MEI A Level Mathematics Year 2

595 Pages·49.005 MB·English

by  Sophie Goldie, Val Hanrahan, Cath Moore, Jean-Paul Muscat, Susan Whitehouse

This resource is endorsed by OCR for use with specification OCR Level 3 Advanced Subsidary GCE
in Mathematics B (MEI) (H630) and OCR Level 3 Advanced GCE in Mathematics B (MEI) (H640).
In order to gain OCR endorsement, this resource has undergone an independent quality check.
Any references to assessment and/or assessment preparation are the publisher’s interpretation of the
specification requirements and are not endorsed by OCR. OCR recommends that a range of teaching
and learning resources are used in preparing learners for assessment. OCR has not paid for the production
of this resource, nor does OCR receive any royalties from its sale. For more information about the
endorsement process, please visit the OCR website, www.ocr.org.uk.
Acknowledgements
The Publishers would like to thank the following for permission to reproduce copyright material.
Questions from past AS and A Level Mathematics papers are reproduced by permission of MEI and OCR.
Practice questions have been provided by Chris Little (p316–317), Neil Sheldon (p407–410), Rose Jewell
(p515–517) and MEI (p125–127 and p235–237).
p35 Figure 3.1 data from United Nations Department of Economics and Social Affairs, Population Division.
World Population prospects: The 2015 Revisions, NewYork, 2015.
p348 Table source: adapted from Table Q1.6(i), Executive summary tables: June 2013, Criminal justice statistics
quarterly: June 2013
p352 Data from TableNTS0905 published by gov.uk, reproduced under the Open Government Licence www.
nationalarchives.gov.uk/doc/open-government-licence/version/3/
p401 Figure 17.13 data from The World Factbook 2013–14. Washington, DC: Central Intelligence Agency, 2013.
https://www.cia.gov/library/publications/the-world-factbook/index.html.
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ISBN: 978 1 4718 5298 5
ISBN: 978 1 5104 2990 1
© Sophie Goldie, Val Hanrahan, Jean-Paul Muscat, Roger Porkess, Susan Whitehouse and MEI 2017
First published in 2017 by
Hodder Education,
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Contents
Getting the most from this book v Review: The sine and cosine rules  128
Prior knowledge vii 1 Working with triangles 128
Problem solving: Triples 132
1    Proof 1
6    Trigonometric functions 135
1.1 Problem solving 2
6.1 Reciprocal trigonometric
1.2 Methods of proof 5
functions 135
6.2 Working with trigonometric
2    Trigonometry 12
equations and identities 140
2.1 Radians 12
6.3 Solving equations involving
2.2 Circular measure 17 radians 145
2.3 Small-angle approximations 22
7    Further algebra 148
Review: Algebra 1 27
Review: Pascal’s triangle and
the binomial expansion 149
R.1 Surds and indices 27
7.1 The general binomial expansion 151
R.2 Exponentials and logarithms 29
7.2 Simplifying algebraic expressions 156
3    Sequences and series 35
7.3 Partial fractions 161
3.1 Definitions and notation 36
8    Trigonometric identities 167
3.2 Arithmetic sequences and series 43
8.1 Compound angle formulae 168
3.3 Geometric sequences and series 47
8.2 Double angle formulae 172
Review: Algebra 2 56
8.3 The forms rcos (θ ± α),
rsin (θ ± α) 175
R.1 Equations and inequalities 56
R.2 Polynomials 61 9    Further differentiation 182
4    Functions 64 9.1 Differentiating exponentials and
logarithms 183
Review: Graphs and transformations 64
9.2 Differentiating trigonometric
4.1 The language of functions 71
functions 187
4.2 Composite functions 80
9.3 Implicit differentiation 191
4.3 The modulus function 90
10  Integration 195
5   Differentiation 96
Review: Integration 196
Review: Differentiation 97
10.1 Finding areas 200
5.1 The shape of curves 102
10.2 Integration by substitution 209
5.2 The chain rule 110
10.3 Integrating other functions 214
5.3 Connected rates of change 116
10.4 Integration involving the natural
5.4 The product and quotient rules 118 logarithmic function 218
10.5 Further integration by
Practice questions: Pure
substitution 225
Mathematics 1 125
10.6 Integration by parts 228
iii
Practice questions: Pure 16  Statistical distributions 351
Mathematics 2 235
Review: The binomial distribution      351
16.1 Discrete random variables 353
Review: Coordinate geometry 238
16.2 The Normal distribution 360
s
R.1 Line segments 238
t
n 17  Statistical hypothesis testing      376
e R.2 Circles 241
t
n
o Problem solving: Eggs 246 Review 376
C
17.1 Interpreting sample data using
11  Parametric equations 248
the Normal distribution 382
11.1 Graphs from parametric 17.2 Bivariate data: correlation and
equations 250 association 392
11.2 Finding the equation by
Practice questions: Statistics 407
eliminating the parameter 253
11.3 Parametric differentiation 259 18  Kinematics 411
12  Vectors 264
Review: Motion in one dimension 411
12.1 Vectors 264 18.1 Motion in two or three dimensions 420
12.2 Using vectors to solve problems    271
19  Forces and motion 435
13  Differential equations 276
Review: Forces and motion 435
19.1 Forces in equilibrium 443
13.1 First order differential equations 277
19.2 Finding resultant forces 453
13.2 Solving differential equations by
separating the variables 282 19.3 Newton’s second law in two
dimensions 459
14  Numerical methods 289
20  Moments of forces 467
14.1 Solving equations numerically 290
20.1 Rigid bodies 468
14.2 The Newton–Raphson method  302
