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Executive
Preview
PL
E
Cambridge IGCSE™ and O Level
Additional
Mathematics
SA
M
MULTI-COMPONENT SAMPLE
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Dear Cambridge Teacher, PL
E
Welcome to the new, third edition of our Cambridge IGCSE™ and O Level Additional Mathematics series,
which supports the revised Cambridge IGCSE and O Level Additional Mathematics syllabuses (0606/4037)
for examination from 2025. We have developed this new edition through extensive research with teachers
around the world to provide you and your learners with the support you need, where you need it. You can be
confident that this series supports all aspects of the revised syllabuses.
This Executive Preview contains sample content from the series, including:
•
•
•
A guide explaining how to use the series
A guide explaining how to use each resource
The table of contents from each resource
‘Worked examples’ demonstrate a step-by-step process of working through a question or problem, and act as
an entry to engaging exercise sets. Class discussion questions allow students to articulate their understanding
of a skill or topic to a partner, a group or the whole class. We are pleased to include a series of investigative
projects authored by NRICH (a collaboration between the Faculties of Mathematics and Education at the
University of Cambridge). Four of these projects appear in the coursebook to further facilitate pair, small
group and whole-class work. The digital teacher’s resource provides full support and guidance on these
projects. A Practice Book for learners is offered to give extra opportunity to consolidate skills using additional
questions. Our digital teacher’s resource supports with pedagogical approaches and ideas for how to teach
the content in the syllabus – find out more in our resource guide pages.
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We are happy to introduce Cambridge Online Mathematics, hosted on our Cambridge GO platform, to the
Additional Mathematics resources. Cambridge Online Mathematics provides enhanced teacher and student
support; it can be used to create virtual classrooms allowing you to blend print and digital resources into your
teaching, in the classroom or as homework. Cambridge Online Mathematics contains all coursebook content
in a digital format, additional quiz questions that can be automarked, worksheets, guided walkthroughs of
new skills and reporting functionality for teachers. The platform is easy to use, tablet-friendly and flexible.
We hope you enjoy this new series of resources. Visit our website to view the full series or speak to your local
sales representative. You can find their details here:
cambridge.org/gb/education/find-your-sales-consultant
With best wishes from the Cambridge team,
Thomas Carter
Commissioning Editor for Cambridge IGCSE™ and O Level Additional Mathematics
Cambridge University Press
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
Cambridge Online
Mathematics
Discover our enhanced digital mathematics support
for Cambridge Lower Secondary, Cambridge
IGCSE™ and Cambridge International AS & A Level
Mathematics – endorsed by Cambridge Assessment
International Education.
Available in 2023
New content to support
the following syllabuses:
• Cambridge IGCSE
Mathematics
• Cambridge IGCSE
International
Mathematics
• Cambridge IGCSE
and O Level Additional
Mathematics
Features can include:
• Guided walkthroughs of key mathematical concepts for students
• Teacher-set tests and tasks with auto–marking functionality
• A reporting dashboard to help you track student progress quickly
and easily
• A test generator to help students practise and refine their skills – ideal
for revision and consolidating knowledge
Free trials
A free trial will be available for Cambridge IGCSE Mathematics
in 2023. In the meantime, please visit https://bit.ly/3TUGl4l for
a free trial of our Cambridge Lower Secondary and Cambridge
International AS & A Level Mathematics versions.
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E
Cambridge IGCSE™ and O Level
PL
Additional
Mathematics
COURSEBOOK
SA
M
Sue Pemberton
Third edition
Digital Access
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Contents
Contents
Introduction00
How to use this book
00
1 Functions00
Past-paper questions
PL
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1.1
Mappings00
1.2
Definition of a function
00
Composite functions
00
1.3
Modulus functions
00
1.4
1.5
Graphs of y = |f(x)| where f(x) is linear 00
1.6
Inverse functions
00
1.7
The graph of a function and its inverse
Summary00
4.1
Solving equations of the type
|ax + b| = |cx + d|
4.2
Solving modulus inequalities
00
4.3
Sketching graphs of cubic polynomials
and their moduli
00
Solving cubic inequalities graphically
4.4
4.5
Solving more complex quadratic
equations00
Summary00
00
2 Simultaneous equations and
quadratics00
2.1
4 Equations, inequalities and
graphs00
Past-paper questions
3 Factors and polynomials
00
00
3.1
Adding, subtracting and multiplying
polynomials00
Division of polynomials
00
3.2
3.3
The factor theorem
00
3.4
Cubic expressions and equations
00
3.5
The remainder theorem
00
Summary00
Past-paper questions
00
5 Logarithmic and exponential
functions00
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Simultaneous equations (one linear
and one non-linear)
00
2.2
Maximum and minimum values of a
quadratic function
00
2.3
Graphs of y = |f(x)| where f(x) is
quadratic00
2.4
Quadratic inequalities
00
2.5
Roots of quadratic equations
00
2.6
Intersection of a line and a curve
00
Summary00
Past-paper questions
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Logarithms to base 10
00
Logarithms to base a00
The laws of logarithms
00
Solving logarithmic equations
00
Solving exponential equations
00
Change of base of logarithms
00
Natural logarithms
00
Practical applications of exponential
equations00
5.9
The graphs of simple logarithmic and
00
exponential functions
nx
5.10
The graphs of y = k e + a and
y = k ln (ax + b) where n, k, a and b
are integers
00
5.11
The inverse of logarithmic and
exponential functions
00
Summary00
Past-paper questions
00
00
i
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
6 Straight-line graphs
00
10 Permutations and combinations
6.1
Past-paper questions
10.1
Factorial notation
00
Arrangements00
10.2
10.3
Permutations00
10.4
Combinations00
Summary00
Past-paper questions
11.1
Pascal’s triangle
00
11.2
The binomial theorem
00
11.3
Arithmetic progressions
00
Geometric progressions
00
11.4
11.5
Infinite geometric series
00
11.6
Further arithmetic and geometric series 00
Summary00
00
7 Coordinate geometry of the
circle00
7.1
7.2
The equation of a circle
00
Problems involving intersection of lines
and circles
00
Summary00
Past-paper questions
Past-paper questions
12 Calculus – Differentiation 1
00
00
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8 Circular measure
8.1
Circular measure
00
8.2
Length of an arc
00
8.3
Area of a sector
00
Summary00
Past-paper questions
00
9 Trigonometry00
9.1
9.2
9.3
9.4
9.5
Angles between 0° and 90°
00
The general definition of an angle
00
Trigonometric ratios of general angles 00
Graphs of trigonometric functions
00
Graphs of y = |f(x)|, where f(x) is a
00
trigonometric function
9.6
Trigonometric equations
00
9.7
Trigonometric identities
00
Further trigonometric equations
00
9.8
9.9
Further trigonometric identities
00
Summary00
Past-paper questions
00
11 Series00
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Problems involving length of a line and
midpoint00
Parallel and perpendicular lines
00
6.2
6.3
Equations of straight lines
00
6.4
Areas of rectilinear figures
00
Converting from a non-linear equation to
6.5
00
linear form
6.6
Converting from linear form to a nonlinear equation
00
6.7
Finding relationships from data
00
Summary00
00
00
00
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
The gradient function
00
The chain rule
00
The product rule
00
The quotient rule
00
Tangents and normals
00
Small increments and approximations 00
Rates of change
00
Second derivatives
00
Stationary points
00
Practical maximum and minimum
problems00
Summary00
Past-paper questions
00
13 Vectors00
13.1
Further vector notation
00
13.2
Position vectors
00
13.3
Vector geometry
00
13.4
Constant velocity problems
00
Summary00
Past-paper questions
00
00
ii
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Contents
14 Calculus – Differentiation 2
15.7
15.8
15.9
15.10
15.11
00
Further indefinite integration
00
Definite integration
00
Further definite integration
00
Area under a curve
00
Area of regions bounded by a line and a
curve00
Summary00
14.1
Derivatives of exponential functions
00
Derivatives of logarithmic functions
00
14.2
14.3
Derivatives of trigonometric functions 00
14.4
Further applications of differentiation 00
Summary00
15 Calculus – Integration
15.1
15.2
15.3
15.4
15.5
00
Past-paper questions
00
Differentiation reversed
00
Indefinite integrals
00
Integration of functions of the form
(ax + b)n00
Integration of exponential functions
00
Integration of sine and cosine
functions00
1
Integration of functions of the form __
​​ x
​​
and ______
​​  1  ​​
ax + b
16 Kinematics00
16.1
Applications of differentiation in
kinematics00
16.2
Applications of integration in kinematics
00
Summary00
Past-paper questions
00
Answers000
Index000
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15.6
00
PL
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Past-paper questions
iii
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
Introduction
This highly illustrated coursebook covers the Cambridge IGCSE TM Additional
Mathematics and O Level syllabuses (0606/4037). The course is aimed at students who
are currently studying or have previously studied Cambridge IGCSE TM Mathematics
(0580/0980) or Cambridge O Level Mathematics (4024).
PL
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Where the content in one chapter includes topics that should have already been covered
in previous studies, a prerequisite knowledge section has been provided so that you can
build on your prior knowledge.
‘Discussion’ sections have been included to provide you with the opportunity to discuss
and learn new mathematical concepts with your classmates.
‘Challenge’ questions have been included at the end of most exercises to challenge and
stretch you.
Towards the end of each chapter, there is a summary of the key concepts to help you
consolidate what you have just learnt. This is followed by a ‘Past paper’ questions
section, which contains questions taken from past papers for this syllabus.
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A Practice Book is also available in the IGCSE TM Additional Mathematics series, which
offers you further targeted practice. This book closely follows the chapters and topics
of the coursebook, offering additional exercises to help you to consolidate concepts
you have learnt and to assess your learning after each chapter.
iv
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We are working with Cambridge Assessment International Education towards endorsement of this title.
How to use this book
How to use this book
Throughout this book, you will notice lots of different features that will help your learning. These are explained below.
ACTIVITY
These set the scene for each chapter, help with
navigation through the Coursebook and indicate
the important concepts in each topic.
Activities give you an opportunity to apply your
understanding of a concept to a practical task.
When activities have answers, you can find these in
the digital version of the Coursebook.
PL
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THIS SECTION WILL SHOW YOU HOW TO:
PRE-REQUISITE KNOWLEDGE
This feature shows how your understanding or use
of a topic covered in another area of the book will
help you with the concepts in this chapter.
TIP
These boxes show you the step-by-step process to
work through an example question or problem, giving
you the skills to work through questions yourself.
CLASS DISCUSSION
At certain points in the chapters you will be given
opportunities to talk about your learning and
understanding of the topic in a small group or
with a partner.
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The information in this feature will help you
complete the exercises, and give you support in
areas that you might find difficult.
WORKED EXAMPLE
KEY WORDS
REFLECTION
The key vocabulary appears in a box at the
start of each chapter, and is highlighted in the
text when it is first introduced. You will also find
definitions of these words in the Glossary at the
back of this book.
These activities ask you to think about the
approach that you take to your work, and how
you might improve this in the future.
Exercises
Appearing throughout the text, exercises give you
a chance to check that you have understood the
topic you have just read about and practice the
mathematical skills you have learned. You can find
the answers to these questions in the digital version
of the Coursebook.
CHALLENGE QUESTIONS
These exercises will stretch your skills in the topic you
have just learned. You can find the answers to these
questions in the digital version of the Coursebook.
Past paper questions
Questions at the end of each chapter provide a
variety of past paper questions, some of which may
require use of knowledge from previous chapters.
Answers to these questions can be found in the digital
version of the Coursebook.
SUMMARY
There is a summary of key points at the end of
each chapter.
This icon shows you where you should complete
an exercise without using your calculator.
v
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
How to use this series
Cambridge IGCSE™ and O Level
Additional
Mathematics
COURSEBOOK
The coursebook contains sixteen chapters that together offer
complete coverage of the syllabus. We have worked with NRICH
to provide a variety of project activities, designed to engage
learners and strengthen their problem-solving skills. Each chapter
contains opportunities for formative assessment, differentiation
and peer and self-assessment offering learners the support needed
to make progress. Cambridge Online Mathematics is available
through the digital/print bundle option or on its own without the
print coursebook. Learners can review content digitally, explore
worked examples and test their knowledge with quiz questions
and answers. Teachers benefit from the ability to set tests and
tasks with the added auto-marking functionality and a reporting
dashboard to help track learner progress quickly and easily.
SA
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Sue Pemberton
PL
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This suite of resources supports learners and teachers following the Cambridge
IGCSE™ and O Level Additional Mathematics syllabuses (0606/4037). Up-to-date
metacognition techniques have been incorporated throughout the resources to meet the
changes in the syllabus content and develop a complete understanding of mathematics
for learners. All of the components in the series are designed to work together.
Third edition
Digital Access
The digital teacher’s resource provides extensive guidance
on how to teach the course, including suggestions for
differentiation, formative assessment and language
support, teaching ideas and PowerPoints. The Teaching
Skills Focus shows teachers how to incorporate a
variety of key pedagogical techniques into teaching,
including differentiation, assessment for learning, and
metacognition. Answers for all components are accessible
to teachers for free on the Cambridge GO platform.
Cambridge IGCSE™ and O Level
Additional
Mathematics
TEACHER’S
RESOURCE
COURSEBOOK
Sue Pemberton
Third edition
Digital Access
vi
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We are working with Cambridge Assessment International Education towards endorsement of this title.
1
Additional
Mathematics
PRACTICE BOOK
Muriel James
Digital Access
A Practice Book is available for learners that wish to have
extra questions to work through. This resource which can be
used in class or assigned as homework, provide a wide variety
of extra maths activities and questions to help learners
consolidate their learning and prepare for assessment. ‘Tips’
are also regularly featured to give learners extra advice and
guidance on the different areas of maths they encounter.
Access to the digital versions of the practice books is
included, and answers can be found either here or in the back
of the books.
PL
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Cambridge IGCSE™ and O Level
Characteristics & classification
How
ofto
living
use organisms
this series
Cambridge IGCSE™ and O Level
Additional
Mathematics
SA
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A Worked Solutions Manual has been introduced to the
series. This offers a fully worked solution, with annotated
comments, to a selection of questions for teachers or
learners to use as they work through the content.
WORKED SOLUTIONS MANUAL
Muriel James
Digital Access
vii
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Chapter 2
SA
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PL
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Simultaneous
equations and
quadratics
THIS SECTION WILL SHOW YOU HOW TO:
•
•
•
•
•
•
solve simultaneous equations in two unknowns by elimination or substitution
find the maximum and minimum values of a quadratic function
sketch graphs of quadratic functions and find their range for a given domain
sketch graphs of the function y = | f (x) | where f (x) is quadratic and solve associated equations
determine the number of roots of a quadratic equation and the related conditions for a line to intersect,
be a tangent or not intersect a given curve
solve quadratic equations for real roots and find the solution set for quadratic inequalities.
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
PRE-REQUISITE KNOWLEDGE
Before you start…
Where it comes from
You should be able to...
Check your skills
Cambridge IGCSE/O Level
Mathematics
Solve simultaneous
equations using the
elimination method.
1
Use the elimination method to solve:
a ​4x + 3y = 1​
​2x − 3y = 14​
Cambridge IGCSE/O Level
Mathematics
PL
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b​
3x + 2y = 19​
x
​ + 2y = 13​
Solve simultaneous
equations using the
substitution method.
2
Use the substitution method to solve:
a ​y = 3x − 10​
​x + y = − 2​
b
Cambridge IGCSE/O Level
Mathematics
Solve quadratic equations
by completing the square.
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Cambridge IGCSE/O Level
Mathematics
Solve quadratic equations
using the factorisation
method.
Cambridge IGCSE/O Level
Mathematics
3
4
Solve quadratic equations
5
using the quadratic formula
​ + 2y = 11​
x
4y
​ − x = − 2​
Solve by factorisation:
a ​​x​​ 2​ + x − 6 = 0​
b
​​x​​ 2​− 10x + 16 = 0​
c
​6​x​​ 2​+ 11x − 10 = 0​
a
Express ​2​x​​ 2​+ 7x + 3​in the form
​a​(x + b)​​ 2​ + c​.
b Use your answer to part a to solve
the equation ​2​x​​ 2​+ 7x + 3 = 0​.
Solve ​2​x​​ 2​− 9x + 8 = 0​.
Give your answers correct to 2
decimal places.
CLASS DISCUSSION
KEY WORDS
Solve each pair of simultaneous equations.
