Table Of ContentMathematics
Higher Level
for the IB Diploma
Paul Fannon, Vesna Kadelburg,
Ben Woolley and Stephen Ward
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cambridge university press
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Cambridge University Press
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Information on this title: www.cambridge.org/9781107661738
© Cambridge University Press 2012
Th is publication is in copyright. Subject to statutory exception
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no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2012
Printed and bound in Great Britain by the MPG Books Group
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ISBN 978-1-107-66173-8 Paperback with CD-ROM for Windows® and Mac®
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purchasing institution.
Not for printing, sharing or distribution.
Contents
iii
Contents
Introduction
vii
Algebra
(Topic 1: Algebra)
1 Counting principles
1
1A The product principle and the addition
principle
1
1B Counting arrangements
6
1C Algebra of factorials
9
1D Counting selections
11
1E Exclusion principle
15
1F Counting ordered selections
18
1G Keeping objects together or separated
21
Algebra, functions and equations
(Topics 1: Algebra & 2: Functions & equations)
2 Exponents and logarithms
28
2A Laws of exponents
28
2B Exponential functions
33
2C The value e
39
2D Introducing logarithms
40
2E Laws of logarithms
44
2F Graphs of logarithms
49
2G Solving exponential equations
50
3 Polynomials
58
3A Working with polynomials
58
3B Remainder and factor theorems
63
3C Sketching polynomial functions
68
3D The quadratic formula and the discriminant
76
4 Algebraic structures
87
4A Solving equations by factorising
87
4B Solving equations by substitution
89
4C Features of graphs
91
4D Using a graphical calculator to solve equations 94
4E Solving simultaneous equations
by substitution
97
4F Systems of linear equations
101
4G Solving inequalities
110
4H Working with identities
113
5 The theory of functions
118
5A Relations, functions and graphs
118
5B Function notation
123
5C Domain and range
125
5D Composite functions
131
5E Inverse functions
133
5F Rational functions
141
6 Transformations of graphs
154
6A Translations
155
6B Stretches
157
6C Refl ections
161
6D Modulus transformations
163
6E Consecutive transformations
167
6F Reciprocal transformations
174
6G Symmetries of graphs and functions
178
7 Sequences and series
190
7A General sequences
190
7B General series and sigma notation
193
7C Arithmetic sequences
195
7D Arithmetic series
199
7E Geometric sequences
201
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iv
Contents
7F Geometric series
205
7G Infi nite geometric series
207
7H Mixed questions
211
8 Binomial expansion
217
8A Introducing the binomial theorem
217
8B Applying the binomial theorem
219
8C Products of binomial expansions
223
8D Binomial expansions as approximations
226
Geometry
(Topics 3: Circular functions & trigonometry & 4: Vectors)
9 Circular measure and
trigonometric functions
232
9A Radian measure
232
9B Defi nitions and graphs of sine and cosine
functions
239
9C Defi nition and graph of the tangent
function
248
9D Exact values of trigonometric functions
251
9E Transformations of trigonometric graphs
254
9F Modelling using trigonometric functions
261
9G Inverse trigonometric functions
264
10 Trigonometric equations
and identities
273
10A Introducing trigonometric equations
273
10B Harder trigonometric equations
281
10C Trigonometric identities
291
10D Using identities to solve equations
297
11 Geometry of triangles
and circles
304
11A Right-angled triangles
305
11B The sine rule
307
11C The cosine rule
312
11D Area of a triangle
318
11E Trigonometry in three dimensions
321
11F Length of an arc
326
11G Area of a sector
330
11H Triangles and circles
333
12 Further trigonometry
346
12A Double angle identities
347
12B Compound angle identities
354
12C Functions of the form a sin x + b cos x
359
12D Reciprocal trigonometric functions
364
13 Vectors
375
13A Positions and displacements
375
13B Vector algebra
384
13C Distances
389
13D Angles
393
13E Properties of the scalar product
397
13F Areas
402
13G Properties of the vector product
404
14 Lines and planes in space
413
14A Vector equation of a line
413
14B Solving problems with lines
420
14C Other forms of equation of a line
432
14D Equation of a plane
439
14E Angles and intersections between lines
and planes
449
14F Intersection of three planes
457
14G Strategies for solving problems with lines
and planes
463
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Contents