14.3 Numerical integration 306 21  Projectiles 479
Problem solving: Numerical integration 314
21.1 Equations for projectile motion    480
Practice questions: Pure 21.2 Projectile problems 483
Mathematics 3 316 21.3 Further examples 487
21.4 The path of a projectile 495
Review: Working with data 318
21.5 General equations 496
R.1 Statistical problem solving 318 Problem solving: Fireworks and
Problem solving: Trains 330 aeroplanes 502
15  Probability 332 22  A model for friction 504
Review: Probability 332 22.1 A model for friction 505
15.1 The probability of events from
Practice questions: Mechanics 515
two experiments 337
15.2 Conditional probability 342 Data set 518
Answers 520
Index 581
iv
Getting the most from this book
Mathematics is not only a beautiful and exciting subject in its own right but also one that underpins many
other branches of learning. It is consequently fundamental to our national wellbeing.
This book covers the content of A2 Mathematics and so provides a complete course for the second of the
two years of Advanced Level study. The requirements of the first year are met in the first book.
Between 2014 and 2016 A level Mathematics and Further Mathematics were very substantially revised, for
first teaching in 2017. Major changes include increased emphasis on
Problem solving
■
Proof
■
■ Use of ICT
■ Modelling
Working with large data sets in statistics.
■
This book embraces these ideas. The first section of Chapter 1 is on problem solving and this theme is
continued throughout the book with several spreads based on the problem solving cycle. In addition a large
number of exercise questions involve elements of problem solving; these are identified by the   icon
PS
beside them. The ideas of mathematical proof and rigorous logical argument are also introduced in
Chapter 1 and are then involved in suitable exercise questions throughout the book. The same is true of
modelling; the modelling cycle is introduced in the first chapter and the ideas are reinforced through the
rest of the book.
The use of technology, including graphing software, spreadsheets and high specification calculators,
is encouraged wherever possible, for example in the Activities used to introduce some of the topics
in Pure mathematics, and particularly in the analysis and processing of large data sets in Statistics.
Places where ICT can be used are highlighted by a     icon. A large data set is provided at the end
 T
of the book but this is essentially only for reference. It is also available online as a spreadsheet
(www.hoddereducation.co.uk/MEIMathsYear2) and it is in this form that readers are expected to store
and work on this data set, including answering the exercise questions that are based on it. These are found
at the end of each exercise in the Statistics chapters and identified with a purple bar. They illustrate, for
each topic, how a large data set can be used to provide the background for examination-type questions.
Throughout the book the emphasis is on understanding and interpretation rather than mere routine
calculations, but the various exercises do nonetheless provide plenty of scope for practising basic
techniques. The exercise questions are split into three bands. Band 1 questions (indicated by a green bar)
are designed to reinforce basic understanding. Band 2 questions (yellow bar) are broadly typical of what
might be expected in an examination: some of them cover routine techniques; others are designed to
provide some stretch and challenge for readers. Band 3 questions (red bar) explore round the topic and
some of them are rather more demanding. Questions in the Statistics chapters that are based on the large
data set are identified with a purple bar. In addition, extensive online support, including further questions,
is available by subscription to MEI’s Integral website, http://integralmaths.org.
In addition to the exercise questions, there are five sets of questions, called Practice questions, covering
groups of chapters. All of these sets include identified questions requiring problem solving  ,
  PS
mathematical proof  , use of ICT     and modelling  .
     MP T    M
This book follows on from A Level Mathematics for Year 1 (AS) and most readers will be familiar with
the material covered in it. However, there may be occasions when they want to check on topics in the
earlier book; the parts entitled Review allow them to do this without having to look elsewhere. The
five short Review chapters provide a condensed summary of the work that was covered in the earlier
book, including one or more exercises; in addition there are nine chapters that begin with a Review
v
section and exercise, and then go on to new work based on it. Confident readers may choose to miss out the
Review material, and just refer to these parts of the book when they are uncertain about particular topics.
Others, however, will find it helpful to work through some or all of the Review material to consolidate their
understanding of the first year work.
k There are places where the work depends on knowledge from earlier in the book and this is flagged up in
o
o the margin in Prior knowledge boxes. This should be seen as an invitation to those who have problems with
b
the particular topic to revisit it earlier in the book. At the end of each chapter there is a summary of the new
s
i
h knowledge that readers should have gained.
t