8x + 3y = 7
3x + y = 10
2x + 5 = 3y
3x + 5y = −9
2y = 15 − 6x
10 − 6y = −4x
Discuss your answers with your classmates.
Discuss what the graphs would be like for each pair of equations.
parabola
CLASS DISCUSSION
Solve each of these quadratic equations.
x2 − 8x + 15 = 0
x2 + 4x + 4 = 0
x2 + 2x + 4 = 0
Discuss your answers with your classmates. Discuss what the graphs would be
like for each of the functions y = x2 − 8x + 15, y = x2 + 4x + 4 and y = x2 + 2x + 4.
minimum point
maximum point
turning point
stationary point
completing the
square
roots
discriminant
tangent
28
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We are working with Cambridge Assessment International Education towards endorsement of this title.
2 Simultaneous equations and quadratics
2.1 Simultaneous equations (one linear
and one non-linear)
In this section you will learn how to solve simultaneous equations where one equation
is linear and the second equation is not linear.
2
The diagram shows the graphs of y = x + 1 and y = x − 5.
The coordinates of the points of intersection of the two graphs are (−2, −1) and (3, 4).
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y
y=x+1
(3, 4)
1
(–2, –1)
x
O
y = x2 – 5
–5
We say that x = −2, y = −1 and x = 3, y = 4 are the solutions of the simultaneous equations
y = x + 1 and y = x2 − 5.
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The solutions can also be found algebraically:
y=x+1
y = ​x​​ 2​− 5
(1)
(2)
Substitute for y from (1) into (2):
x + 1 = x2 − 5
rearrange
x2 − x − 6 = 0
factorise
(x + 2)(x − 3) = 0
x = − 2 or x = 3
Substituting x = −2 into (1) gives y = −2 + 1 = −1.
Substituting x = 3 into (1) gives y = 3 + 1 = 4.
The solutions are: x = −2, y = −1 and x = 3, y = 4.
29
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
WORKED EXAMPLE 1
Solve the simultaneous equations.
2x + 2y = 7
2
2
x + 4y = 8
Answers
2x + 2y = 7
2
(1)
2
(2)
7
−
2y
From (1), x = ​ ______
​
2
Substitute for x in (2):
7 − 2y 2
2
​
​​​​  ​​ − 4y = 8
​​​(______
2 )
2
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​x​​  ​− 4​y​​  ​= 8
expand brackets
49 − 28y + 4y
2
​​ _____________
​​ − 4y = 8
4
multiply both sides by 4
49 − 28y + 4y2 − 16y2 = 32
rearrange
2
12y + 28y − 17 = 0
factorise
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(6y + 17)(2y − 1) = 0
5
1
​​   ​​
y = − 2 ​​ __ ​​ or y = __
2
6
5
1
Substituting y = − 2 ​ __ ​ into (1) gives x = 6 ​ __ ​
3
6
1
Substituting y = __
​   ​ into (1) gives x = 3
2
5
1
1
The solutions are: x = 6 ​ __ ​, y = − 2 ​ __ ​ and x = 3, y = __
​   ​
3
2
6
Exercise 2.1
Solve the following simultaneous equations.
2 y=x−6
1 y = ​x​​ 2​
​
​ ​
​ ​
​
y=x+6
​x​​ 2​ + xy = 8
6
3y = 4x − 5
​2
​ ​
​x​​  ​+ 3xy = 10
7
2x + y = 7
​  ​
​
xy = 6
y=x−1
​
​ ​
​x​​ 2​+ ​y​​ 2​= 25
4 xy = 4
​ ​ ​
y = 2x + 2
8
x−y=2
​
​2
​
2​x​​  ​− 3​y​​ 2​= 15
9
14 x + y = 4
​2
​ ​
​x​​  ​+ ​y​​ 2​= 10
3
11 xy = 2
​
​ ​
x+y=3
12 y​ ​​ 2​= 4x
​ ​
​
2x + y = 4
13 x + 3y = 0
​ 2
​ ​
2​x​​  ​+ 3y = 1
16 x − 2y = 1
​
​ ​
4​y​​ 2​− 3​x​​ 2​= 1
17 3 + x + xy = 0
​
​
​
2x + 5y = 8
18 xy = 12
​
​
​
(​ x − 1)​(​ y + 2)​= 15
x + 2y = 7
​
​ ​
​x​​ 2​+ ​y​​ 2​= 10
5
2
​ ​​  ​ − xy = 0
x
​
​
​
x+y=1
10 y = 2x
​2
​
​
​x​​ ​+ ​y​​2​= 3
15 y = 3x
​ 2 ​
​
2​y​​  ​ − xy = 15
19 Calculate the coordinates of the points where the line y = 1 − 2x cuts the curve x
​ ​​2​+ ​y​​ 2​= 2.
30
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2 Simultaneous equations and quadratics
20 The sum of two numbers x and y is 11.
The product of the two numbers is 21.25.
a
Write down two equations in x and y.
b
Solve your equations to find the possible values of x and y.
21 The sum of the areas of two squares is 818 cm2.
The sum of the perimeters is 160 cm.
Find the lengths of the sides of the squares.
Find the length of the line AB.
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22 The line y = 2 − 2x cuts the curve 3x2 − y2 = 3 at the points A and B.
23 The line 2x + 5y = 1 meets the curve x2 + 5xy − 4y2 + 10 = 0 at the points A and B.
Find the coordinates of the midpoint of AB.
24 The line y = x − 10 intersects the curve x2 + y2 + 4x + 6y − 40 = 0 at the points A
and B. Find the length of the line AB.
25 The straight line y = 2x − 2 intersects the curve x2 − y = 5 at the points A and B.
Given that A lies below the x-axis and the point P lies on AB such that
AP : PB = 3 : 1, find the coordinates of P.
26 The line x − 2y = 2 intersects the curve x + y2 = 10 at two points A and B.
Find the equation of the perpendicular bisector of the line AB.
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2.2 Maximum and minimum values of
a quadratic function
The general equation of a quadratic function is f(x) = ax2 + bx + c,
where a, b and c are constants and a ≠ 0.
2
The graph of the function y = ax + bx + c is called a parabola.
The orientation of the parabola depends on the value of a,
the coefficient of x2.
If a . 0, the curve has a minimum point which occurs
at the lowest point of the curve.
If a , 0, the curve has a maximum point which occurs
at the highest point of the curve.
The maximum and minimum points are also called
turning points or stationary points.
Every parabola has a line of symmetry that passes through
the maximum or minimum point.
31
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
WORKED EXAMPLE 2
f (x) = x2 − 3x − 4
x∈ℝ
a
Find the axis crossing points for the graph of y = f (x).
b
Sketch the graph of y = f (x) and use the symmetry of the curve to find the
coordinates of the minimum point.
c
State the range of the function f (x).
a
y = x2 − 3x − 4
When x = 0, y = −4
When y = 0,
PL
E
Answers
2
x − 3x − 4 = 0
(x + 1) (x − 4) = 0
x = − 1 or x = 4
Axes crossing points are: (0, −4), (−1, 0) and (4, 0).
b
y
The line of symmetry cuts the
x-axis midway between −1 and 4.
–1
c
y = x2 – 3x – 4
4
O
SA
M
So, the line of symmetry is x = 1.5
2
When x = 1.5, y = 1.5 − 3(1.5) − 4
y = − 6.25
Minimum point = (1.5, − 6.25)
x = 1.5
x
The range is f(x) > −6.25
–4
(1.5, −6.25)
Completing the square
If you expand the expressions (x + d )2 and (x − d )2 you obtain the results:
2
2
2
2
2
2
(x + d ) = x + 2dx + d and (x − d ) = x − 2dx + d
Rearranging these give the following important results:
2
2
2
x + 2dx = (x + d ) − d
x2 − 2dx = (x − d )2 − d 2
This is known as completing the square.
32
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2 Simultaneous equations and quadratics
To complete the square for x2 + 8x:
8÷2=4
​ ​​ 2​+ 8x = ​(x + 4)​​ 2​ − ​42​​  ​
x
​x​​ 2​+ 8x = ​(x + 4)​​ 2​− 16
2
To complete the square for x + 10x − 3:
10 ÷ 2 = 5
PL
E
​ ​​ 2​+ 10x − 3 = ​(x + 5)​​ 2​ − ​52​​  ​− 3
x
​x​​ 2​+ 10x − 3 = ​(x + 5)​​ 2​− 28
2
To complete the square for 2x − 8x − 14 you must first take a factor of 2 out of
the expression:
2
2
2​x​​  ​− 8x + 14 = 2[ ​x​​  ​− 4x + 7 ]
4÷2=2
​x​​ 2​− 4x + 7 = ​(x − 2)​​ 2​ − ​22​​  ​+ 7
​x​​ 2​− 4x + 3 = ​(x − 2)​​ 2​+ 3
2
2
2
So, 2​x​​  ​− 8x + 6 = 2​[ ​​(x − 2)​​​  ​+ 3 ]​ = 2​​(x − 2)​​​  ​+ 6
You can also use an algebraic method for completing the square, as shown in
Worked example 3.
WORKED EXAMPLE 3
SA
M
Express 2x2 − 4x + 5 in the form p(x − q)2 + r, where p, q and r are constants to
be found.
Answers
2x2 − 4x + 5 = p(x − q)2 + r
Expanding the brackets and simplifying gives:
2
2
2
2x − 4x + 5 = px − 2pqx + pq + r
2
Comparing coefficients of x , coefficients of x and the constant gives:
2 = p (1) −4 = −2pq
2
(2) 5 = pq + r
(3)
Substituting p = 2 in equation (2) gives q = 1.
Substituting p = 2 and q = 1 in equation (3) gives r = 3.
2
2
So 2x − 4x + 5 = 2(x − 1) + 3.
33
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
Completing the square for a quadratic expression or function enables you to:
•
write down the maximum or minimum value of the expression
•
write down the coordinates of the maximum or minimum point of the function
•
sketch the graph of the function
•
write down the line of symmetry of the function
•
state the range of the function.
In Worked example 3 you found that:
This part of the expression is a square so
it will always be ⩾ 0. The smallest value it
can be is 0. This occurs when x = 1.
PL
E
2
2
2x − 4x + 5 = 2 (x − 1) + 3
The minimum value of the expression is 2 × 0 + 3 = 3 and this minimum occurs when
x = 1.
So, the function y = 2x2 − 4x + 5 will have a minimum at the point (1, 3).
When x = 0, y = 5.
2
The graph of y = 2x − 4x + 5 can now be sketched:
y
SA
M
5
y = 2x 2 – 4x + 5
x=1
(1, 3)
O
x
The line of symmetry is x = 1.
The range is y > 3.
The general rule is:
For a quadratic function f (x) = ax2 + bx + c that is written in the form
f (x) = a(x − h)2 + k,
i
if a > 0, the minimum point is (h, k)
ii
if a < 0, the maximum point is (h, k).
34
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2
Simultaneous equations and quadratics
WORKED EXAMPLE 4
f (x) = 2 + 8x − 2x2
x∈ ℝ
Find the value of a, the value of b and the value of c for which f (x) = a − b (x + c)2.
b
Write down the coordinates of the maximum point on the curve y = f (x).
c
Sketch the graph of y = f (x), showing the coordinates of the points where the graph intersects the
x and y-axes.
d
State the range of the function f (x).
Answers
a
2 + 8x − 2x2 = a − b(x + c)2
PL
E
a
2 + 8x − 2x2 = a − b(x2 + 2cx + c2)
2 + 8x − 2x2 = a − bx2 − 2bcx − bc2
Comparing coefficients of x2, coefficients of x and the constant gives:
−2 = −b (1)
8 = −2bc (2)
2 = a − bc2 (3)
Substituting b = 2 in equation (2) gives c = −2.
Substituting b = 2 and c = −2 in equation (3) gives a = 10.
So, a = 10, b = 2 and c = −2.
y = 10 − 2 (x − 2)2
This part of the expression is a square so
it will always be ⩾ 0. The smallest value it
can be is 0. This occurs when x = 2.
SA
M
b
The maximum value of the expression is 10 − 2 × 0 = 10 and this maximum occurs when x = 2.
So, the function y = 2 + 8x − 2x2 will have maximum at the point (2, 10).
c
y = 2 + 8x − 2x2
When x = 0, y = 2.
When y = 0,
y
(2, 10)
y = 2 + 8x – 2x 2
2
10 − 2(x − 2) = 0
2(x − 2)2 = 10
(x − 2)2 = 5
2
__
x − 2 = ±√5
__
x = 2 ± √5
__
__
x = 2 − √ 5 or x = 2 + √ 5
O
2– 5
x
2+
5
(x = − 0.236 or x = 4.24 to 3 sf)
__
__
Axes crossing points are: (0, 2), (2 + √ 5 , 0) and (2 − √ 5 , 0)
d
The range is f (x) < 10.
35
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
Exercise 2.2
2
3
4
5
Use the symmetry of each quadratic function to find the maximum or minimum points.
Sketch each graph, showing all axis crossing points.
a
2
y = x − 5x − 6
b
y = x2 − x − 20
c
y = x2 + 4x − 21
d
2
y = x + 3x − 28
e
y = x2 + 4x + 1
f
y = 15 + 2x − x2
x2 − 5x
d
Express each of the following in the form (x − m)2 + n.
a
2
x − 8x
b
x2 − 10x
c
e
2
x + 4x
f
x2 + 7x
g
x2 + 9x
h
x2 + 3x
x2 − 10x − 5
c
x2 − 6x + 2
d
x2 − 3x + 4
x2 + 6x + 9
g
x2 + 4x − 17
h
x2 + 5x + 6
2x2 − 12x + 1
c
3x2 − 12x + 5
d
2x2 − 3x + 2
2x2 + 7x − 3
g
2x2 − 3x + 5
h
3x2 − x + 6
c
3x − x2
d
8x − x2
c
10 − 5x − x2
d
7 + 3x − x2
c
7 + 8x − 2x2
d
2 + 5x − 3x2
Express each of the following in the form (x − m)2 + n.
a
2
x − 8x + 15
b
e
2
x + 6x + 5
f
Express each of the following in the form a(x − p)2 + q.
a
2
2x − 8x + 3
b
e
2
2x + 4x + 1
f
Express each of the following in the form m − (x − n)2.
a
6
6x − x
2
b
10x − x2
Express each of the following in the form a − (x + b)2.
5 − 2x − x
2
b
8 − 4x − x2
SA
M
a
7
8
9
x2 − 3x
PL
E
1
Express each of the following in the form a − p(x +q)2.
2
b
1 − 4x − 2x2
a
9 − 6x − 2x
a
Express 4x2 + 2x + 5 in the form a (x + b)2 + c, where a, b and c are constants.
b
2
Does the function y = 4x + 2x + 5 meet the x-axis?
Explain your answer.
f (x) = 2x2 − 8x + 1
a
2
2
Express 2x − 8x + 1 in the form a(x + b) + c, where a and b are integers.
b
Find the coordinates of the stationary point on the graph of y = f (x).
10 f (x) = x2 − x − 5 for x ∈ ℝ
a
Find the minimum value of f (x) and the corresponding value of x.
b
Hence write down a suitable domain for f (x) in order that f
−1 (x) exists.
11 f (x) = 5 − 7x − 2x2 for x ∈ ℝ
a
2
Write f(x) in the form p − 2(x − q) , where p and q are constants to be found.
b
Write down the range of the function f (x).
36
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2 Simultaneous equations and quadratics
12 f (x) = 14 + 6x − 2x2 for x ∈ ℝ
a
2
2
Express 14 + 6x − 2x in the form a + b (x + c) , where a, b and c are constants.
b
Write down the coordinates of the stationary point on the graph of y = f (x).
c
Sketch the graph of y = f (x).
13 f (x) = 7 + 5x − x2 for 0 < x < 7
2
2
Express 7 + 5x − x in the form a − (x + b) , where a, and b are constants.
b
Find the coordinates of the turning point of the function f (x), stating
whether it is a maximum or minimum point.
c
Find the range of f.
d
State, giving a reason, whether or not f has an inverse.
PL
E
a
14 The function f is such that f (x) = 2x2 − 8x + 3.
a
2
Write f (x) in the form 2(x + a) + b, where a and b are constants to be found.
b
Write down a suitable domain for f so that f
−1
exists.
15 f (x) = 4x2 + 6x − 8 where x > m
Find the smallest value of m for which f has an inverse.