v
Calculus
(Topic 6: Calculus)
Algebra
(Topic 1: Algebra)
16 Basic differentiation and
its applications
527
16A Sketching derivatives
527
16B Differentiation from fi rst principles
535
16C Rules of differentiation
538
16D Interpreting derivatives and second
derivatives
541
16E Trigonometric functions
547
16F The exponential and natural logarithm
functions
549
16G Tangents and normals
550
16H Stationary points
554
16I General points of infl exion
560
16J Optimisation
562
17 Basic integration and
its applications
569
17A Reversing differentiation
569
17B Constant of integration
571
17C Rules of integration
572
17D Integrating x–1 and ex
575
17E Integrating trigonometric functions
576
17F Finding the equation of a curve
577
17G Defi nite integration
579
17H Geometrical signifi cance of defi nite
integration
581
17I The area between a curve and the y-axis
588
17J The area between two curves
591
18 Further differentiation methods 599
18A Differentiating composite functions
using the chain rule
599
18B Differentiating products using the
product rule
604
18C Differentiating quotients using the
quotient rule
607
18D Implicit differentiation
611
18E Differentiating inverse trigonometric
functions
616
19 Further integration methods
622
19A Reversing standard derivatives
622
19B Integration by substitution
626
19C Using trigonometric identities in
integration
634
19D Integration using inverse trigonometric
functions
641
19E Other strategies for integrating
quotients
644
19F Integration by parts
649
20 Further applications of
calculus
659
20A Related rates of change
659
20B Kinematics
663
20C Volumes of revolution
668
20D Optimisation with constraints
674
15 Complex numbers
476
15A Defi nition and basic arithmetic of i
476
15B Geometric interpretation
482
15C Properties of complex conjugates
487
15D Complex solutions to polynomial
equations
493
15E Sums and products of roots of polynomials
498
15F Operations in polar form
504
15G Complex exponents
510
15H Roots of complex numbers
512
15I Using complex numbers to derive
trigonometric identities
517
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vi
Contents
Probability and statistics
(Topic 5: Statistics & probability)
Algebra
(Topic 1: Algebra)
21 Summarising data
689
21A Some important concepts in statistics
689
21B Measures of spread
691
21C Frequency tables and grouped data
695
22 Probability
705
22A Introduction to probability
705
22B Combined events and Venn diagrams
711
22C Tree diagrams and fi nding the intersection
715
22D Independent events
720
22E Counting principles in probability
722
22F Conditional probability
724
22G Further Venn diagrams
727
22H Bayes’ theorem
732
23 Discrete probability distributions 743
23A Random variables
743
23B Expectation, median and variance of a discrete
random variable
747
23C The binomial distribution
751
23D The Poisson distribution
758
24 Continuous distributions
769
24A Continuous random variables
769
24B Expectation and variance of continuous random
variables
773
24C The normal distribution
777
24D Inverse normal distribution
783
25 Mathematical induction
791
25A The principle of mathematical induction
791
25B Induction and series
793
25C Induction and sequences
795
25D Induction and differentiation
799
25E Induction and divisibility
801
25F Induction and inequalities
803
26 Questions crossing all chapters
809
Examination tips
820
Answers
821
Index
911
Acknowledgements
917
Terms and conditions of use for the CD-ROM
918
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Introduction
vii
Structure of the book
Th e book is split roughly into four blocks: chapters 2 to 8 cover algebra and functions; chapters
9 to 14 cover geometry; chapters 16 to 20 cover calculus; and chapters 21 to 24 cover probability
and statistics. Chapters 1, 15 and 25; on counting principles, complex numbers and induction
(respectively), bring together several areas of the course and chapter 26 includes questions that mix
diff erent parts of the course – a favourite trick in International Baccalaureate® (IB) examinations.
You do not have to work through the book in the order presented, but (given how much the
International Baccalaureate® likes to mix up topics) you will fi nd that several questions refer to
material in previous chapters. In these cases a ‘rewind panel’ will tell you that the material has been
covered previously so that you can either remind yourself or decide to move on.
We have tried to include in the book only the material that will be examinable. Th ere are many
proofs and ideas which are useful and interesting, and these are on the CD-ROM if you would like
to explore them.