m
Two common features of the book are Activities and Discussion points. These serve rather different
o
r
purposes. The Activities are designed to help readers get into the thought processes of the new work that
f

t
s they are about to meet; having done an Activity, what follows will seem much easier. The Discussion points
o
m invite readers to talk about particular points with their fellow students and their teacher and so enhance

e their understanding. Callout boxes and Note boxes are two other common features. Callout boxes provide
h
t explanations for the current work. Note boxes set the work in a broader or deeper context. Another feature is

g
n a Caution icon , highlighting points where it is easy to go wrong.
i
t
t
e The authors have taken considerable care to ensure that the mathematical vocabulary and notation are
G
used correctly in this book, including those for variance and standard deviation, as defined in the OCR
specification for A Level in Mathematics B (MEI). In the paragraph on notation for sample variance and
sample standard deviation (page 46), it explains that the meanings of ‘sample variance’, denoted by s 2, and
‘sample standard deviation’, denoted by s, are defined to be calculated with divisor (n – 1). In early work in
statistics it is common practice to introduce these concepts with divisor n rather than (n – 1). However there
is no recognised notation to denote the quantities so derived. Students should be aware of the variations
in notation used by manufacturers on calculators and know what the symbols on their particular models
represent.
Answers to all exercise questions and practice questions are provided at the back of the book, and also online
at www.hoddereducation.co.uk/MEIMathsYear2. Full step-by-step worked solutions to all of the practice
questions are available online at www.hoddereducation.co.uk/MEIMathsYear2. All answers are also available
on Hodder Education’s Dynamic Learning platform.
Finally a word of caution. This book covers the content of Year 2 of A Level Mathematics and is designed
to provide readers with the skills and knowledge they will need for the examination. However, it is not the
same as the specification, which is where the detailed examination requirements are set out. So, for example,
the book uses the data set of cycling accidents to give readers experience of working with a large data set, but
this is not the data set that will form the basis of any examination questions. Similarly, in the book cumulative
binomial tables are used in the explanation of the output from a calculator, but such tables will not be
available in examinations. Individual specifications will also make it clear how standard deviation is expected
to be calculated. So, when preparing for the examination, it is essential to check the specification.
This is a 4th edition MEI textbook so much of the material is well tried and tested. However, as a
consequence of the changes to A Level requirements in mathematics, large parts of the book are either new
material or have been very substantially rewritten.
Catherine Berry
Roger Porkess
vi
Prior knowledge
This book builds on work from AS/Year 1 A level Mathematics. AS work is reviewed either in
sections at the start of chapters, or in separate review chapters in this A2 book.
The order of the chapters has been designed to allow later ones to use and build on work in earlier
chapters. The list below identifies cases where the dependency is particularly strong.
The Statistics and Mechanics chapters are placed in separate sections of the book for easy reference, but it
is expected that these will be studied alongside the Pure mathematics work rather than after it.
The work in Chapter 1: Proof pervades the whole book. It builds on the work on problem solving
■
and proof covered in Chapter 1 of AS/Year 1 Mathematics.
Chapter 2: Trigonometry builds on the trigonometry work in Chapter 6 of AS/Year 1