16 f (x) = 1 + 4x − x2 for x > 2
2
2
Express 1 + 4x − x in the form a − (x + b) , where a and b are constants to
be found.
b
Find the coordinates of the turning point of the function f (x), stating
whether it is a maximum or minimum point.
SA
M
a
c
−1
Explain why f (x) has an inverse and find an expression for f (x) in terms of x.
2.3 Graphs of y = |f (x)| where f (x)
is quadratic
To sketch the graph of the modulus function y = | ax2 + bx + c |, you must:
•
2
first sketch the graph of y = ax + bx + c
•
2
reflect in the x-axis the part of the curve y = ax + bx + c that is below the x-axis.
WORKED EXAMPLE 5
Sketch the graph of y = |x2 − 2x − 3|.
Answers
First sketch the graph of y = x2 − 2x − 3.
When x = 0, y = −3.
So, the y-intercept is −3.
37
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
CONTINUED
When y = 0,
2
x − 2x − 3 = 0
(x + 1)(x − 3) = 0
x = − 1 or x = 3.
The x-intercepts are −1 and 3.
PL
E
−1 + 3
​   ​= 1.
The x-coordinate of the minimum point = ______
2
2
The y-coordinate of the minimum point = (1) − 2(1) − 3 = −4.
The minimum point is (1, −4).
y = x 2 – 2x – 3
y
–1
O
3
x
–3
(1, – 4)
Now reflect in the x-axis the part of the curve y = x2 − 2x − 3 that is below the
x-axis.
y = |x 2 – 2x – 3|
SA
M
y
(1, 4)
3
–1
x
3
O
A sketch of the function y = | x2 + 4x − 12 | is shown below.
y
y = |x 2 + 4x – 12|
(–2, 16)
12
–6
O
2
x
38
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2 Simultaneous equations and quadratics
Now consider using this graph to find the number of solutions of the equation
| x2 + 4x − 12 | = k, where k > 0.
y
y = |x 2 + 4x – 12|
y = 20 |x 2 + 4x – 12| = 20 has 2 solutions
(–2, 16)
y = 16 |x 2 + 4x – 12| = 16 has 3 solutions
12
–6
O
|x 2 + 4x – 12| = 7 has 4 solutions
PL
E
y=7
|x 2 + 4x – 12| = 0 has 2 solutions
x
2
The conditions for the number of solutions of the equation | x2 + 4x − 12 | = k are:
Value of k
k=0
Number of solutions
2
0 , k , 16
k = 16
k . 16
4
3
2
Equations involving | f (x) |, where f (x) is quadratic, can be solved algebraically:
To solve | x2 + 4x − 12 | = 16:
2
x + 4x − 12 = 16
2
x + 4x − 28 = 0
________________
2
or
x2 + 4x − 12 = − 16
or
x2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
SA
M
4 − 4 × 1 × (− 28) ​
− 4 ± √​
x = ______________________
​​
​​ or
2×1
____
− 4 ± √​ 128 ​
x = __________
​​
​​
or
2
__
x = − 2 ± 4​​√2 ​​
x = −2
(x = 3.66 or x = − 7.66 to 3 sf )
__
__
The exact solutions are x = − 2 − 4​√2 ​ or x = −2 or x = − 2 + 4​√2 ​.
TIP
​The graph of y = |​ ​x​​ 2​+ 4x − 12 |​is sketched here showing these three solutions.​
y
y = |x 2 + 4x – 12|
(–2, 16)
12
–2 – 4
2
–6
–2
O
2
–2 + 4
2
x
39
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
Exercise 2.3
1
Sketch the graphs of each of the following functions.
a
2
y = ​|​ x​​  ​− 4x + 3 |​
b
y = |​​ x​​ 2​− 2x − 3 |​
c
y = ​| ​x​​ 2​− 5x + 4 |​
d
2
y = ​|​ x​​  ​− 2x − 8 |​
e
y = |​ 2​x​​ 2​− 11x − 6 |​
f
y = ​| 3​x​​ 2​+ 5x − 2 |​
f (x) = 1 − 4x − x2
a Write f (x) in the form a − (x + b)2, where a and b are constants.
b Sketch the graph of y = f (x).
c
Sketch the graph of y = | f (x) |.
3
f (x) = 2x2 + x − 3
a Write f(x) in the form a (x + b)2 + c, where a, b and c are constants.
b Sketch the graph of y = | f (x) |.
4
a
b
c
Find the coordinates of the stationary point on the curve y = | (x − 7) (x + 1) |.
Sketch the graph of y = | (x − 7) (x + 1) |.
Find the set of values of k for which | (x − 7) (x + 1) | = k has four solutions.
5
a
b
Find the coordinates of the stationary point on the curve y = | (x + 5) (x + 1) |.
Find the set of values of k for which | (x + 5) (x + 1) | = k has two solutions.
6
a
b
Find the coordinates of the stationary point on the curve y = | (x − 8) (x − 3) |.
Find the value of k for which | (x − 8) (x − 3) | = k has three solutions.
7
Solve these equations.
PL
E
2
|​​ x​​ 2​− 6 |​= 10
|​ x​​ 2​− 2 |​= 2
b​
| ​x​​ 2​− 5x |​= 6
c​
d
2
​|​ x​​  ​+ 2x |​= 24
|​ x​​ 2​− 5x + 1 |​= 3
e​
|​ x​​ 2​+ 3x − 1 |​= 3
f​
g
2
​|​ x​​  ​+ 2x − 4 |​= 5
| 2​x​​ 2​− 3 |​= 2x
h​
|​ x​​ 2​− 4x + 7 |​= 4
i​
SA
M
a
8
CHALLENGE QUESTION
Solve these simultaneous equations.
a y=x+1
b 2y = x + 4
c
y = 2x
1
2
2
y = | x − 2x − 3 |
y = ​​ ​ __ ​x − x − 3 ​​
y = | 2x2 − 4 |
2
|
|
2.4 Quadratic inequalities
You should already know how to solve linear inequalities.
Two examples are shown below.
Solve 2x − 5
2x − 10
2x
x
,9
,9
, 19
, 9.5
Solve 5 − 3x > 17
− 3x > 12
x < − 4
expand brackets
add 10 to both sides
divide both sides by 2
subtract 5 from both sides
divide both sides by −3
40
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2 Simultaneous equations and quadratics
TIP
It is very important that you remember the rule that when you multiply or
divide both sides of an inequality by a negative number then the inequality
sign must be reversed. This is illustrated in the second of these examples,
when both sides of the inequality were divided by −3.
CLASS DISCUSSION
PL
E
7x + 12
Robert is asked to solve the inequality _______
​  x ​
> 3.
He writes: 7x + 12 ⩾ 3x
4x ⩾ −12
So x ⩾ −3
Anna checks his answer using the number −4.
She writes: When x = −4,
(7 × (−4) + 12) ÷ (−4) = (−16) ÷ (−4) = 4
Hence x = −4 is a value of x that satisfies
So Robert’s answer must be incorrect!
the original inequality
Discuss Robert’s working out with your classmates and explain Robert’s error.
7x + 12
Now solve the inequality _______
​
​
> 3 correctly.
x
SA
M
Quadratic inequalities can be solved by sketching a graph and considering when the
graph is above or below the x-axis.
WORKED EXAMPLE 6
Solve x2 − 3x − 4 . 0.
Answers
Sketch the graph of y = x2 − 3x − 4.
2
When y = 0, x − 3x − 4 = 0
(x + 1) (x − 4) = 0
x = − 1 or x = 4
So, the x-axis crossing points are −1 and 4.
2
For x − 3x − 4 . 0 you need to find
the range of values of x for which the
curve is positive (above the x-axis).
The solution is x , −1 and x . 4.
y = x 2 – 3x – 4
y
+
–1
O
+
4
x
–
41
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
WORKED EXAMPLE 7
Solve 2x2 < 15 − x.
Answers
Rearranging: 2x2 + x − 15 < 0.
2
When y = 0, 2x + x − 15 = 0
(2x − 5) (x + 3) = 0
x = 2.5 or x = − 3
PL
E
2
Sketch the graph of y = 2x + x − 15.
So, the x-axis crossing points are −3 and 2.5
2
For 2x + x − 15 < 0 you need to
find the range of values of x for
which the curve is either zero or
negative (below the x-axis).
The solution is −3 < x < 2.5
y
y = 2x 2 + x – 15
+
+
–3
2.5
O
x
–
SA
M
Exercise 2.4
1
Solve.
a
(x + 3) (x − 4) . 0
b
(x − 5) (x − 1) < 0
c
(x − 3) (x + 7) > 0
d
x(x − 5) , 0
e
(2x + 1) (x − 4) , 0
g
2
3
(3 − x) (x + 1) > 0
h
(x − 5) > 0
i
(x − 3)2 < 0
Solve.
a
x2 + 5x − 14 , 0
b
x2 + x − 6 > 0
c
x2 − 9x + 20 < 0
d
x2 + 2x − 48 . 0
e
2x2 − x − 15 < 0
f
5x2 + 9x + 4 . 0
b
12x , x2 + 35
c
x(3 − 2x) < 1
e
(x + 3) (1 − x) , x − 1
f
(4x + 3) (3x − 1) , 2x (x + 3)
Solve.
a
d
4
(2x + 3) (x − 5) , 0
f
2
x2 , 18 − 3x
2
x + 4x , 3(x + 2)
Find the set of values of x for which
a
x2 − 11x + 24 , 0 and 2x + 3 , 13
b
x2 − 4x < 12
and 4x − 3 . 1
c
x(2x − 1) , 1
and 7 − 2x , 6
2
d
x − 3x − 10 , 0 and x2 − 10x + 21 , 0
e
x2 + x − 2 . 0
and x2 − 2x − 3 > 0.
42
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2
5
Simultaneous equations and quadratics
Solve.
| x2 + 2x − 2 | , 13
a
6
b
| x2 − 8x + 6 | , 6
c
| x2 − 6x + 4 | , 4
CHALLENGE QUESTION
4
, 0.
Find the range of values of x for which ___________
2
3x − 2x − 8
REFLECTION
Look back at this exercise.
How confident do you feel in your understanding of this section?
b
What can you do to increase your level of confidence?
PL
E
a
2.5 Roots of quadratic equations
The solutions of an equation are called the roots of the equation.
Consider solving the following three quadratic equations using the quadratic formula
________
2
x=
− b ± √b − 4ac
______________
2a
.
x2 + 2x − 8_______________
=0
2
− 2 ± √2 − 4 × 1 × (− 8)
x=
2×1
___
√
36
−
2
±
x = _________
2
x = 2 or x = − 4
____________
2
x=
− 6 ± √6 − 4 × 1 × 9
__________________
2×1
√
0
−
6
±
x = ________
2
x = − 3 or x = − 3
__
2
x + 2x + 6 = 0
____________
2
x=
2 distinct roots
2 equal roots
− 2 ± √2 − 4 × 1 × 6
__________________
2×1
____
√
−
20
−
2
±
x = __________
2
no solutions
SA
M
_____________________
2
x + 6x + 9 = 0
0 roots
The part of the quadratic formula underneath the square root sign is called
the discriminant.
discriminant = b2 − 4ac
The sign (positive, zero or negative) of the discriminant tells you how many roots there
are for a particular quadratic equation.
b2 − 4ac
Nature of roots
.0
2 real distinct roots
=0
2 real equal roots
,0
0 real roots
2
There is a connection between the roots of the quadratic equation ax + bx + c = 0 and
2
the corresponding curve y = ax + bx + c.
43
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
b2 − 4ac Nature of roots of
ax2 + bx + c = 0
.0
Shape of curve y = ax2 + bx + c
2 real distinct roots
a>0
x
or
or
a<0
x
The curve cuts the x-axis at 2 distinct points.
=0
x
2 real equal roots
a>0
a<0
PL
E
oror
x
The curve touches the x-axis at 1 point.
,0
0 real roots
x
a>0
or
a<0
or
x
The curve is entirely above or entirely below the
x-axis.
WORKED EXAMPLE 8
SA
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Find the values of k for which x2 − 3x + 6 = k(x − 2) has two equal roots.
Answers
x2 − 3x + 6 = k(x − 2)
x2 − 3x + 6 − kx + 2k = 0
x2 − (3 + k)x + 6 + 2k = 0
For two equal roots, b2 − 4ac = 0.
2
(3 + k) − 4 × 1 × (6 + 2k) = 0
k2 + 6k + 9 − 24 − 8k = 0
k2 − 2k − 15 = 0
(k + 3) (k − 5) = 0
So k = −3 or k = 5.
44
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2 Simultaneous equations and quadratics
WORKED EXAMPLE 9
Find the values of k for which x2 + (k − 2)x + 4 = 0 has two distinct roots.
Answers
x2 + (k − 2)x + 4 = 0
Critical values are −2 and 6.
+
–2
6
k
–
So k , −2 or k . 6.
Exercise 2.5
1
+
PL
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2
For two distinct roots b − 4ac . 0
(k − 22) − 4 × 1 × 4 . 0
k2 − 4k + 4 − 16 . 0
k2 − 4k − 12 . 0
(k + 2) (k − 6) . 0
State whether these equations have two distinct roots, two equal roots or no roots.
a
x2 + 4x + 4 = 0
b
d
x2 − 3x + 15 = 0
e
g
3x2 + 2x + 7 = 0
h
x2 + 4x − 21 = 0
c
x2 + 9x + 1 = 0
x2 − 6x + 2 = 0
f
4x2 + 20x + 25 = 0
5x2 − 2x − 9 = 0
Find the values of k for which x2 + kx + 9 = 0 has two equal roots.
3
Find the values of k for which kx2 − 4x + 8 = 0 has two distinct roots.
4
Find the values of k for which 3x2 + 2x + k = 0 has no real roots.
5
Find the values of k for which (k + 1)x2 + kx − 2k = 0 has two equal roots.
6
Find the values of k for which kx2 + 2(k + 3)x + k = 0 has two distinct roots.
7
Find the values of k for which 3x2 − 4x + 5 − k = 0 has two distinct roots.
8
Find the values of k for which 4x2 − (k − 2)x + 9 = 0 has two equal roots.
9
Find the values of k for which 4x2 + 4 (k − 2)x + k = 0 has two equal roots.
SA
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2
10 Show that the roots of the equation x2 + (k − 2)x − 2k = 0 are real and distinct for
all real values of k.
11 Show that the roots of the equation kx2 + 5x − 2k = 0 are real and distinct for all
real values of k.
45
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
2.6 Intersection of a line and a curve
When considering the intersection of a straight line and a parabola, there are three
possible situations.
2 points of intersection
The line cuts the curve at two
distinct points.
Situation 2
Situation 3
PL
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Situation 1
1 point of intersection
The line touches the curve at one point.
This means that the line is a tangent to the curve.
0 points of intersection
The line does not intersect
the curve.
You have already learned that to find the points of intersection of the line y = x − 6
with the parabola y = x2 − 3x − 4 you solve the two equations simultaneously.
This would give x2 − 3x − 4 = x − 6
x2 − 4x + 2 = 0.
The resulting quadratic equation can then be solved using the quadratic formula:
________
SA
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2
− b ± √b − 4ac
x = _____________
2a
The number of points of intersection will depend on the value of b2 − 4ac.
The different situations are given in the table below.
b2−4ac
.0
=0
,0
Nature of roots
2 real distinct roots
2 real equal roots
0 real roots
Line and curve
2 distinct points of intersection
1 point of intersection (line is a tangent)
no points of intersection
The condition for a quadratic equation to have real roots is b2 − 4ac > 0.
WORKED EXAMPLE 10
Find the value of k for which y = 2x + k is a tangent to the curve y = x2 −4x + 4.
Answers
2
x − 4x + 4 = 2x + k
2
x − 6x + (4−k) = 0
2
Since the line is a tangent to the curve, b − 4ac = 0.
2
(− 6 ) − 4 × 1 × (4 − k) = 0
36 − 16 + 4k = 0
k = −5
46
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2 Simultaneous equations and quadratics
WORKED EXAMPLE 11
Find the range of values of k for which y = x − 5 intersects the curve y = kx2 − 6
at two distinct points.
Answers
kx2 − 6 = x − 5
kx − x − 1 = 0
2
2
(− 1) − 4 × k × (− 1) . 0
1 + 4k . 0
1
k . − ​​ __ ​​
4
WORKED EXAMPLE 12
PL
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2
Since the line intersects the curve at two distinct points, b − 4ac . 0.