Each chapter starts with a list of learning objectives to give you an idea about what the chapter
contains. Th ere is also an introductory problem that illustrates what you will be able to do aft er you
have completed the chapter. Some introductory problems relate to ‘real life’ situations, while others
are purely mathematical. You should not expect to be able to solve the problem, but you may want
to think about possible strategies and what sort of new facts and methods would help you. Th e
solution to the introductory problem is provided at the end of the chapter, aft er the summary of
the chapter contents.
Key point boxes
Th e most important ideas and formulae are emphasised in the ‘KEY POINT’ boxes. When the
formulae are given in the Formula booklet, there will be an icon:
; if this icon is not present, then
the formulae are not in the Formula booklet and you may need to learn them or at least know how
to derive them.
Worked examples
Each worked example is split into two columns. On the right is what you should write down.
Sometimes the example might include more detail then you strictly need, but it is designed to give
you an idea of what is required to score full method marks in examinations. However, mathematics
is about much more than examinations and remembering methods. So, on the left of the worked
examples are notes that describe the thought processes and suggest which route you should use to
tackle the question. We hope that these will help you with any exercise questions that diff er from
the worked examples. It is very deliberate that some of the questions require you to do more than
repeat the methods in the worked examples. Mathematics is about thinking!
Introduction
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viii
Introduction
Signposts
Th ere are several boxes that appear throughout the book.
Theory of knowledge issues
Every lesson is a Th eory of knowledge lesson, but sometimes the links may not
be obvious. Mathematics is frequently used as an example of certainty and truth,
but this is oft en not the case. In these boxes we will try to highlight some of the
weaknesses and ambiguities in mathematics as well as showing how mathematics
links to other areas of knowledge.
From another perspective
Th e International Baccalaureate® encourages looking at things in diff erent ways.
As well as highlighting some international diff erences between mathematicians
these boxes also look at other perspectives on the mathematics we are covering:
historical, pragmatic and cultural.
Research explorer
As part of your course, you will be asked to write a report on a mathematical
topic of your choice. It is sometimes diffi cult to know which topics are suitable as
a basis for such reports, and so we have tried to show where a topic can act as a
jumping-off point for further work. Th is can also give you ideas for an Extended
essay. Th ere is a lot of great mathematics out there!
Exam hint
Although we would encourage you to think of mathematics as more than just
learning in order to pass an examination, there are some common errors it is
useful for you to be aware of. If there is a common pitfall we will try to highlight
it in these boxes. We also point out where graphical calculators can be used
eff ectively to simplify a question or speed up your work, oft en referring to the
relevant calculator skills sheet on the CD-ROM.
Fast forward / rewind
Mathematics is all about making links. You might be interested to see how
something you have just learned will be used elsewhere in the course, or you may
need to go back and remind yourself of a previous topic. Th ese boxes indicate
connections with other sections of the book to help you fi nd your way around.
How to use the questions
The colour-coding
Th e questions are colour-coded to distinguish between the levels.
Black questions are drill questions. Th ey help you practise the methods described
in the book, but they are usually not structured like the questions in the
examination. Th is does not mean they are easy, some of them are quite tough.
exam hint
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Introduction
ix
Each diff erently numbered drill question tests a diff erent skill. Lettered subparts of a question
are of increasing diffi culty. Within each lettered part there may be multiple roman-numeral parts
((i), (ii),...) , all of which are of a similar diffi culty. Unless you want to do lots of practice we would
recommend that you only do one roman-numeral part and then check your answer. If you have
made a mistake then you may want to think about what went wrong before you try any more.
Otherwise move on to the next lettered part.
Green questions are examination-style questions which should be accessible to students on
the path to getting a grade 3 or 4.
Blue questions are harder examination-style questions. If you are aiming for a grade 5 or 6
you should be able to make signifi cant progress through most of these.
Red questions are at the very top end of diffi culty in the examinations. If you can do these
then you are likely to be on course for a grade 7.
Gold questions are a type that are not set in the examination, but are designed to provoke
thinking and discussion in order to help you to a better understanding of a particular
concept.
At the end of each chapter you will see longer questions typical of the second section of
International Baccalaureate® examinations. Th ese follow the same colour-coding scheme.