■
Mathematics.
Review: Algebra 1 reviews the work on surds, indices, exponentials and logarithms from
■
Chapters 2 and 13 of AS/Year 1 Mathematics.
Chapter 3: Sequences and series requires some use of logarithms, covered in Review: Algebra 1.
■
Review: Algebra 2 reviews the work on equations, inequalities and polynomials from Chapters 3,
■
4 and 7 of AS/Year 1 Mathematics.
Chapter 4: Functions begins with a review of the work on transformations covered in Chapter 8
■
of AS/Year 1 Mathematics.
Chapter 5: Differentiation begins with a review of the work on differentiation covered in
■
Chapter 10 of AS/Year 1 Mathematics.
Review: The sine and cosine rules reviews the work on triangles covered in part of Chapter 6
■
of AS/Year 1 Mathematics.
Chapter 6: Trigonometric functions builds on the work in Chapter 2, and uses ideas about
■
functions from Chapter 4.
Chapter 7: Further algebra starts with a review of the work on the binomial expansion from
■
Chapter 9 of AS/Year 1 Mathematics. It also builds on work on the factor theorem and algebraic
division, covered in Review: Algebra 2.
Chapter 8: Trigonometric identities builds on the work in Chapter 2 and Chapter 6.
■
Chapter 9: Further differentiation builds on the work in Chapter 5. It also requires the use of
■
radians, covered in Chapter 2.
Chapter 10: Integration starts with a review of the work on integration covered in Chapter 11
■
of AS/Year 1 Mathematics. It follows on from the differentiation work in Chapter 9, and also
requires the use of radians, covered in Chapter 2, and partial fractions, covered in Chapter 7.
Review: Coordinate geometry reviews the work in Chapter 5 of AS/Year 1 Mathematics.
■
Chapter 11: Parametric equations uses trigonometric identities covered in Chapter 6 and
■
Chapter 8. You should also recall the equation of a circle, covered in Review: Coordinate
geometry, and be confident in the differentiation techniques covered in Chapter 5 and Chapter 9.
Chapter 12: Vectors builds on the vectors work in Chapter 12 of AS/Year 1 Mathematics.
■
Chapter 13: Differential equations uses integration work covered in Chapter 10.
■
Chapter 14: Numerical methods requires some simple differentiation and knowledge of how
■
integration relates to the area under a graph.
vii
Review: Working with data reviews the work in Chapters 14 and 15 of AS/Year 1 Mathematics.
■
Chapter 15: Probability starts with a review of the probability work in Chapter 16 of AS/Year 1
■
Mathematics.
Chapter 16: Statistical distributions starts with a review of the work on the binomial distribution
■
covered in Chapter 17 of AS/Year 1 Mathematics. It involves use of probability covered in
Chapter 15.
Chapter 17: Statistical hypothesis testing starts with a review of the work on hypothesis testing
■
covered in Chapter 18 of AS/Year 1 Mathematics. It requires use of the Normal distribution
covered in Chapter 16.
Chapter 18: Kinematics starts with a review of the work on kinematics covered in Chapters 19 and
■
21 of AS/Year 1 Mathematics. You should be confident in working with vectors in two dimensions
(reviewed in Chapter 12) and in working with parametric equations (Chapter 11).
Chapter 19: Forces and motion starts with a review of the work on force covered in Chapter 20 of
■
AS/Year 1 Mathematics. It requires the use of vectors in two dimensions (reviewed in Chapter 12).
Chapter 20: Moments of forces uses work on force covered in Chapter 19, and the use of vectors in
■
two dimensions (reviewed in Chapter 12).
Chapter 21: Projectiles uses trigonometric identities from Chapter 6 and Chapter 8, and work on
■
parametric equations from Chapter 11. It also requires use of vectors in two dimensions (reviewed in
Chapter 12).
Chapter 22: A model for friction uses work on force and moments covered in Chapters 19 and 20,
■
as well as vectors in two dimensions (reviewed in Chapter 12).
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viii