Find the values of k for which y = kx − 3 does not intersect the curve y = x2 − 2x + 1.
Answers
x2 − 2x + 1 = kx − 3
x − x(2 + k) + 4 = 0
2
2
Since the line and curve do not intersect, b − 4ac , 0.
SA
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2
(2 + k) − 4 × 1 × 4 , 0
k2 + 4k + 4 − 16 , 0
k2 + 4k − 12 , 0
(k + 6) (k − 2) , 0
Critical values are −6 and 2.
So −6 , k , 2.
+
–6
+
2
k
–
Exercise 2.6
1
Find the values of k for which y = kx + 1 is a tangent to the curve y = 2x2 + x + 3.
2
Find the value of k for which the x-axis is a tangent to the curve
y = x2 + (3 − k)x − (4k + 3).
3
Find the values of the constant c for which the line y = x + c is a tangent to the
2
​   ​.
curve y = 3x + __
x
Find the set of values of k for which the line y = 3x + 1 cuts the curve
y = x2 + kx + 2 in two distinct points.
4
5
The line y = 2x + k is a tangent to the curve x2 + 2xy + 20 = 0.
a
Find the possible values of k.
b
For each of these values of k, find the coordinates of the point of contact of
the tangent with the curve.
47
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
6
Find the set of values of k for which the line y = k − x cuts the curve
y = x2 − 7x + 4 in two distinct points.
7
Find the values of k for which the line y = kx − 10 intersects the curve
x2 + y2 = 10x.
8
Find the set of values of m for which the line y = mx − 5 does not intersect
the curve y = x2 − 5x + 4.
9
The line y = mx + 6 is a tangent to the curve y = x2 − 4x + 7.
SUMMARY
Completing the square
PL
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Find the possible values of m.
For a quadratic function f (x) = ax2 + bx + c that is written in the form f (x) = a(x − h)2 + k,
i
if a . 0, the minimum point is (h, k)
ii if a , 0, the maximum point is (h, k).
Quadratic equation (ax2 + bx + c = 0) and corresponding curve ( y = ax2 + bx + c)
b2 − 4ac
.0
Nature of roots of ax2 + bx + c = 0
2 real distinct roots
Shape of curve y = ax2 + bx + c
a>0
x
or
or
a<0
x
SA
M
The curve cuts the x-axis at 2 distinct points.
=0
x
2 real equal roots
a>0
oror
a<0
x
The curve touches the x-axis at 1 point.
,0
0 real roots
x
a>0
or
a<0
or
x
The curve is entirely above or entirely below the x-axis.
48
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2 Simultaneous equations and quadratics
CONTINUED
Intersection of a quadratic curve and a straight line
Situation 2
2 points of intersection
The line cuts the curve at two
distinct points.
Interpreting the discriminant
Situation 3
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Situation 1
1 point of intersection
0 points of intersection
The line touches the curve at one
point. This means that the line is a
tangent to the curve.
The line does not intersect
the curve.
Solving simultaneously the equation of the curve with the equation of the line will give a quadratic equation of
2
2
the form ax + bx + c = 0. The discriminant b − 4ac, gives information about the roots of the equation and also
about the intersection of the curve with the line.
b2−4ac
Nature of roots
Line and curve
2 real distinct roots
2 distinct points of intersection
=0
2 real equal roots
1 point of intersection (line is a tangent)
,0
no real roots
no points of intersection
SA
M
.0
2
The condition for a quadratic equation to have real roots is b − 4ac > 0.
Past paper questions
Worked example
a
Express 5​x​​ 2​− 14x − 3 in the form p​(x + q)​​ 2​ + r, where p, q and r are constants.
[3]
b
Sketch the graph of y = ​| 5​x​​  ​− 14x − 3 |​on the axes below. Show clearly any points where your graph
meets the coordinate axes.
[4]
2
y
O
x
49
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
c
State the set of values of k for which ​| 5​x​​ 2​− 14x − 3 |​ = k has exactly four solutions.
[2]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2018
Answers
a
3
14
5​x​​ 2​− 14x − 3 = 5​​(​x​​ 2​ − ___
​   ​x − __
​   ​)​​
5
5
7 2
7 2 3
= 5​[​​(x − __
​   ​)​​​  ​ − (
​​ __
​   ​)​​​  ​ − __
​   ​ ]​
5
5
5
PL
E
7 2 64
= 5​[​​(x − __
​   ​)​​​  ​ − ___
​   ​ ]​
5
25
7 2 64
= 5​​(x − __
​   ​)​​​  ​ − ___
​   ​
5
5
b
2
First sketch the graph of y = 5​x​​  ​− 14x − 3.
When x = 0, y = − 3.
So the y-intercept is −3.
When y = 0,
7 2 64
5​​(x − __
​   ​)​​​  ​ − ___
​   ​= 0
5
5
7 2 64
​   ​)​​​  ​ = ___
​   ​
5​​(x − __
5
5
SA
M
7 2 64
​   ​)​​​  ​ = ___
​   ​
​​(x − __
5
25
8
7
x − ​ __ ​ = ± ​ __ ​
5
5
1
x = 3 or x = − ​ __ ​
5
1
So, the x-intercepts are − ​ __ ​and 3.
5
7
64
Using the answer to part i, the minimum point on the curve is ​(__
​   ​, − ___
​   ​)​.
5
5
Graph of y = 5​x​​ 2​− 14x − 3 is:
Graph of y = | 5​x​​ 2​− 14x − 3 | is:
y
y
–0.2
O
3
–3
64
)
5
x
3
( 75 , – 64
)
5
c
7
(5 ,
–0.2 O
3
x
–3
64
The values of k for which ​| 5​x​​ 2​− 14x − 3 |​ = k has exactly four solutions are 0 , k , ___
​   ​.
5
50
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2 Simultaneous equations and quadratics
1
Find the set of values of k for which the line y = k (4x − 3) does not intersect the curve y = 4x2 + 8x − 8.
[5]
Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q4 Jun 2014
2
Find the set of values of x for which x(x + 2) , x.[3]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q1 Jun 2014
3
a
b
Express 2x2 − x + 6 in the form p(x − q)2 + r, where p, q and r are constants to be found.
[3]
2
Hence state the least value of 2x − x + 6 and the value of x at which this occurs.
[2]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q5 Jun 2014
Find the range of values of k for which the equation kx2 + k = 8x − 2xk has 2 real distinct roots.
PL
E
4
[4]
Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q1 Nov 2015
5
a
b
Find the set of values of x for which 4x2 + 19x − 5 < 0.
i
Express x2 + 8x − 9 in the form (x + a)2 + b, where a and b are integers.
ii Use your answer to part i to find the greatest value of 9 − 8x − x2 and the value of x at which
this occurs.
iii Sketch the graph of y = 9 − 8x − x2, indicating the coordinates of any points of intersection with
the coordinate axes.
[3]
[2]
[2]
[2]
Adapted from Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2015
2
The curve 3​x​​  ​ + xy − ​y​​  ​+ 4y − 3 = 0 and the line y = 2(1 − x) intersect at the points A and B.
i
Find the coordinates of A and B.[5]
ii Find the equation of the perpendicular bisector of the line AB, giving your answer in the form
ax + by = c, where a, b and c are integers.
[4]
SA
M
6
2
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q9 Jun 2017
7
a
b
2
Write 9​x​​  ​− 12x + 5 in the form p​(x − q)​​ 2​ + r, where p, q and r are constants.
Hence write down the coordinates of the minimum point of the curve y = 9​x​​ 2​− 12x + 5.
[3]
[1]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q2 Jun 2020
8
The line y = 5x + 6 meets the curve xy = 8 at the points A and B.
a Find the coordinates of A and B. [3]
b Find the coordinates of the point where the perpendicular bisector of the line AB meets the line y = x.[5]
Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q6 Jun 2020
9
Solve the inequality (x − 1) (x − 5) . 12.
[4]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q1 Nov 2017
10 Solve the equations
y−x=4
​x​​ 2​+ ​y​​ 2​− 8x − 4y − 16 = 0
[5]
Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q1 June 2018
11 Find the values of k for which the line y = kx + 3 is a tangent to the curve y = 2​x​​ 2​+ 4x + k − 1.
[5]
Cambridge IGCSE Additional Mathematics 0606 Paper 12 Q2 Mar 2020
51
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CAMBRIDGE IGCSE™ AND O LEVEL ADDITIONAL MATHEMATICS: COURSEBOOK
12 Find the values of the constant k for which the equation k​x​​ 2​− 3(k + 1)x + 25 = 0 has equal roots.
[4]
Cambridge IGCSE Additional Mathematics 0606 Paper 22 Q2 Mar 2021
13 Do not use a calculator in this question. The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B.
The mid-point of AB is the point C. Show that the point C lies on the line x + y = 11. [7]
Cambridge IGCSE Additional Mathematics 0606 Paper 21 Q6 Nov 2019
Show that 2​x​​  ​+ 5x − 3 can be written in the form a​(x + b)​​ 2​ + c, where a, b and c are constants.
Hence, write down the coordinates of the stationary point on the curve with equation y = 2​x​​ 2​+ 5x − 3.
On the axes below, sketch the graph of y = |​ 2​x​​ 2​+ 5x − 3 |​, stating the coordinates of the intercepts with
the axes.
PL
E
14 a
b
c
2
y
[3]
[2]
x
O
[3]
d
2
Write down the value of k for which the equation |​ 2​x​​  ​+ 5x − 3 |​ = k has exactly 3 distinct solutions.
[1]
Cambridge IGCSE Additional Mathematics 0606 Paper 12 Q4 Mar 2021
Write ​x​​ 2​− 9x + 8 in the form ​(x − p)​​ 2​ − q, where p and q are constants.
Hence write down the coordinates of the minimum point on the curve y = ​x​​ 2​− 9x + 8.
On the axes below, sketch the graph of y = ​| ​x​​ 2​− 9x + 8 |​, showing the coordinates of the points where
the curve meets the coordinate axes.
SA
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15 i
ii
iii
[2]
[1]
y
20
16
12
8
4
–2
O
–4
2
4
6
8
10 x
–8
–12
–16
iv
[3]
Write down the value of k for which ​| ​x​​  ​− 9x + 8 |​ = k has exactly 3 solutions.
2
[1]
Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q4 Nov 2018
52
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PL
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Cambridge IGCSE™ and O Level
Additional
Mathematics
PRACTICE BOOK
SA
M
Muriel James
Digital Access
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Contents
Contents
Introduction00
How to use this book
00
00
How to use this series
PL
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1 Functions00
2 Simultaneous equations and quadratics
00
3 Factors and polynomials
00
4 Equations, inequalities and graphs
00
5 Logarithmic and exponential functions
00
6 Straight-line graphs
00
7 Coordinate geometry of the circle
00
8 Circular measure
00
SA
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9 Trigonometry00
10 Permutations and combinations
00
11 Series00
12 Calculus – Differentiation 1
00
13 Vectors00
14 Calculus – Differentiation 2
00
15 Calculus - Integration
00
16 Kinematics00
Answers00
Glossary00
Index00
i
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
Introduction
This practice book supports the Cambridge IGCSE TM Additional Mathematics and
O Level syllabus (0606). It has been written by a highly experienced author, who is very
familiar with the syllabus. The course is aimed at students who are currently studying
or have previously studied Cambridge IGCSE TM Mathematics (0580/0980).
PL
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The practice book has been written to closely follow the chapters and topics of the
coursebook, offering additional exercises to help you to consolidate what you have
learnt.
At the start of each chapter, there is a list of learning intentions which tell you what
you will learn in the chapter.
Worked examples are used throughout to demonstrate the methods for selected topics
using typical workings and thought processes. These present methods to you in a
practical and easy-to-follow way
The exercises offer plenty of opportunities for you to practice methods that have just
been introduced.
Towards the end of each chapter, there is a summary of the key concepts to help you
consolidate what you have learnt. This is followed by a questions section which brings
together the methods and concepts from the whole chapter.
SA
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A Coursebook is available in the Additional Mathematics series, which includes class
discussion activities, worked examples for every method, exercises and a ‘Past paper’
questions section. A digital Teacher’s Resource, to offer support and advice, is available
on the Cambridge GO platform.
ii
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How to use this book
How to use this book
Throughout this book, you will notice lots of different features that will help your
learning. These are explained below.
LEARNING INTENTIONS
KEY WORDS
PL
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These set the scene for each exercise and indicate the important concepts.
Definitions for useful vocabulary are given in bold throughout each chapter. You will
also find definitions for these words in the Glossary at the back of this book.
Exercises
These help you to practise skills that are important for studying Cambridge IGCSE
Mathematics.
There are two types of exercise:
Exercises which let you practice the mathematical skills you have learned.
•
Exercises which bring together all the mathematical concepts in a chapter,
pushing your skills further.
SA
M
•
WORKED EXAMPLE
Wherever you need to know how to use a formula to carry out a calculation,
there are worked examples boxes to show you how to do this.
REMINDER
This feature highlights key concepts from the corresponding chapter in the
coursebook.
TIP
This feature contains key equations or formulae that you will need to know.
SUMMARY
There is a summary of key points at the end of each chapter.
iii
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Chapter 1: Functions
LEARNING INTENTIONS
This section will show you how to
•
•
•
•
•
•
understand and use the terms: function, domain, range (image set), one-one function, inverse function
and composition of functions
use the notation f (x) = 2x3 + 5, f : x ↦ 5x − 3, f −1(x) and f 2(x)
understand the relationship between y = f (x) and y = | f (x) |
solve graphically or algebraically equations of the type | ax + b | = c and | ax + b | = cx + d
explain in words why a given function is a function or why it does not have an inverse
find the inverse of a one-one function and form composite functions
sketch graphs to show the relationship between a function and its inverse.
PL
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•
1.1 Mappings
REMINDER
The table below shows one-one, many-one and one-many mappings.
one-one
many-one
y
one-many
x
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y
f(x) = ± x
f(x) = x + 1
f(x)= x 2
O
O
For one input value there is just
one output value.
O
x
x
x
For two input values there is one
output value.
For one input value there are two
output values.
Exercise 1.1
Determine whether each of these mappings is one-one, many-one or one-many.
1
3
5
7
x ↦ 2x + 3
3
x ↦ 2x
−1
x ↦ ___
x
2
x ↦ __
x
x∈ ℝ
2
2
x↦x +4
x
x∈ ℝ
x∈ ℝ
4
x↦3
x∈ ℝ
x ∈ ℝ, x . 0
6
2
x↦x +1
x ∈ ℝ, x > 0
x ∈ ℝ, x . 0
8
x ↦ ± √x
__
x ∈ ℝ, x . 0
1
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
1.2 Definition of a function
REMINDER
A function is a rule that maps each x value to just one y value for a defined
set of input values.
⎧ one-one
This means that mappings that are either ⎨ many-one are called functions.
⎩
The mapping x ↦ x + 1, where x ∈ ℝ, is a one-one function.
PL
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The function can be defined as f : x ↦ x + 1, x ∈ ℝ or f (x) = x + 1, x ∈ ℝ.
The set of input values for a function is called the domain of the function.
The set of output values for a function is called the range (or image set) of
the function.
WORKED EXAMPLE 1
The function f is defined by f (x) = (x − 1)2 + 4, for 0 < x < 5.
Find the range of f.
Answers
f (x) = (x − 1)2 + 4 is a positive quadratic function so the graph will be of the form
This part of the expression is a square so it will always be > 0.
The smallest value it can be is 0. This occurs when x = 1.
SA
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2
(x − 1) + 4
The minimum value of the expression is 0 + 4 = 4 and this minimum occurs when
x = 1.
So the function f (x) = (x − 1)2 + 4 will
have a minimum at the point (1, 4).
y
(5, 20)
When x = 5, y = (5 − 1)2 + 4 = 20.
The range is 1 < f (x) < 20.
Range
When x = 0, y = (0 − 1)2 + 4 = 5.
5
(1, 4)
O
x
Domain
Exercise 1.2
1
Which of the mappings in Exercise 1.1 are functions?
2
Find the range for each of these functions.
a
f (x) = x − 9,
−2 < x < 8
b
c
f (x) = 7 − 2x,
−3 < x < 5
d
e
f (x) = 3x,
−4 < x < 3
f
f (x) = 2x − 2,
2
f (x) = 2x ,
−1
f (x) = ___
x,
0<x<6
−4 < x < 3
1<x<6
2
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Chapter 1: Functions
3
The function g is defined as g (x) = x2 − 5 for x > 0.