Of course, these are just guidelines. If you are aiming for a grade 6, do not be surprised if you fi nd
a green question you cannot do. People are never equally good at all areas of the syllabus. Equally,
if you can do all the red questions that does not guarantee you will get a grade 7; aft er all, in the
examination you have to deal with time pressure and examination stress!
Th ese questions are graded relative to our experience of the fi nal examination, so when you fi rst
start the course you will fi nd all the questions relatively hard, but by the end of the course they
should seem more straightforward. Do not get intimidated!
Calculator versus non-calculator questions
In the fi nal examination there will be one paper in which calculators are not allowed. Some
questions specifi cally need a calculator but most could appear in either the calculator or the non-
calculator paper.
Some questions are particularly common in the non-calculator paper and you must be able
to know how to deal with these. Th ey are highlighted by the non-calculator symbol.
Some questions can be done in a clever way using a calculator or cannot be realistically
done without using a calculator. Th ese questions are marked with a calculator symbol.
In the fi nal examination you will not get this indication, so you must make sure you learn which
types of questions have an easy calculator method. Th e calculator skills sheets on the CD-ROM can
help with this.
For the questions that do not have either calculator icon, you should mix up practising with and
without a calculator. Be careful not to become too reliant on your calculator. Half of the core
examination is done without one!
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x
Introduction
On the CD-ROM
On the CD-ROM there are various materials that you might fi nd useful.
Prior learning
Th e International Baccalaureate® syllabus lists what candidates should know
before taking the examination. Th e topics on the list are not all explicitly covered
in the syllabus, but their knowledge may be required to answer examination
questions. Don’t worry, you do not need to know all this before starting the
course, and we have indicated in the rewind boxes where a particular concept or
skill is required. On the CD-ROM you will fi nd a self-assessment test to check
your knowledge, and some worksheets to help you learn any skills that you might
be missing or need to revise.
Option chapters
Each of the four options is covered by several chapters on the CD-ROM.
Coursebook support
Supporting worksheets include:
• calculator skills sheets that give instructions for making optimal use of some
of the recomended graphical calculators
• extension worksheets that go in diffi culty beyond what is required at
International Baccalaureate®
• fi ll-in proof sheets to allow you to re-create proofs that are not required in the
examination
• self-discovery sheets to encourage you to investigate new results for yourself
• supplementary sheets exploring some applications, international and
historical perspectives of the mathematics covered in the syllabus.
e-version
A fl at pdf of the whole coursebook (for days when you don’t want to carry the
paperback!).
We hope you fi nd Higher Level Mathematics for the IB diploma an interesting
and enriching course. You might also fi nd it quite challenging, but do not get
intimidated, frequently topics only make sense aft er lots of revision and practice.
Persevere and you will succeed.
Th e author team.
© Cambridge University Press 2012
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1 Counting principles
1
1
Counting is one of the fi rst things we learn in mathematics
and at fi rst it seems very simple. If you were asked to count
how many people there are in your school, this would not be
too tricky. If you were asked how many chess matches would
need to be played if everyone were to play everyone else, this
would be a little more complicated. If you were asked how
many diff erent football teams could be chosen, you might
fi nd that the numbers become far too large to count without
coming up with some clever tricks. Th is chapter aims to help
develop strategies for counting in such diffi cult situations.
1A The product principle and the
addition principle
Counting very small groups is easy. So, we need to break down
more complicated problems into counting small groups. But
how do we then combine these together to come up with an
answer to the overall problem? Th e answer lies in using the
product principle and the addition principle, which can be
illustrated using the following menu.
Counting sometimes
gets extremely diffi cult.
Are there more whole
numbers or odd numbers;
fractions or decimals? Have
a look at the work of Georg
Cantor, and the result may
surprise you!
Counting
principles
Introductory problem
If a computer can print a line containing all 26 letters
of the alphabet in 0.01 seconds, estimate how long it
would take to print all possible permutations of
the alphabet.
In this chapter you
will learn:
how to break down
•
complicated questions
into parts that are
easier to count, and
then combine them
together
how to count the
•
number of ways to
arrange a set of objects
the algebraic
•
properties of a useful
new tool called the
factorial function
in how many ways you
•
can choose objects
from a group
strategies for applying
•
these tools to harder
problems.
