 155
7.7 Finding relationships from data 159
Summary 165
Examination questions 165
8 Circular measure 170
8.1 Circular measure 171
8.2 Length of an arc 174
8.3 Area of a sector 177
Summary ISO
Examination questions 181
9 Trigonometry 186
9.1 Angles between 0° and 90° 187
9.2 The general definition of an angle 190
9.3 Trigonometric ratios of general angles 192
9.4 Graphs of trigonometric functions 195
9.5 Graphs of y = |f(v)| , where f(x) is a trigonometric function 205
9.6 Trigonometric equations 209
9.7 Trigonometric identities 214
9.8 Further trigonometric equations 216
9.9 Further trigonometric identities 218
Summai7 220
Examination questions 221
10 Permutations and combinations 224
10.1 Factorial notation 225
10.2 Arrangements 226
10.3 Permutations 229
10.4 Combinations 234
Summary 237
Examination questions 238
11 Series 243
11.1 Pascal's triangle 244
11.2 The binomial theorem 249
11.3 Arithmetic progressions 252
11.4 Geometric progressions 257
11.5 Infinite geometric series 262
11.6 Further arithmetic and geometric series 267
Summary 270
Examination questions 271
12 Differentiation 1 274
12.1 The gradient function 275
12.2 The chain rule 280
12.3 The product rule 282
12.4 The quotient rule 285
12.5 Tangents and normals 287
12.6 Small increments and approximations 291
12.7 Rates of change 294
12.8 Second derivatives 298
12.9 Stationary points 300
12.10 Practical maximum and minimum problems 305
Summary 310
Examination questions 311
13 Vectors 315
13.1 Further vector notation 317
13.2 Position vectors 319
13.3 Vector geometry 323
13.4 Constant velocity problems 327
Summary 331
Examination questions 335
14 Differentiation 2 336
14.1 Derivatives of exponential functions 337
14.2 Derivatives of logarithmic functions 341
14.3 Derivatives of trigonometric functions 345
14.4 Further applications of differentiation 350
Summary 356
Examination questions 357
15 Integration 361
15.1 Differentiation reversed 362
15.2 Indefinite integrals 365
15.3 Integration of functions of the form {ax+ b)n 367
15.4 Integration of exponential functions 368
15.5 Integration of sine and cosine functions 370
15.6 Integration of functions of the form — and 372
° X ax+ b
15.7 Further indefinite integration 375
15.8 Definite integration 378
15.9 Further definite integration 383
15.10 Area under a curve 385
15.11 Area of regions bounded by a line and a curve 392
Summary 397
Examination questions
398
16 Kinematics 402
16.1 Applications of differentiation in kinematics
404
16.2 Applications of integration in kinematics
412
Summary
418
Examination questions
419
Answers 422
Index 454
Cambridge IGCSE and 0 Level Additional Mathematics
Acknowledgements
Past examination paper questions throughout are reproduced by permission of Cambridge Assessment
International Education.
Thanks to thef ollowingf or permission to reproduce images:
Cover artwork; Shestakovych/Shutterstock
Chapter 1 Fan jianhua/Shutterstock; Chapter 2 zhu difeng/Shutterstock; Chapter 3 LACUNA
DESIGN/Getty Images; Chapter 4 Michael Dechev/Shutterstock; Fig. 4.1 Steve Bower/Shutterstock;
Fig. 4.2 Laboko/Shutterstock; Fig. 4.3 irin-k/Shutterstock; Chapter 5 zentilia/Shutterstock; Chapter
6 Peshkova/Shutterstock; Chapter 7 ittipon/Shutterstock; Chapter 8 Zhu Qiu/FyeFm/Getty Images;
Chapter 9 paul downing/Getty Images; Fig. 9.1 aarrows/Shutterstock; Chapter 10 Gino Santa Maria/
Shutterstock; Fig. 10.1snake3d/Shutterstock; Fig. 10.2 Keith Publicover/Shutterstock; Fig. 10.3
Aleksandr Kurganov/Shutterstock; Fig. 10.4 Africa Studio/Shutterstock; Chapter 11 elfinadesign/
Shutterstock; Chapter 12 AlenKadr/Shutterstock; Chapter 13 muratart/Shutterstock; Chapter 14
Neamov/Shutterstock; Chapter 15 Ahuli Labutin/Shutterstock; Chapter 16 AlexLMX/Getty
Introduction
This highly illustrated coursebook covers the Cambridge IGCSl^ and O Level Additional Mathematics
syllabuses (0606 and 4037). The course is aimed at students who are currendy studying or have
previously studied Cambridge ICCS^ Mathematics( 0580) or Cambridge O Level Mathematics (4024).