Find the range of g.
4
The function f is defined by f (x) = 4 − x2 for x ∈ ℝ.
Find the range of f.
The function f is defined by f (x) = 3 − (x − 1)2 for x > 1.
5
Find the range of f.
1
The function f is defined by f (x) = (4x + 1)2 − 2 for ​x > − _
​   ​​.
4
Find the range of f.
7
The function f is defined by f : x ↦ 8 − (x − 3)2 for 2 < x < 7.
Find the range of f.
PL
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6
_
8
The function f is defined by f (x) = 3 − ​√x − 1 ​ for x > 1.
Find the range of f.
9
Find the largest possible domain for the following functions.
a
d
g
1
f (x) = ____
​
​
x+3
1
f (x) = _____
​  2
​
​x​​ ​− 4
1
_ ​
g : x ↦ ​ _____
√
​ x − 2 ​
b
3
f (x) = ____
​
​
x−2
c
4
f (x) = ___________
​     ​
(​ x − 3)(​​ x + 2)​
e
f : x ↦ √​ ​x​​ 3​− 4 ​
f
f : x ↦ ​√x + 5 ​
h
x
_ ​
f : x ↦ ​ ______
√
​ 3 − 3x ​
i
f : x ↦ 1 − x2
_
_
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1.3 Composite functions
REMINDER
•
•
•
When one function is followed by another function, the resulting function
is called a composite function.
fg (x) means the function g acts on x first, then f acts on the result.
f 2(x) means ff (x), so you apply the function f twice.
WORKED EXAMPLE 2
f : x ↦ 4x + 3, for x ∈ ℝ
g : x ↦ 2x2 − 5, for x ∈ ℝ
Find fg (3).
Answers
fg (3) = f (2 × 32 − 5)
= f (13)
= 4 × 13 + 3
= 55
g acts on 3 first and g (3) = 2 × 32 − 5 = 13.
3
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
WORKED EXAMPLE 3
g (x) = 2x2 − 2, for x ∈ ℝ
h (x) = 4 − 3x, for x ∈ ℝ
Solve the equation hg (x) = −14.
Answers
hg (x) = − 14
− 14 = 10 − 6x2
24 = 6x2
4 = x2
x = ± 2
g acts on x first and g (x) = 2x2 − 2.
h is the function ‘triple and take from 4’
expand the brackets
set up and solve the equation.
Exercise 1.3
f (x) = 2 − x2, for x ∈ ℝ
x
​g (x) = __
​   ​+ 3, ​for x ∈ ℝ
2
Find the value of gf (4).
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1
PL
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hg (x) = h (2x2 − 2)
= 4 − 3(2x2 − 2)
= 4 − 6x2 + 6
= 10 − 6x2
2
f (x) = (x − 2)2 − 2, for x ∈ ℝ
Find f 2(3).
3
The function f is defined by​f (x) = 1 + ​√x − 3 ​ ,​for x > 3.
−3
The function g is defined by ​g (x) = ​ ___ ​− 1,​for x . 0.
x
Find gf (7).
4
The function f is defined by f (x) = (x − 2)2 + 3, for x . −2.
3x + 4
The function g is defined by ​g (x) = ​ ______ ​,​ for x . 2.
x+2
Find fg (6).
5
f : x ↦ 3x − 1, for x . 0
__
​g : x ↦ √​ x ​,​ for x . 0
_
Express each of the following in terms of f and g.
a
6
__
​x ↦ 3​√x ​ − 1​
_
b​
x ↦ ​√3x − 1 ​​
The function f is defined by f : x ↦ 2x − 1, for x ∈ ℝ.
8
The function g is defined by ​g : x ↦ ​ _____ ​,​ for x ≠ 4.
4−x
Solve the equation gf (x) = 5.
4
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Chapter 1: Functions
f (x) = 2x2 + 3, for x . 0
5
​g (x) = __
​   ​, ​for x . 0
x
Solve the equation fg (x) = 4.
7
2x − 1
The function f is defined by ​f : x ↦ _____
​
​​, for x ∈ ℝ, x ≠ 3.
x−3
x+1
The function g is defined by ​g : x ↦ ​ ____
​, ​for x ∈ ℝ, x ≠ 1.
2
Solve the equation fg (x) = 4.
8
PL
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The function g is defined by g (x) = 1 − 2x2 for x > 0.
9
The function h is defined by h (x) = 3x − 1 for x > 0.
Solve the equation gh (x) = −3, giving your answer(s) as exact value(s).
10 The function f is defined by f : x ↦ x2, for x ∈ ℝ.
The function g is defined by g : x ↦ x + 2, for x ∈ ℝ.
Express each of the following as a composite function, using only f and g.
a
x ↦ (x + 2)
2
b
x ↦ x2 + 2
c
x↦x+4
d
x ↦ x4
__
11 The functions f and g are defined by f : x ↦ x + 3 and ​g : x ↦ √​ x ​​, for x . 0.
Express in terms of f and g.
a
_
x
​ ↦ ​√x + 3 ​​
b
__
x↦x+6
c​
x ↦ ​√x ​ + 3​
__
12 Functions f and g are defined as ​f (x) = ​√x ​​ and​ g (x) = ​
Find the domain and range of g.
b
Solve the equation g (x) = 0.
c
Find the domain and range of fg.
​​.
2x + 1
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a
x−5
_____
TIP
Before writing
your final answers,
compare your
solutions with the
domains of the
original functions.
1.4 Modulus functions
REMINDER
•
•
The modulus (or absolute value) of a number is the magnitude of the
number without a sign attached.
The modulus of x, written as | x |, is defined as
⎧ x
|x| = ⎨ 0
⎩ −x
•
if
if
if
x.0
x=0
x,0
The statement | x | = k, where k > 0, means that x = k or x = −k.
5
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
WORKED EXAMPLE 4
Solve.
a | 4x + 3 | = x + 18
b
| 2x2 − 7 | = 9
Answers
a
| 4x + 3 | = x + 18
4x + 3 = − x − 18
5x = − 21
21
x = 5  x = − ​​ ___ ​​
5
21
Solution is: x = 5 or − ​ __ ​
5
| 2x2 − 7 | = 9
b
2
or
2x − 7 = 9
2x2 = 16
x2 = 8
__
x = ± 2​​√2 ​​
PL
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4x + 3 = x + 18
3x = 15
or
2x2 − 7 = − 9
2x2 = − 2
x2 = − 1 no real solution
__
Solution is: x = ± 2 √​​ 2 ​​
Exercise 1.4
1
Solve.
| 2x − 1 | = 11
b
| 2x + 4 | = 8
c
|
|
|
|
3x + 4
e​
​ ​ _____
​ ​​ = 4
3
2x + 5 __
2x
h​
​ ​ _____
​
+ ​   ​ ​​ = 3
5
3
|
|
9 − 2x
f​
​ _____
​
​ ​​ = 4
3
SA
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a
d
g
2
3
4
x−2
​​ ____
​   ​ ​​ = 6
5
x
​​ __
​   ​ − 6 ​​ = 1
3
|
|
i
| 6 − 3x | = 4
|
|
| 2x − 6 | = x
Solve.
a
​​ ​
| x + 4 ​ |​= 3​
b​
​ ​
​ ​= 3​
x+3
c ​​1 + ​
d
| 2x − 3 | = 3x
e
f
2x − 5
_____
|
4x + 2
_____
|
2x + | 3x − 4 | = 5
|
2x + 5
_____
|
​ ​= 4​
x+3
7 − | 1 − 2x | = 3x
Solve, giving your answers as exact values if appropriate.
a
| x2 − 4 | = 5
b
| x2 + 5 | = 11
c
| 9 − x2 | = 3 − x
d
2
| x − 3x | = 2x
e
| x2 − 16 | = 2x + 1
f
| 2x2 − 1 | = x + 2
g
2
| 3 − 2x | = x
h
| x2 − 4x | = 3 − 2x
i
| 2x2 − 2x + 5 | = 1 − x
TIP
Remember to check
your answers to make
sure that they satisfy
the original equation.
Solve each pair of simultaneous equations.
a
y=x+4
b y=1−x
y = | x2 − 2 | y = | 4x2 − 4x |
6
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Chapter 1: Functions
1.5 Graphs of y = |f (x) | where f (x)
is linear
Exercise 1.5
2
Sketch the graphs of each of the following functions, showing the coordinates of
the points where the graph meets the axes.
a
y = |x − 2|
d
1
​   ​ x − 3 ​​
​y = ​ _
3
a
Complete the table of values for y = 3 − | x − 1 | .
|
x
|
−2
y
b
3
y = | 3x − 3 |
e
y = | 6 − 3x |
−1
0
1
c
y = |3 − x|
|
1
2
3
4
3
Draw the graph of y = 3 − | x − 1 |, for −2 < x < 4.
y = | 2x | + 2
b
y = |x| − 2
c
y = |x − 1| + 3
e
| |
y = | 3x − 6 | − 2
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a
6
7
8
y = 4 − | 3x|
1
f​
y = 4 − ​_
​   ​ x ​
2
Given that each of these functions is defined for the domain −3 < x < 4, find the
range of
d
5
|
1
f​
y = ​5 − _
​   ​ x ​
2
Draw the graphs of each of the following functions.
a
4
b
PL
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1
f : x ↦ 6 − 3x
b
g : x ↦ | 6 − 3x |
c
h : x ↦ 6 − | 3x |
c
h : x ↦ 2 − | 2x |
Find the range of each function for −1 < x < 5.
a
f : x ↦ 2 − 2x
a
Sketch the graph of y = | 3x − 2 | for −4 < x < 4, showing the coordinates of
the points where the graph meets the axes.
b
On the same diagram, sketch the graph of y = x + 3.
c
Solve the equation | 3x − 2 | = x + 3.
b
g : x ↦ | 2 − 2x |
A function f is defined by f (x) = 2 − | 3x − 1 |, for −1 < x < 3.
a
Sketch the graph of y = f (x).
b
State the range of f.
c
Solve the equation f (x) = −2.
a
On a single diagram, sketch the graphs of x + 3y = 6 and y = | x + 2 |.
1
Solve the inequality | x + 2 | , ​​ __​​ ​(6 − x)​.
3
b
7
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
1.6 Inverse functions
REMINDER
The inverse of the function f (x) is written as f −1(x).
The domain of f −1(x) is the range of f (x).
The range of f −1(x) is the domain of f (x).
It is important to remember that not every function has an inverse.
An inverse function f −1(x) can exist if, and only if, the function f (x) is a
one-one mapping.
WORKED EXAMPLE 5
f (x) = (x + 3)2 −1, for x . −3
PL
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•
•
•
•
•
a
−1
Find an expression for f (x).
b
−1
Solve the equation f (x) = 3.
Answers
f (x) = (x + 3)2 −1, for x . −3
Step 1: Write the function as y = …
y = (x + 3)2 − 1
Step 2: Interchange the x and y variables.
x = (y + 3)2 − 1
Step 3: Rearrange to make y the subject. x + 1 = (y + 3)2
_____
​​√x + 1 ​​ = y + 3
_____
y = ​​√x + 1 ​​− 3
_
​f​​  −1(​​ x)​= √​ x + 1 ​− 3
SA
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a
b
−1
f (x) = 3.
_____
√
​​ x
+ 1 ​​− 3 = 3
_____
√
​​ x + 1 ​​= 6
x + 1 = 36
x = 35
Exercise 1.6
1
2
3
f (x) = (x + 2)2 − 3, for x > −2.
Find an expression for f −1(x).
5
​f (x) = ____
​
​,​ for x > 0.
x−2
−1
Find an expression for f (x).
2
2
​   ​​.
f (x) = (3x − 2) + 3, for ​x > _
3
−1
Find an expression for f (x).
8
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Chapter 1: Functions
_
4
f​ (x) = 4 − √​ x − 2 ​,​for x > 2.
Find an expression for f −1(x).
5
f : x ↦ 3x − 4, for x . 0.
4
​g : x ↦ ​ _____ ​,​ for x ≠ 4.
4−x
−1
−1
Express f (x) and g (x) in terms of x.
f (x) = (x − 2)2 + 3, for x . 2.
a
7
Solve the equation f −1(x) = f (4).
−1
Solve the equation g (x) = 6.
x
f​ (x) = __
​   ​ − 2,​for x ∈ ℝ
2
a Find f −1(x).
b
9
b
3x + 1
​g (x) = _____
​
​,​ for x . 3
x−3
a Find an expression for g−1(x) and comment on your result.
b
8
−1
Find an expression for f (x).
PL
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6
g (x) = x2 − 4x, for x ∈ ℝ
−1
Solve fg (x) = f (x), leaving answers as exact values.
3x + 1
x−2
​f : x ↦ ​ _____ ​,​for x ≠ 1​g : x ↦ ____
​   ​
,​ for x . −2
3
x−1
Solve the equation f (x) = g−1(x).
​x​​ 2​− 9
10 If ​f (x) = _____
​  2
​, ​x ∈ ℝ, find an expression for f −1(x).
​x​​ ​+ 4
SA
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__
11 If ​f (x) = 2​√x ​​ and g (x) = 5x, solve the equation f −1g (x) = 0.01.
2x − 4
12 Find the value of the constant k such that ​f (x) = _____
​
​​is a self-inverse function.
x+k
3
13 The function f is defined by f (x) = x . Find an expression for g (x) in terms of x
for each of the following:
a
fg (x) = 3x + 2
b
gf (x) = 3x + 2
x+1
14 Given that f (x) = 2x + 1 and g​ (x) = ​ ____
​
,​find the following.
2
a f −1
b g−1
c
(fg)−1
d (gf )−1
e f −1g−1
f
TIP
A self-inverse
function is one for
which f (x) = f −1(x),
for all values of x in
the domain.
g−1f −1
Write down any observations from your results.
x+2
15 Given that​ fg (x) = ____
​   ​
​ and g (x) = 2x + 5, find f (x).
3
16 Functions f and g are defined for all real numbers.
2
2
g (x) = x + 7 and gf (x) = 9x + 6x + 8.
Find f (x).
9
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
1.7 The graph of a function and
its inverse
REMINDER
y
6
This is true for all one-one functions and their
inverse functions.
4
y=x
f
PL
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The graphs of f and f −1 are reflections of each
other in the line y = x.
2
−1
−1
This is because: ff (x) = x = f f (x)
f –1
Some functions are called self-inverse
–4
–2 O
2
4
6 x
−1
functions because f and its inverse f are
–2
the same.
1
1
−1
​   ​,​ for x ≠ 0.
​   ​​​,​for x ≠ 0, then ​​f​​  ​(x) = __
If ​f (x) = __
–4
x
x
1
So ​f (x) = __
​   ​,​ for x ≠ 0​,​is an example of a self-inverse function.
x
When a function f is self-inverse, the graph of f will be symmetrical about the
line y = x.
Exercise 1.7
y
On a copy of the grid, draw the graph of
the inverse of the function y = 2−x.
6
f (x) = x2 + 5, x > 0.
4
SA
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1
2
On the same axes, sketch the graphs of
y = f (x) and y = f −1(x), showing the
coordinates of any points where the curves
meet the coordinate axes.
3
–8
–6
–4
y = 2 –x
2
–2
O
2
4 x
–2
1
​g (x) = __
​   ​ ​x​​ 2​− 4,​ for x > 0.
2
−1
Sketch, on a single diagram, the graphs of y = g (x) and y = g (x), showing the
coordinates of any points where the curves meet the coordinate axes.
4
The function f is defined by f (x) = 3x − 6 for all real values of x.
a Find the inverse function f −1(x).
b Sketch the graphs of f (x) and f −1(x) on the same axes.
c
Write down the point of intersection of the graphs f (x) and f −1(x).
5
2
The function f is defined as: f (x) = x − 2x, for x > 1.
a Explain why f −1(x) exists and find f −1(x).
b State the range of the function f −1(x).
c
Sketch the graphs of f (x) and f −1(x) on the same axes.
d Write down where f −1(x) crosses the y-axis.
10
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
Chapter 1: Functions
6
a
b
c
3x − 1
3
By finding f −1(x), show that ​f (x) = ​ ______ ​,​ x ∈ ℝ, ​x ≠ __
​   ​, ​is a
2
2x − 3
self-inverse function.
Sketch the graphs of f (x) and f −1(x) on the same axes.
Write down the coordinates of the intersection of the graphs with the
coordinate axes.