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2
Topic 1: Algebra
Mains
Pizza
Hamburger
Paella
Desserts
Ice-cream
Fruit salad
ad
MENU
mburger
aella
essert
ger
a
erts
rger
la
erts
Pizza
Anna would like to order a main course and a dessert. She can
choose one of three main courses and one of two desserts. How
many diff erent choices could she make? Bob would like to order
either a main course or a dessert. He can choose one of the three
main courses or one of the two desserts; how many diff erent
orders can he make?
We can use the notation n(A) to represent the number of ways
of making a choice about A.
Th e product principle tells us that when we want to select one
option from A and one option from B we multiply the individual
possibilities together.
KEY POINT 1.1
KEY POINT 1.1
Th e Product principle (AND rule)
Th e number of ways of in which both choice A and choice
B can be made is the product of the number of options for
A and the number of options for B.
n(A AND B) = n(A) × n(B)
Th e addition principle tells us that when we wish to select
one option from A or one option from B we add the individual
possibilities together.
Th e addition principle has one essential restriction. You can
only use it if there is no overlap between the choices for A and
the choices for B. For example, you cannot apply the addition
principle to counting the number of ways of getting an odd
The word analysis
literally means
‘breaking up’.
When a problem is
analysed it is broken down
into simpler parts. One of
the purposes of studying
mathematics is to develop an
analytical mind, which is
considered very useful in
many different disciplines.
n(A) means the size
of the set of options
of A. See Prior
Learning Section G
on the CD-ROM.
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1 Counting principles
3
number or a prime number on a die. If there is no overlap between
the choices for A and for B, the two events are mutually exclusive.
KEY POINT 1.2
KEY POINT 1.2
Th e Addition Principle (OR rule)
Th e number of ways of in which either choice A or choice
B can be made is the sum of the number of options for A
and the number of options for B.
If A and B are mutually exclusive then
n(A OR B) = n(A) + n(B)
Th e hardest part of applying either the addition or product
principle is breaking the problem down and deciding which
principle to use. You must make sure that you have included all
of the cases and checked that they are mutually exclusive. It is
oft en useful to rewrite questions to emphasise what is required,
‘AND’ or ‘OR’.
Worked example 1.1
An examination has ten questions in section A and four questions in section B. How many
diff erent ways are there to choose questions if you must:
(a) choose one question from each section?
(b) choose a question from either section A or section B?
Describe the problem accurately
(a) Choose one question from A
(10 ways)
AND
one from B
(4 ways)
‘AND’ means we should apply the
product principle: n (A) × n (B)
Number of ways = 10 × 4
= 40
Describe the problem accurately
(b) Choose one question from A
(10 ways)
OR
one from B
(4 ways)
‘OR’ means we should apply the
addition principle: n (A) + n (B)
Number of ways = 10 + 4
= 14
In the example above we cannot answer a question twice so there
are no repeated objects; however, this is not always the case.
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4
Topic 1: Algebra
Th is leads us to a general idea.
KEY POINT 1.3
KEY POINT 1.3
Th e number of ways of selecting something r times from n
objects is nr.
Exercise 1A
1. If there are 10 ways of doing A, 3 ways of doing B and 19 ways of
doing C, how many ways are there of doing
(a) (i) both A and B?
(ii) both B and C?
(b) (i) either A or B?
(ii) either A or C?
2. If there are 4 ways of doing A, 7 ways of doing B and 5 ways of
doing C, how many ways are there of doing
(a) all of A, B and C?
(b) exactly one of A, B or C?
3. How many diff erent paths are there
(a) from A to C?
(b) from C to E?
(c) from A to E?
A
B
C
D
E
Worked example 1.2
In a class there are awards for best mathematician, best sportsman and nicest person. Students
can receive more than one award. In how many ways can the awards be distributed if there are
twelve people in the class?
Describe the problem
accurately
Choose one of 12 people for the best mathematician (12 ways)
AND
one of the 12 for best sportsmen
(12 ways)
AND
one of the 12 for nicest person.
(12 ways)
Apply the product principle
12 × 12 × 12 = 1728
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