Where the content in one chapter includes topics that should have already been covered in previous
studies, a recap section has been provided so that students can build on their prior knowledge.
'Class discussion' sections have been included to provide students with the opportunity to discuss
and learn new mathematical concepts with their classmates, with their class teacher acting as the
facilitator. The aim of these class discussion sections is to improve the student's reasoning and ofal
communication skills.
'Challenge' questions have been included at the end of most exercises to challenge and stretch high-
ability students.
Towards the end of each chapter, there is a summary of the key concepts to help students consolidate
what they have just learnt. This is followed by a 'Past paper' questions section, which contains real
questions taken from past examination papers.
A Practice Book is also available in the Cambridge IGCSL^ Additional Mathematics series, which offers
students further targeted practice. This book closely follows the chapters and topics of the coursebook
offering additional exercises to help students to consolidate concepts learnt and to assess their learning
after each chapter. A Teacher's Resource, to offer support and advice, is also available.
Cambridge IGCSE and 0 Level Additional Mathematics
How to use this book
Chapter - each chapter begins with a set of learning objectives to explain what you
will learn in this chapter.
Chapter 2
Simultaneous equations and quadratics
This section will show you how to:
n solve simultaneous equations in two unknowns by elimination or substitution
n find the maximum and minimum values of a quadratic function
n sketch graphs of quadratic functions and find their range for a given domain
n sketch graphs of the function y |= f{: c)t where ffx) is quadratic and solve associated equations
n determine the numberof roots of a quadratic equation and the related conditions for a line to
intersect, be a tangent or not intersect a given curve
n solve quadratic equations for real roots and find the solution set for quadratic inequalities^
Recap - check that you are familiar with the introductory skills required for
the chapter.
You should already know how to solve linear inequalities.
Two examples are shown below.
Solve 2(x - 5) < 9 expand brackets Solve 5 - 3x ^ 17 subtract 5 fiom both sides
2* - 10 < 9 add 10 to both sides -3x 12 divide both sides by -3
2x < 19 divide both sides by 2 X « -4
X < 9.5
Class Discussion - additional activities to be done in the classroom for enrichment.
CLASS DISCUSSION
Solve each of these three pairs of simultaneous equations.
8x + = 7 Sx + y = 10 2*+ 5 = .3)1
Sx+by = -9 2)1 = 15 -6 x 10 - 6)= -4x
Discuss your answers with your classmates.
Discuss what the graphs would be like for each pair of equations.
Worked Example - detailed step-by-step approaches to help students solve problems.
Find the value of;
11!
® 5!
Answers
g 8 ! 8x7x6x/x^x^x^)0
si ^x/x/x/yl
=8x7x6
= 336
b ill = HxlOx9xXx/x^x/xy^>0x^yl
813! /x /x^x.^x/x/xix/ X 3 X 2 X 1
_ 990
6
= 165
Note - quick suggestions to remind you about key facts and highlight important
points.
o Note:
logic 30 can also be written as Ig 30 or log30.
Challenge- challenge yourself with tougher questions that stretch your skills
CHALLENGEQ
6 This design is made from
1 blue circle, 4 orange circles
and 16 green circles.
The circles touch each other.
Given that the radius of each
green circle is 1 unit, find the
exact radius of
a t he orange circles,
b the blue circle.
Summary - at the end of each chapter to review what you have learnt.
Summary
One radian (b) is the size of the angle subtended
at the centre of a circle, radius r, by an arc of length r.
When 9 is measured in radians:
n t he length of arc AB = rO
m t he area of sector AOB = -A O.
2