SUMMARY
Functions
PL
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A function is a rule that maps each x-value to just one y-value for a defined set of input values.
Mappings that are either one-one or many-one are called functions.
The set of input values for a function is called the domain of the function.
The set of output values for a function is called the range (or image set) of the function.
Modulus function
The modulus of x, written as | x |, is defined as
⎧ x if x . 0
| x | = ⎨ 0 if x = 0
⎩ − x if x < 0
Composite functions
fg (x) means the function g acts on x first, then f acts on the result.
2
f (x) means ff (x).
SA
M
Inverse functions
The inverse of a function f (x) is the function that undoes what f (x) has done.
The inverse of the function f (x) is written as f −1(x).
−1
The domain of f (x) is the range of f (x).
The range of f −1(x) is the domain of f (x).
An inverse function f −1(x) can exist if, and only if, the function f (x) is a one-one mapping.
The graphs of f and f
−1
are reflections of each other in the line y = x.
Exercise 1.8
1
A one-one function f is defined by f (x) = (x − 2)2 − 3, for x > k.
a State the least value that x can take.
b For this least value of k, write down the range of f.
2
The function f (x) = x2 − 4ax (where a is a positive constant) is defined for all real
values of x.
Given that the range is > −8, find the exact value of a.
11
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: PRACTICE BOOK
f (x) = (2x − 1)2 + 3, for x . 0
5
g (x) = ​​ ___ ​​, for x . 0
2x
Solve the equation fg (x) = 7.
4
The function f is defined by f (x) = 1 − x2, for x ∈ ℝ.
The function g is defined by g (x) = 2x − 1, for x ∈ ℝ.
Find the values of x (in exact form) which solve the equation fg (x) = gf (x).
5
Solve these simultaneous equations.
y = 2x + 5
y = | 3 − x2|
6
a
Sketch the graph of y = | 2x + 1 | for the domain −3 , x , 3, showing the
coordinates of the points where the graph meets the axes.
b
On the same diagram, sketch the graph of y = 3x.
c
Solve the equation 3x = | 2x + 1 |
a
Sketch the graph of y = | x + 3 |.
b
Solve the inequality | x + 3 | . 2x + 1.
7
8
f (x) = x2 − 3, for x ∈ ℝ
g (x) = 3x + 2, for x ∈ ℝ
Solve the equation gf (x) = g−1(8).
1
f (x) = 2x + 3 and g​ (x) = ​ ____ ​, ​x ∈ ℝ, x ≠ 1.
x+1
a Find an expression for the inverse function f −1(x).
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b
Find an expression for the composite function gf (x).
c
−1
Solve the equation f (x) = gf (x) − 1.
2x + 1
10 The function f is defined as: ​f (x) = ​ _____ ​, x ≠ − 2​.
x+2
−1
a Find f (x).
b
−1
Find the points of intersection of the graphs of f (x) and f (x).
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Cambridge IGCSE™ and O Level
Additional
Mathematics
TEACHER’S
RESOURCE
COURSEBOOK
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Sue Pemberton
Third edition
Digital Access
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Contents
Welcome
About the authors
How to use this series
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How to use this Teacher’s Resource
How to use this Teacher’s Resource for professional development
About the syllabus
About the assessment
Approaches to teaching and learning
Functions
2
Simultaneous equations and quadratics
3
Factors and polynomials
4
Equations, inequalities and graphs
5
Logarithmic and exponential functions
6
Straight-line graphs
7
Coordinate geometry of the circle
8
Circular measure
9
Trigonometry
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10 Permutations and combinations
11 Series
12 Calculus – Differentiation 1
13 Vectors
14 Calculus – Differentiation 2
15 Calculus – Integration
16 Kinematics
Acknowledgements
Copyright
Additional downloadable resources
Syllabus correlation grid
Lesson plan template
Active learning
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Assessment for learning
Developing language skills
Differentiation
Language awareness
Metacognition
Glossary
Coursebook answers
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Skills for life
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Practice Book answers
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Introduction
Welcome to the third edition of our very popular Cambridge IGCSE™ Additional Mathematics series.
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This new series has been designed around extensive research interviews and lesson observations with teachers
and students around the world following the course. As a result of this research, some changes have been made
to the new series, with the aim of solving and supporting your biggest classroom challenges and developing your
students’ passion and excitement for Mathematics
As well as targeted support in the Coursebook, we have produced an updated Practice book, with exercises
for each topic to provide more opportunities for students to consolidate their learning and develop their
knowledge application skills. We are introducing a Worked Solutions Manual to provide additional support for
teachers and students to work through selected Coursebook questions.
As we develop new resources, we ensure that we are keeping up-to-date with best practice in pedagogies. For
this new series we have added new features to the Coursebook, such as engaging projects to develop students’
collaborative skills and ‘pre-requisite knowledge’ guides to unlock students’ prior learning and help you to
evaluate students’ learning starting points.
Finally, we have updated this Teacher’s Resource to make it as useful and relevant as possible to your dayto-day teaching needs. From teaching activity, assessment and homework ideas, to how to tackle common
misconceptions in each topic, to a new feature developing your own teaching skills, we hope that this handy
resource will inspire you, support you and save you much-needed time.
We hope that you enjoy using this series and that it helps you to continue to inspire and excite your students
about this vital and ever-changing subject. Please don’t hesitate to get in touch if you have any questions for us,
as your views are essential for us to keep producing resources that meet your classroom needs.
Thomas Carter
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Head of Mathematics, Cambridge University Press
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
About the authors
Julianne Hughes
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Julianne Hughes has a first class honours degree in Pure Mathematics (Cardiff University 1991) and is qualified
to teach mathematics in secondary and further education in the UK.
She has been a teacher, tutor, mathematics consultant, author and resource creator since 1996.
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Julianne is now retired from teaching and tutoring and her main focus is resource creation, authoring materials
and assessing.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
How to use this
Teacher’s Resource
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This Teacher’s Resource contains both general guidance and teaching notes that help you to deliver the content
in our Cambridge resources.
There are teaching notes for each unit of the Coursebook. Each set of teaching notes contains the following
features to help you deliver the chapter.
At the start of each chapter there is a teaching plan (Figure 1). This summarises the topics covered in the chapter,
including the number of learning hours recommended for each topic, an outline of the learning content, and the
Cambridge resources from this series that can be used to deliver the topic.
Topic
Order in
chapter
Learning content
Resources
Each chapter also includes information on any background knowledge that students should have before studying
this chapter, advice on helpful language support and a selection of useful links to digital resources.
At the beginning of the teaching notes for the individual sections there is an outline of the learning objectives
(Figure 2) for that section, as well as any common misconceptions that students may have about the topic and
how you can overcome these.
Success criteria
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Syllabus learning objectives / learning
intentions
For each section, there is a selection of starter ideas, main activities and plenary ideas. You can pick out
individual ideas and mix and match them depending on the needs of your class. The activities include
suggestions for how they can be differentiated or used for assessment.
Homework ideas give suggestions for tasks, along with advice for how to assess students’ work.
You will find answers to the Coursebook and Workbook questions and exercises at the end of each chapter in
this Teacher’s Resource and answers to the Practical Workbook questions at the end of this resource.
This Teacher’s Resource also includes a set of PowerPoint presentations which include the worked examples
from each chapter, plus some extra material, explanations and definitions. Every PowerPoint slide has additional
explanatory notes and observations which are designed to help support your teaching.
Downloadable resources include differentiated Worksheets for each chapter, a sample lesson plan for each
chapter which demonstrate how certain elements within each topic may be approached, and additional printed
resource sheets.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
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How to use this
Teacher’s Resource to
supplement PD
We regularly hear from teachers that the Continuous Professional Development (CPD) they feel they get the
most out of is face-to-face training. However, we also hear that not all teachers have the time or budget to get
out of the classroom, so here’s some handy suggestions and information about how to use this teacher’s resource
for your own professional development. After all, we are all lifelong learners!
Approaches to teaching and learning
Our teacher resources now contain guidance on the key pedagogies underpinning our course content and how
we understand and define them. You can find detailed information for you to read in your own time about active
learning, assessment for learning, metacognition, differentiation, language awareness and skills for life taken
from our ‘Approaches to Learning and Teaching’ series.
Teaching activity ideas
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This Teacher’s Resource contains a range of starter, main and plenary activity ideas for you to try in your
classroom. Use them to support your creativity, breathe new life into a topic and build upon them with your
own ideas.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
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How to use this
Teacher’s Resource
as CPD
We regularly hear from teachers that the Continuous Professional Development (CPD) they feel they get the
most out of is face-to-face training. However, we also hear that not all teachers have the time or budget to get
out of the classroom, so here’s some handy suggestions and information about how to use this teacher’s resource
for your own professional development. After all, we are all lifelong learners!
Teaching skills focus
We have created a new ‘Teaching skills focus’ feature that appears once every chapter, covering a different
teaching skill with suggestions of how you can implement it in the teaching of the topic. From differentiation, to
assessment for learning, to metacognition, this feature aims to support you with trying out a new technique or
approach in your classroom and reflecting upon your own practices.
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Try it out once per teaching topic, or when you have time, and develop your skills in a supported and
contextualised way.
Approaches to learning and teaching
Our teacher resources now contain guidance on the key pedagogies underpinning our course content and how
we understand and define them. You can find detailed information for you to read in your own time about active
learning, assessment for learning, metacognition, differentiation, language awareness and skills for life taken
from our ‘Approaches to learning and teaching’ series.
Why not try reading each support document alongside the relevant Teaching skills focus for an extra bit of
bedtime reading?
Teaching activity ideas
This teacher's resource provides plenty of engaging teaching ideas - from suggestions for starters, mains and
plenaries to NRICH project guidance. You can choose what works best for your learners.
We want to include you in the Cambridge community of teachers. In this new resource, we have utilised up-todate pedagogy and our research in schools to cater to teachers and learners. Our authors are skilled teachers and
we hope you enjoy their suggestions for activities to engage your learners.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Approaches to
learning and teaching
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The following are the pedagogical practices underpinning our course content and how we understand and
define them.
Active learning
Active learning is a pedagogical practice that places student learning at its centre. It focuses on how students
learn, not just on what they learn. We, as teachers, need to encourage students to ‘think hard’, rather than
passively receive information. Active learning encourages students to take responsibility for their learning and
supports them in becoming independent and confident students in school and beyond.
Assessment for Learning
Assessment for Learning (AfL) is a pedagogical practice that generates feedback which can be used to improve
students’ performance. Students become more involved in the learning process and, from this, gain confidence
in what they are expected to learn and to what standard. We, as teachers, gain insights into a student’s level of
understanding of a particular concept or topic, which helps to inform how we support their progression.
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Differentiation
Differentiation is usually presented as a pedagogical practice where teachers think of students as individuals and
learning as a personalised process. Whilst precise definitions can vary, typically the core aim of differentiation
is viewed as ensuring that all students, no matter their ability, interest or context, make progress towards their
learning intentions. It is about using different approaches and appreciating the differences in students to help
them make progress. Teachers therefore need to be responsive, and willing and able to adapt their teaching to
meet the needs of their students.
Language awareness
For many students, English is an additional language. It might be their second or perhaps their third language.
Depending on the school context, students might be learning all or just some of their subjects through English.
For all students, regardless of whether they are learning through their first language or an additional language,
language is a vehicle for learning. It is through language that students access the learning intentions of the
lesson and communicate their ideas. It is our responsibility, as teachers, to ensure that language doesn’t present a
barrier to learning.
Metacognition
Metacognition describes the processes involved when students plan, monitor, evaluate and make changes to
their own learning behaviours. These processes help students to think about their own learning more explicitly
and ensure that they are able to meet a learning goal that they have identified themselves or that we, as teachers,
have set.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Skills for Life
These six areas are:
•
•
•
•
•
•
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How do we prepare students to succeed in a fast-changing world? To collaborate with people from around the
globe? To create innovation as technology increasingly takes over routine work? To use advanced thinking skills
in the face of more complex challenges? To show resilience in the face of constant change? At Cambridge, we are
responding to educators who have asked for a way to understand how all these different approaches to life skills
and competencies relate to their teaching. We have grouped these skills into six main Areas of Competency that
can be incorporated into teaching, and have examined the different stages of the learning journey and how these
competencies vary across each stage.
Creativity – finding new ways of doing things, and solutions to problems
Collaboration – the ability to work well with others
Communication – speaking and presenting confidently and participating effectively in meetings
Critical thinking – evaluating what is heard or read, and linking ideas constructively
Learning to learn – developing the skills to learn more effectively
Social responsibilities – contributing to social groups, and being able to talk to and work with people from
other cultures.
Cambridge learner and teacher attributes
This course helps develop the following Cambridge learner and teacher attributes.
Cambridge learners
Cambridge teachers
Confident in teaching their subject and engaging
each student in learning.
Responsible for themselves, responsive to and
respectful of others.
Responsible for themselves, responsive to and
respectful of others.
Reflective as learners, developing their ability to
learn.
Reflective as learners themselves, developing their
practice.
Innovative and equipped for new and future
challenges.
Innovative and equipped for new and future
challenges.
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Confident in working with information and ideas –
their own and those of others.
Engaged intellectually and socially, ready to make a Engaged intellectually, professionally and socially,
difference.
ready to make a difference.
Reproduced from Developing the Cambridge learner attributes with permission from Cambridge Assessment
International Education.
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
2 Simultaneous
equations and quadratics
Topic
Order in
chapter
Solving
quadratic
equations for
real roots
First
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Scheme of work
Resources
Solve quadratic equations for real
roots by
factorising,
formula,
completing the square.
Coursebook:
Sections 2.2 and 2.5
Find the maximum or minimum
value of the quadratic function
f : x ↦ ax2 + bx + c
by completing the square.
Coursebook:
Section 2.2
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Applications
After solving
of completing quadratic
the square
equations
Learning content
PowerPoints:
2 recap b Factorising and
quadratic formula
2.2b Completing the square recap
2.2c The parabola and quadratic
function forms
PowerPoints:
2.2a Worked examples 2 & 4
Use the maximum or minimum
value of f(x) = ax2 + bx + c to sketch
the graph or determine the range
for a given domain.
Modulus
functions
Solving
quadratic
inequalities
After solving
quadratic
equations;
could be
studied later
in the course
Understand the relationship
between y = f(x) and y = |f(x)|,
where f(x) is quadratic.
After solving
quadratic
equations
Find the solution set for quadratic
inequalities.
Coursebook:
Section 2.3
PowerPoints:
2.3a Modulus of a quadratic
including Worked example 5
Coursebook:
Section 2.4
PowerPoints:
2.4 Worked examples 6 & 7
Pdf files:
Chapter 2 Teacher notes, class
discussion section 2.4
This resource is printable
and/or editable
Chapter 2 Lesson Plan
(Continued)
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Nature of
roots and
intersections
of lines and
curves
After solving
quadratic
equations
After solving
quadratics
and
inequalities
and
simultaneous
equations
Learning plan
Solve simple simultaneous
equations in two unknowns,
with one linear, by elimination or
substitution.
Coursebook:
Section 2.1
PowerPoints:
2 recap a Solving two linear
simultaneous equations
2.1a Solving two simultaneous
equations with one linear including
Worked example 1
2.1b Solving two simultaneous
equations both non linear
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Solving
simultaneous
equations
Know the conditions for
Coursebook:
ax2 + bx + c = 0 to have: (i) two real Sections 2.5 and 2.6
roots, (ii) two equal roots, (iii) no
real roots
PowerPoints:
2.3b Roots and intersections
and the related conditions for a
2.5a Worked examples 8 to 12
given line to (i) intersect a given
2.5b Connecting the nature of roots
curve, (ii) be a tangent to a
with intersections of graphs
given curve, (iii) not intersect
2.6 Simultaneous equations and
a given curve.
quadratics further practice
Success criteria
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Syllabus learning objectives / learning
intentions
Solve quadratic equations for real roots
and find the solution set for quadratic
inequalities.
Make a simple sketch of the graph of a
quadratic function using any roots and
the y-intercept.
Students can solve quadratic equations using an appropriate
method for the problem being considered. They can use this
information to make a sketch of the graph of the quadratic
function. They understand how to use this skill to find
the critical values needed to solve quadratic inequalities.
They are also able to write the solution set for quadratic
inequalities in the correct form.
Students are able to complete the square for expressions of
Find the maximum or minimum value of
the quadratic function f : x ↦ ax2 + bx + c the form ax2 + bx + c where a is positive or negative and can
by completing the square.
interpret the results correctly.
Students can apply the methods of
finding roots and completing the
square to sketching graphs and to
finding domains and ranges of
quadratic functions.
Students can apply the methods of finding roots and
completing the square to sketching graphs and to finding
domains and ranges of quadratic functions.
Understand the relationship between
y = f (x) and y = | f (x) |, where f (x)
is quadratic.
Students can successfully apply the method of finding roots
and sketch or draw accurately the graph of y = | ax2 + bx + c |.
They can also use this to solve simple problems.
Solve simple simultaneous equations in
two unknowns, with at least one linear,
by elimination or substitution.
Students choose an appropriate method of solution and
show the method of solution in full. They are able to
understand that two lines can only intersect once and how
the number of points of intersection changes when one of
the equations is not linear.
(Continued)
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Success criteria
Know the conditions for ax2 + bx + c = 0
to have:
(i) two real roots, (ii) two equal roots, (iii)
no real roots
and the related conditions for a given
line to:
(i) intersect a given curve, (ii) be a
tangent to a given curve, (iii) not intersect
a given curve.
Students understand the relevance of the discriminant and
are able to apply knowledge of the appropriate condition to
solve simple algebraic problems.
Students are able to combine all the necessary skills to solve
simultaneous equations and connect the conditions for the
nature of the roots of a quadratic equation to determine how
a line intersects with a curve.
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Syllabus learning objectives / learning
intentions
BACKGROUND KNOWLEDGE
• The following table details what knowledge it is assumed that students already have from
studying Cambridge IGCSE or O Level Mathematics. In Additional Mathematics, it is expected
that students will be able to use these skills as part of a solution in a multi-step process, and the
interpretation needed to do this should be of a greater challenge than that generally expected in
the mathematics course.
Examples
Solve simultaneous equations using the
elimination method.
Use the elimination method to solve these
simultaneous equations.
a 4x + 3y = 1; 2x – 3y = 14
b 3x + 2y = 19; x + 2y = 13
Solve simultaneous equations using the
substitution method.
Use the substitution method to solve these
simultaneous equations.
a y = 3x – 10; x + y = –2
b x + 2y = 11; 4y – x = –2
Solve quadratic equations using the
factorisation method.
Factorise and solve these equations.
a x2 + x – 6 = 0
b x2 – 10x + 16 = 0
c
6x2 + 11x – 10 = 0
Solve quadratic equations by completing
the square.
a
Write 2x2 + 7x + 3 in the form a ( x + b ) 2 + c.
b
Use your answer to part a to solve the
equation 2x2 + 7x + 3 = 0.
Solve quadratic equations using the
quadratic formula.
Solve 2x2 – 9x + 8 = 0.
Give your answers correct to 2 decimal places.
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What your students should be able to do
• The work in this chapter is essential to the whole course. The skill of solving quadratic equations
or of factorising a quadratic expression is required in several other syllabus areas. It is highly
recommended that this chapter is covered as soon as possible in the course. The skill of solving
a pair of simultaneous equations also appears in other syllabus areas. For example, the work on
the straight line, in equations, inequalities and graphs and in sequences and series. Some of the
questions in this chapter require the use of skills that are considered in Chapter 6, Straight-line
graphs. It may be sensible, therefore, to have looked at this chapter first or to work on them in
sections, together.
(Continued)
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
BACKGROUND KNOWLEDGE
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• This chapter starts with solving pairs of simultaneous linear equations by elimination and
substitution. Solving quadratic equations is briefly recapped before solving pairs of equations in
which only one of the equations is linear is studied. Quadratic expressions and functions are then
considered more fully, including the shape of the graphs, maximum and minimum values,
symmetry and modulus of the quadratic function. This is all essential to what comes after, that is,
the solution of quadratic inequalities and using the discriminant to study the nature of the roots
of quadratic equations and the points of intersections of graphs.
LANGUAGE SUPPORT
The definitions of key words and phrases are
given in the glossary.
When considering the nature of the roots of
quadratic equations it is important to model
the correct language for the possible cases.
These cases are:
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• two roots that are real and distinct
(sometimes written as real and different)
• two roots that are equal (sometimes written
as real and equal or repeated)
• two roots that are real (this includes those
that are real and distinct and real and equal)
• no real roots.
Model this language and these ideas for
students as much as possible so that the
interpretation needed to be successful is
instinctive for them.
In worked examples 3 and 4, the completing of
the square is done using the algebraic structure
which has been given and then forming and
solving equations. The language used in the
worked examples is such that this method is
fine, as students are simply required to write
down the correct form or find the values of the
constants given. However, if students need to
show that a quadratic expression has a particular
completed square form, then they should not
form and solve equations in this way. Students
should understand that using what you are
trying to show as part of your solution is invalid.
They should derive the correct completed
square form using an approach similar to
that used in the Coursebook prior to worked
example 3 or as demonstrated in PowerPoint
2.2b. This is a key and important difference in
the language used.
Links to Digital Resources
•
•
•
•
WolframAlpha has a systems of equations solver and some step-by-step example solutions.
Purplemath has examples for solving systems of non-linear equations by considering graphs.
There are many useful videos on quadratics to be found at The Khan Academy.
Maths is Fun has some real-world examples of quadratic equations
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
REAL-LIFE CONTEXT
algebra classes. To engage your students in
this topic, it may be helpful to start this section
by looking at the sort of real-world situations
that can be modelled by quadratic equations.
The properties of the parabola give us satellite
dishes and car headlamps, for example. The
Sydney Opera House is distinctly parabolic in
appearance. The motion of a pebble thrown
up in the air and falling to the ground is
also parabolic. Many features in design and
modelling require the skills that are introduced
in this syllabus. Knowing this may help students
understand the importance of what they
are studying.
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Simultaneous equations can be used to
represent and solve a variety of everyday
problems in the real world. For example,
deciding whether one mobile phone deal
is better value than another, or finding the
maximum profit available from making
and selling goods. They are used in many
applications in the study of various sciences and
are an essential tool for any student of science
or engineering, for example.
Quadratic equations are everywhere.
They are used in business and finance, physics,
architecture and the natural world, not just
Common Misconceptions and Issues
Students are expected to have developed proficient algebraic methods of solving equations and inequalities.
Graphs support the learning and help understanding, but algebra is the main key to a successful, efficient and,
most importantly, accurate solution.
How to identify
How to avoid or overcome
Students are too dependent
on their calculator to solve
equations and do not
demonstrate that they have
mastered the techniques in
the syllabus.
For example, students often
find factors by first using their
calculator to find roots and then
working back.
Students commonly reach
for their calculator to solve
quadratic equations. Calculators
are an excellent checking tool,
but no substitute for showing
proper method.
Very often, factors such as
2x − 1 are written as x − 0.5,
which is incorrect.
Some icons have been used in
the PowerPoint presentations
to try to indicate when it is a
useful time to check your
working with a calculator,
to allow you to emphasise
this with your students.
Students often need to make a
sketch of a function. Students
need to be clear that drawing
a sketch is not the same as
drawing an accurate graph.
Students who are not sure
about sketching often plot
points and join them together.
This can result in some very
poor graph shapes.
Students should be clear that
when a sketch is needed, the
command word in the question
will be sketch.
To draw a good sketch, it should
be approximately correctly
positioned. Any key points, such
as intercepts or turning points
should be marked if possible. All
key features of the curve should
be present.
Students should also be clear
that when an accurate drawing is
expected, the command word is
likely to be draw.
This is likely should the graph
then be used to solve an equation
or inequality, for example.
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Misconception/issue
(Continued)
5
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
How to identify
How to avoid or overcome
When considering whether a
quadratic equation has real
roots, students sometimes only
consider either real and equal or
real and different, but not both.
Students will form a quadratic
equation or use an incorrect
inequality sign when forming an
inequality.
This can be checked using
PowerPoint 2.6.
Make sure you model the correct
language when considering the
nature of roots and ensure that
students are experienced in using
b2 − 4ac > 0 when real roots are
specifically required.
As with chapter 1, thinking that,
for y = | f (x) |, the values of x
cannot be negative.
This is very common when
solving equations. Should a
value of x be negative, students
often think it should be rejected
and will indicate this in their
working.
This can be resolved by working
on the graphs of absolute value
functions so that students can
clearly see that x can have
negative values but that y cannot
and then linking the graphs back
to the equations they are solving.
Starter ideas
1 Alpha beta starter
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Misconception/issue
Description and purpose: This is a good starter for any lesson involving the use of factorising quadratic
expressions. It makes students think about products and sums of numbers, the need for which is
relatively clear.
Resources:
•
•
PowerPoint 2 starter: Alpha beta
Pens and paper
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Activity:
There are five questions.
Each question asks for a pair of numbers, alpha and beta, that have a given sum and product.
A timer appears on the screen and runs for 1 minute (30 seconds green and 30 seconds blue).
Students have this time to write down their answers.
As soon as they have done this, they put up their hand.
If all hands are up before the timer runs out, click to reveal the answer.
If not, the answer will appear once the time is up.
Click to move to the next question.
This activity could possibly lead into: any activity that was dependent on factorising quadratics as a tool.
This activity could be adapted: The numbers in each question can be changed if you wish to use the starter as
a review as well as a starter – or if you wish to use it again with the same group for a different lesson.
2 What’s my equation?
Description and purpose: This activity can be used to recap the work on modulus functions when f(x) is
linear, covered in Chapter 1, in readiness for extending to quadratic functions.
Resources:
•
Geogebra or Desmos or other free graphing software.
Activity:
Present the class with the graph of a modulus function and ask them for the possible equations.
To keep this as a short starter, limit the number of functions to one or two (or at the very most, three).
6
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
| |
1x .
Possible functions to draw are y = | x + 2 |, y = | 2x − 4 |, y = __
3
To type these into Geogebra or Desmos, for example, enter y = abs(x + 2).
Answers:
1x
1 x, y = − __
y = x + 2, y = 2 − x; y = 2x − 4 , y = 4 − 2x; y = __
3
3
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This activity could possibly lead into: the study of modulus functions whose initial form was quadratic or
revision of solving modulus equations.
This activity could be adapted: If technology is not available, draw the graph or graphs on a flip chart or a
display board. Also, the exercise could be extended to include simple quadratic functions such as y = x2,
which could be a challenge.
Main teaching ideas
This topic could be taught with or without a calculator.
Much of the simultaneous equation solving students meet in this course requires them to be able to solve a
quadratic equation to be able to complete the task. This is why these syllabus areas have been grouped into one
chapter. You may choose to start with this, as the Coursebook does, and build on skills your students should
already have. Alternatively, you could start with a recap of solving quadratic equations and build on that.
Either approach is well supported by the Coursebook and teacher resources.
Students may have a good understanding of the methods used to solve quadratic equations. It is a good
idea to make sure of this before progressing through the rest of the material in the chapter. Students need a
good foundation on which to build their skills. Some of these ideas will last for more than one lesson. All the
suggestions made have assessment for learning activities embedded within them.
1 Quadratic equations and the parabola
Learning intention:
Solve quadratic equations for real roots.
Make a simple sketch of the graph of a quadratic function using any roots and the y-intercept.
Find the maximum or minimum value of the quadratic function f : x ↦ ax 2 + bx + c by completing
the square.
Use the maximum or minimum value of f (x) = ax 2 + bx + c to sketch the graph or determine the range
for a given domain.
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•
•
•
•
Resources:
•
•
•
•
•
PowerPoint 2 recap b: Factorising and quadratic formula
PowerPoint 2.2a: Worked examples 2 and 4
PowerPoint 2.2b: Completing the square recap
PowerPoint 2.2c: The parabola and quadratic function forms
Coursebook Exercise 2.2
Description and purpose: Solving quadratic equations, which is an essential skill for the solution of the
simultaneous equations, has been split into two sections. In PowerPoint 2 recap b the methods of factorising
and using the quadratic formula are revised. Factorising is demonstrated using a reverse grid approach.
This method reduces the amount of purely mental processing and allows the visual to help with thinking.
Students will be able to ‘see’ the factors of the first and last terms in place in the grid. These skills should
be sufficient for the work on solving simultaneous equations, which could then be studied if you wish.
Before moving on to applications of solving quadratic equations, it is sensible for students to investigate the
possible shapes of the graphs. This, as well as the different ways of presenting a quadratic function (vertex
form, standard form, factorised form) are looked at in PowerPoint 2.2c.
7
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Differentiation:
Support:
•
•
Factorising using a reverse grid approach reduces the amount of purely mental processing and
allows the visual to help with thinking.
The square and compare method of completing the square also is visually supportive for students
who have not engaged with other methods of doing this.
Challenge:
•
•
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The method of completing the square is required as a skill in its own right, as well as being a useful method
of solving equations. It also is useful for drawing graphs and for finding least and greatest values of
functions, for example. A slightly different method to those described in the Coursebook is demonstrated
in PowerPoint 2.2b which recaps the method of solving quadratic equations by completing the square using
the first two terms of a 3-term quadratic expression only. The method used is called the square and compare
method for completing the square. As with all the alternative methods given, it is offered as a useful
alternative to support students who have not engaged with other approaches. Worked examples 2 and 4 have
been put together in one resource in PowerPoint 2.2a. This is to allow you to dip into it, or not, as you wish.
This leads neatly into Exercise 2.2 of the Coursebook.
Rearranging equations which include algebraic fractions and then solving.
Deriving equations first and then solving − in a real-world context such as business. These could be
given as investigation tasks for some students to use to self-study while other students master the
more basic skills.
Assessment for Learning: There are many opportunities for discussion using the discussion points in the
PowerPoints provided. There should also be opportunities for students to ask questions of each other and
of the teacher, whilst working. Many of these skills will be knowledge that students already have, but try
not to assume that they will all easily recall how to apply each technique. Allow students time to revise these
skills and repair any skills that have not been recalled correctly.
2 Modulus functions
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Learning intention:
•
Understand the relationship between y = f(x) and y = |f(x)|, where f(x) is quadratic.
Resources:
•
•
PowerPoint 2.3a: Modulus of a quadratic including Worked example 5
Coursebook Exercise 2.3
Description and purpose: The section on modulus functions builds on the processes studied in Chapter 1.
You may choose to look at it in Chapter 1, when the linear functions are considered, or even later in the
course when other functions may then be included. The PowerPoint available for this section works through
the examples in section 2.3. Students will need to be able to find the roots and y-intercept for a quadratic
function and then apply their knowledge of the modulus to it when drawing graphs. Sometimes, students
will need to be able to find the coordinates of the turning point to solve simple problems about points of
intersection of the graph of the modulus function with another function. This may lead into a problemsolving exercise on solving equations of the type shown in question 7 of Exercise 2.3 in the Coursebook,
where students could find the number of solutions using graphing software or by drawing accurately and
then going on to solve.
Differentiation:
Support:
•
•
•
It is vital that students understand the basic skill of solving a quadratic equation and its application
to finding the roots here.
It is also vital that students understand the shape of a parabola and how the modulus function acts
on this.
Try to use visual support, such as graphing software, whenever possible.
8
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Challenge:
•
•
Coursebook Exercise 2.3, Q7 and Q8 and similar questions should be a good challenge for students.
The Purplemath website offers an example similar to those in Exercise 2.3 Q7 and also gives an
interesting example of nested absolute value functions.
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Assessment for Learning: Some opportunities for discussion arise through the use of the PowerPoint.
Students may also be encouraged to peer mark and assess their answers to Exercise 2.3. Natural
opportunities for Q&A sessions should arise when this exercise is being carried out. Try to ask questions
that allow students to show you that they have understood the mathematics they are studying. You could
review their knowledge using a piece of work that they had to mark and grade. Can they find errors? Can
they discuss the impact of those errors?
3 Quadratic inequalities
Learning intention:
•
Find the solution set for quadratic inequalities.
Resources:
•
•
•
•
PowerPoint 2.4: Worked examples 6 and 7
Chapter 2 Teacher notes: class discussion section 2.4
Chapter 2 Lesson plan: Solving inequalities
Coursebook Exercise 2.4
Description and purpose: A demonstration lesson plan has been given for a possible lesson covering
quadratic inequalities. The lesson incorporates the recap and class discussion in section 2.4. The Chapter
2 teacher notes give some support for managing the discussion direction, if it is needed. Some ideas about
students developing their own explanations are also given. There are links in the document to a video that
may be useful to challenge students and a Wolfram inequality checker tool. The lesson leads into Exercise
2.4 of the Coursebook.
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Differentiation:
Support: The focus should be on the algebraic process here but if students need visual support, access to
graphing software may be supportive for some.
Challenge: The Khan Academy has a video on rearranging inequalities with algebraic fractions and
then solving.
Assessment for Learning: Assessment for learning opportunities should arise naturally through
observation, peer checking of answers, Q&A sessions and whole class discussion as well as the discussion
points which arise in the PowerPoint of worked examples.
4 Simultaneous equations
Learning intention:
•
Solve simple simultaneous equations in two unknowns, with at least one linear, by elimination
or substitution.
Resources:
•
•
•
•
PowerPoint 2 recap a: Solving two linear simultaneous equations
PowerPoint 2.1a: Solving two simultaneous equations with one linear including worked example 1
PowerPoint 2.1b: Solving two simultaneous equations both non-linear
Coursebook Exercise 2.1
9
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Differentiation:
Support:
•
•
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Description and purpose: It is appropriate for the work in this chapter that students know how to find the
points of intersection of a line and a curve, as these will be related to conditions for the nature of the roots
of quadratic equations. The methodologies used to solve simultaneous equations are standard. These
methods are tried and tested and students usually understand them well and use them with proficiency.
However, we all forget things from time to time, and so the Coursebook offers plenty of revision of the key
concepts. PowerPoint 2 recap a animates the recap of solving a pair of linear simultaneous equations at the
start of the chapter in the Coursebook. PowerPoint 2.1a starts with the example in the Coursebook at the
start of section 2.1 and then works through worked example 1. These techniques are sufficient to be able
to solve simultaneous equations where at least one equation is linear and so lead into Exercise 2.1 of the
Coursebook.
Students need to be familiar with solving simple simultaneous equations where neither equation is linear.
PowerPoint 2.1b has some examples of these and includes an example where students are asked to solve a
quartic equation that is quadratic in x 2. This can be used as an introduction for Chapter 5 section 5.5 or
simply as a forerunner to the ideas which are explored more fully in Chapter 5.
Some students try to use the elimination method, when the method of substitution is far simpler
when solving simultaneous equations. If this is the case, try to encourage your students to stick
to one method of solution. Make sure that they know that, at this level, substitution is very much
more useful as a method as it is more universal.
Try to ensure that students use their calculator as a checking tool. If they have made an error,
work with them through their solution to help them find it and correct it successfully.
Challenge: Some more challenging material is also provided in the exercises, to allow some students to
develop their skills; for example, Exercise 2.1 Q19 to Q26.
SA
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Assessment for Learning: As well as the discussion opportunities which will naturally arise through the use
of the resource materials and when your students are working through questions, assessment for learning
can be carried out using a lesson review by students. At the end of the lesson, ask them what they have
learned. Write their responses down for them to refer to in future work. This could then be used as part of a
revision session later in the course.
5 Nature of roots
Learning intention:
•
Know the conditions for ax 2 + bx + c = 0 to have: (i) two real roots, (ii) two equal roots, (iii) no real
roots, and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given
curve, (iii) not intersect a given curve.
Resources:
•
•
•
•
•
PowerPoint 2.3b: Roots and intersections
PowerPoint 2.5a: Worked examples 8 to 12
PowerPoint 2.5b: Connecting the nature of roots with intersections of graphs
Coursebook Exercise 2.5
Coursebook Exercise 2.6
Description and purpose: You may choose to link the final sections on the nature of roots and the
intersections of lines and curves. The resources and the Coursebook enable you to choose to separate or
combine them as you prefer. PowerPoint 2.5b looks at six specific curves, their graphs and hence their roots
and then identifies the discriminant in each case. Students are asked all through to explain what is happening
on the basis of what they can see in the formula. The results are summarised. In the final three slides, more
consideration of the connection between roots and intersections is given. Worked examples 8 to 12 have
been combined in one resource, PowerPoint 2.5a. Again, you may wish to use part of it and come back to it
at another point. It is not suggested that you use it all in one session. Exercise 2.5 of the Coursebook may be
worked through after worked example 9, and Exercise 2.6 follows once the PowerPoint is complete, or both
exercises may be looked at after the full set of worked examples in the PowerPoint has been considered.
10
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Differentiation:
Support: In the problem after worked example 5, some students may struggle to see why the number
of intersections of the line and curve is the same as the number of solutions of the given equation.
PowerPoint 2.3b looks at this very point. A statement is made regarding the roots of an equation and
the points of intersection of two lines as being the same and the question ‘Why?’ is asked.
This could be used as a class investigation. Many students will find it intuitively obvious, but some will
not. Those to whom it is obvious may not find it easy to put their case in a watertight argument. There
is more work on this point later with non-horizontal lines.
Review activities
1 Order review
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Challenge: Worked example 9 is solved using the discriminant. This is the most straightforward
approach. There is an alternative calculus method which some students may prefer. You may wish to set
this as an investigation for your students once you have covered simple differentiation.
Assessment for Learning: As part of assessment for learning for this topic, be careful to check that students
understand the language used and are able to devise a method of solution based on that language. You can
check this by marking written work or through general discussion, for example.
Description and purpose: This review requires students to order the six steps needed to rearrange and
solve a quadratic inequality. They then have to decide whether the solution that has been given is correct.
The purpose of this review is to consolidate the logical steps practised in the lesson and also to remind
students of the importance of sketching the graphs!
Resources:
•
PowerPoint 2 review: Quadratics order order
Activity:
Students are given this scenario:
SA
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Disaster! Neena’s pet bookworm has shredded her homework!
She cannot tell which part of her answer comes first…. help Neena by sorting her work into the correct order!
Now tell her if she was correct!
Allow students to discuss and write down what they think the correct order is.
They can use the A, B, C…. marked alongside.
Answer:
The correct order on the PowerPoint is CEFBDA.
When everyone is ready, click to move the statements to the correct place. Click again to move to a second
slide where the sketch graph of the quadratic is revealed for students to judge the solution.
This activity could possibly lead from: a lesson focused on solving quadratic inequalities.
This activity could be adapted: The six statements that need to be ordered could be written on large pieces
of paper in exactly the same way as in the PowerPoint and pinned onto a board. Labelling them A, B, C
etc. will give students an easy way to describe which order is the correct order. As another alternative, the
statements could be printed or written in regular sized font/print and the class split into pairs or groups and
given a set each which they can then move around to form the correct order.
2 Check my graph review
Description and purpose: This task is designed to make students think about the shape of the graphs they
draw and how accuracy and attention to detail can improve a solution. It can be used as an assessment
for learning exercise. It should be a useful tool in assessing whether they have fully understood what they
have learned.
11
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
Resources:
•
•
Check my graph
Check my graph teacher notes
Activity:
The Check my graph file can either be printed as a handout or displayed on a flip chart or
interactive whiteboard.
It has two graphs for your students to check for accuracy and award marks.
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They should identify and be able to explain how to correct any errors.
They then have to find two good comments about the work and set one target for improvement.
Suggested comments and marks are made in the Check my graph teacher notes file. Your students may think
of others!
This activity could possibly lead from: a lesson on sketching the graphs of the modulus of a quadratic
function.
This activity could be adapted: The graphs could be updated or added to, and this could also be used with
other types of modulus function.
Homework ideas
1 Coursebook: Quadratic equations and the parabola, Exercise 2.2
Completion of this exercise should give students a good amount of practice of finding roots, sketching
graphs and completing the square.
2 PowerPoint 2.6
SA
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Completion of the questions in this PowerPoint should measure the ability of students when solving
problems involving the nature of the roots of a quadratic equation. It will also check whether they can
solve quadratic inequalities successfully and whether they can make the connections between points of
intersection and roots. This PowerPoint has two versions. The first version has no model answers included,
but does have some hints in the teacher notes for each slide. These can be removed if you do not want to
give any hints at all. The second version includes animated answers and is very supportive of those who need
greater modelling of what is needed. Again, hints, and also details of what each animation will reveal, are
included in the Teacher notes for each slide. As well as a useful homework tool, this PowerPoint can be used
as a revision exercise, for self-study, further practice in class or as part of a bank of resources students can
access at any point throughout the course when needed. This practice material is also available as a PDF file
in case technology is not available to your students.
12
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
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Curriculum framework
correlation grid
These learning objectives are reproduced from the Cambridge IGCSE and O Level syllabuses Additional Mathematics (0606/4037) for examination from 2025.
This Cambridge International copyright material is reproduced under licence and remains the intellectual property of Cambridge Assessment International Education.
The following table shows how the learning objectives map to the Cambridge IGCSE Additional Mathematics Coursebook, Workbook, Worked Solutions Manual and
Teacher’s Resource.
Chapter
1
1.
Functions
✓
1.2 find the domain and range of functions
✓
1.3 recognise and use function notations
✓
1.4 understand the relationship between y = f(x) and y = |f(x)|,
where f(x) may be linear, quadratic, cubic or trigonometric
✓
3
SA
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1.1 understand the terms: function, domain, range (image set),
one – one function, many – one function, inverse function, and
composition of functions
2
1.5 explain in words why a given function does not have an inverse
✓
1.6 find the inverse of a one – one function
✓
1.7 form and use composite functions
✓
1.8 use sketch graphs to show the relationship between a function
and its inverse
✓
4
5
6
7
8
9
10
11
12
13
14
15
16
✓
1
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
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Chapter
1
2.
Quadratic functions
2
2.1 find the maximum or minimum value of the quadratic function
f : x ↦ ax2 + bx + c by completing the square or by differentiation
✓
2.2 use the maximum or minimum value of f(x) to sketch the graph
of y =f(x) or determine the range for a given domain
✓
2.3 know the conditions for f(x) = 0 to have:
(i) two real roots
(ii) two equal roots
(iii) no real roots
and the related conditions for a given line to:
(i) intersect a given curve
(ii) be a tangent to a given curve
(iii) not intersect a given curve
✓
2.5 find the solution set for quadratic inequalities either graphically
or algebraically
3.
Factors of polynomials
5
6
7
8
9
10
11
12
13
14
15
16
✓
✓
✓
3.2 find factors of polynomials
✓
SA
3.1 know and use the remainder and factor theorems
3.3 solve cubic equations
4
M
2.4 solve quadratic equations for real roots
3
✓
2
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PL
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Chapter
1
4.
Equations, inequalities and graphs
4.1 solve equations of the type
• |ax + b| = c (c > 0)
• |ax + b| = cx + d
• |ax + b| = |cx + d|
• |ax 2 + bx + c| = d
2
3
5
6
7
8
9
10
11
12
13
14
15
16
✓
Using algebraic or graphical methods
M
4.2 solve graphically or algebraically inequalities of the type
• k|ax + b| . c (c > 0)
• k|ax + b| < c (c > 0)
• k|ax + b| < |cx + d|
where k . 0
• |ax + b| < cx + d
• |ax 2 + bx + c| . d
• |ax 2 + bx + c| < d
4
✓
4.3 use substitution to form and solve a quadratic equation in
order to solve a related equation
✓
4.4 sketch the graphs of cubic polynomials and their moduli, when
given as a product of three linear factors
✓
SA
4.5 solve graphically cubic inequalities of the form
• f(x) > d
• f(x) . d
• f(x) < d
• f(x) , d
✓
where f(x) is a product of three linear factors and d is a constant
3
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PL
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Chapter
1
5.
Simultaneous equations
5.1 solve simultaneous equations in two unknowns by elimination
or substitution
6.
Logarithmic and exponential functions
2
3
4
5
✓
6.2 know and use the laws of logarithms, including change of base
of logarithms
✓
6.3 solve equations of the form ax = b
✓
Straight line graphs
7.1 use the equation of a straight line
8
9
10
11
12
13
14
15
16
✓
✓
7.3 solve problems involving mid-point and length of a line,
including finding and using the equation of a perpendicular bisector
✓
7.4 transform given relationships to and from straight line form,
including determining unknown constants by calculating the
gradient or intercept of the transformed graph
✓
M
7.2 know and use the condition for two lines to be parallel or
perpendicular
Coordinate geometry of the circle+
SA
8.
7
✓
6.1 know and use simple properties and graphs of the logarithmic
and exponential functions, including lnx and ex
7.
6
8.1 know and use the equation of a circle with radius r and centre (h, k)
✓
8.2 solve problems involving the intersection of a circle and a
straight line
✓
8.3 solve problems involving tangents to a circle
✓
8.4 solve problems involving the intersection of two circles
✓
4
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CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
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Chapter
1
9.
Circular measure
2
3
9.1 solve problems involving the arc length and sector area of a
circle, including knowledge and use of radian measure
10. Trigonometry
4
5
6
7
8
9
✓
10.2 understand and use the amplitude and period of a trigonometric
function, including the relationship between graphs of related
trigonometric functions
✓
10.3 draw and use the graphs of
y = a sin bx + c
y = a cos bx + c
y = a tan bx + c
where a is a positive integer, b is a simple fraction or integer, and c
is an integer
✓
10.4 use the relationships:
• sin2 A + cos2 A = 1
• sec2 A = 1 + tan2 A
• cosec2 A = 1 + cot2 A
✓
M
11
12
13
14
15
16
✓
10.1 know and use the six trigonometric functions of angles of any
magnitude
10.5 solve, for a given domain, trigonometric equations involving
the six trigonometric functions
✓
10.6 prove trigonometric relationships involving the six
trigonometric functions
✓
SA
10
5
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We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PL
E
Chapter
1
11. Permutations and combinations
2
3
4
5
6
7
8
9
10
11.1 recognise the difference between permutations and
combinations and know when each should be used’
✓
11.2 know and use the notation n! and the expressions for
permutations and combinations of n items taken r at a time
✓
11.3 answer problems on arrangement and selection using
permutations or combinations
✓
12. Series
12.1 use the binomial theorem for expansion of (a + b)n for positive
integer n
12.2 use the general term (nr)an − rbr, 0 < r < n
11
13
14
15
16
✓
✓
12.3 recognise arithmetic and geometric progressions and understand
the difference between them
✓
12.4 use the formulae for the nth term and for the sum of the first n
terms to solve problems involving arithmetic or geometric progressions
✓
12.5 use the condition for the convergence of a geometric
progression, and the formula for the sum to infinity of a convergent
geometric progression
✓
M
12
13. Vectors in two dimensions
✓
13.2 know and use position vectors and unit vectors
✓
13.3 find the magnitude of a vector; add and subtract vectors and
multiply vectors by scalars
✓
13.4 compose and resolve velocities
✓
SA
13.1 Understand and use vector notation
6
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PL
E
Chapter
1
14. Calculus
14.1 understand the idea of a derived function
14.2 use the notations
dy d2y
d dy
f′(x), f″(x), ___, ____2 = [___(___)]
dx dx
dx dx
dy
dx, δx → 0, ___
dx
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
✓
✓
14.3 know and use the derivatives of the standard functions xn (for
any rational n), sin x, cos x, tan x, ex, ln x.
14.4 differentiate products and quotients of functions
✓
✓
14.5 Use differentiation to find gradients, tangents and normals
✓
14.6 Use differentiation to find stationary points
✓
14.7 apply differentiation to connected rates of change, small
increments and approximations
✓
14.8 apply differentiation to practical problems involving maxima
and minima
✓
14.9 use the first and second derivative tests to discriminate
between maxima and minima
✓
M
✓
✓
1
14.11 integrate sums of terms in powers of x including __
x
✓
SA
14.10 understand integration as the reverse process of
differentiation
7
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.
We are working with Cambridge Assessment International Education towards endorsement of this title.
CAMBRIDGE IGCSE™ ADDITIONAL MATHEMATICS: TEACHER’S RESOURCE
PL
E
Chapter
1
14.12 integrate functions of the form:
• (ax + b)n for any rational n
• sin (ax + b)
• cos (ax + b)
• sec2 (ax + b)
• eax + b.
2
3
14.13 evaluate definite integrals and apply integration to the
evaluation of plane areas
4
5
6
7
8
9
10
11
12
13
14
15
16
✓
✓
✓
14.15 make use of the relationships in 14.14 to draw and use the
following graphs:
• displacement–time
• distance–time
• velocity–time
• speed–time
• acceleration–time.
✓
SA
M
14.14 apply differentiation and integration to kinematics problems
that involve displacement, velocity and acceleration of a particle
moving in a straight line with variable or constant acceleration
8
Original material © Cambridge University Press & Assessment 2023. This material is not final and is subject to further changes prior to publication.