Table Of ContentAQA A-level Mathematics for A-level Year 1 and AS is available as a Whiteboard eTextbook and
Student eTextbook.
Whiteboard eTextbooks are online interactive versions of the printed textbook that enable teachers to:
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Important notice: AQA only approve the Student Book and Student eTextbook. The other resources
referenced here have not been entered into the AQA approval process.
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Integral A-level Mathematics online resources
Our eTextbooks link seamlessly with Integral A-level Mathematics online resources, allowing you to
move with ease between corresponding topics in the eTextbooks and Integral.
These online resources have been developed by MEI and cover the new AQA A-level Mathematics
specifi cations, supporting teachers and students with high quality teaching and learning activities that
include dynamic resources and self-marking tests and assessments.
Integral A-level Mathematics online resources are available by subscription to enhance your use of this
book. To subscribe to Integral visit www.integralmaths.org
AQA
A-level
Mathematics
For A-level Year 1 and AS
1
Authors
Approval message from AQA
Sophie Goldie
The core content of this digital textbook has been approved by AQA for use with our
qualifi cation. This means that we have checked that it broadly covers the specifi cation and Val Hanrahan
that we are satisfi ed with the overall quality. We have also approved the printed version of this
book. We do not however check or approve any links or any functionality. Full details of our Cath Moore
approval process can be found on our website.
Jean-Paul Muscat
We approve print and digital textbooks because we know how important it is for teachers
and students to have the right resources to support their teaching and learning. However, the Susan Whitehouse
publisher is ultimately responsible for the editorial control and quality of this digital book.
Series editors
Please note that when teaching the AQA A-level Mathematics course, you must refer to
AQA’s specifi cation as your defi nitive source of information. While this digital book has been written to
Roger Porkess
match the specifi cation, it cannot provide complete coverage of every aspect of the course.
Catherine Berry
A wide range of other useful resources can be found on the relevant subject pages of our
website: aqa.org.uk
Consultant editor
Heather Davis
Bibliography
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www.hoddereducation.co.uk
ISBN: 978 1 4718 5286 2
© Sophie Goldie, Val Hanrahan, Jean-Paul Muscat, Roger Porkess, Susan Whitehouse and MEI 2017
First published in 2017 by
Hodder Education,
An Hachette UK Company
Carmelite House
50 Victoria Embankment
London EC4Y 0DZ
www.hoddereducation.co.uk
Impression number 10 9 8 7 6 5 4 3 2 1
Year 2021 2020 2019 2018 2017
All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be
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Typeset in Bembo Std, 11/13 pts. by Aptara®, Inc.
Printed in Italy
A catalogue record for this title is available from the British Library.
Contents
Getting the most from this book v 7 Polynomials 130
Prior knowledge vii 7.1 Polynomial expressions 131
7.2 Dividing polynomials 140
1 Problem solving 1
7.3 Polynomial equations 142
1.1 Solving problems 2
8 Graphs and transformations 148
1.2 Writing mathematics 8
1.3 Proof 12 8.1 The shapes of curves 149
Problem solving: Mountain modelling 16 8.2 Using transformations to
sketch curves 155
2 Surds and indices 19
8.3 Using transformations 164
2.1 Using and manipulating surds 20 8.4 Transformations and graphs of
trigonometric functions 167
2.2 Working with indices 24
9 The binomial expansion 172
3 Quadratic functions 32
9.1 Binomial expansions 173
3.1 Quadratic graphs and equations 33
9.2 Selections 180
3.2 The completed square form 42
3.3 The quadratic formula 47 Practice questions: Pure
mathematics 2 186
4 Equations and inequalities 53
4.1 Simultaneous equations 54 10 Differentiation 190
4.2 Inequalities 59
10.1 The gradient of the tangent
as a limit 191
5 Coordinate geometry 65
10.2 Differentiation using
5.1 Working with coordinates 66 standard results 194
5.2 The equation of a straight line 71 10.3 Tangents and normals 198
5.3 The intersection of two lines 78 10.4 Increasing and decreasing
5.4 The circle 80 functions, and turning points 201
5.5 The intersection of a line 10.5 Sketching the graphs of
and a curve 87 gradient functions 206
Problem solving: Integer point circles 92 10.6 Extending the rule 210
10.7 Higher order derivatives 213
Practice questions: Pure
10.8 Practical problems 218
mathematics 1 96
10.9 Finding the gradient from
first principles 221
6 Trigonometry 99
Problem solving: Proofs 226
6.1 Trigonometric functions 100
11 Integration 229
6.2 Trigonometric functions for
angles of any size 104 11.1 Integration as the reverse
of differentiation 230
6.3 Solving equations using graphs
of trigonometric functions 112 11.2 Finding areas 234
6.4 Triangles without right angles 118 11.3 Areas below the x axis 238
6.5 The area of a triangle 126 11.4 Further integration 241
iii
12 Vectors 247 19 Kinematics 403
12.1 Vectors 247 19.1 The language of motion 403
12.2 Working with vectors 253 19.2 Speed and velocity 406
o 12.3 Vector geometry 259 19.3 Acceleration 411
s
rs
ent 13 Exponentials and logarithms 264 19.4 Using areas to find distances
d vte and displacement 415
an 13.1 Exponential functions 265
eo 19.5 The constant acceleration
hC
g 13.2 Logarithms 268 formulae 421
in 13.3 The exponential function 273 19.6 Further examples 426
n
n 13.4 The natural logarithm function 278
u
20 Forces and Newton’s laws
R
13.5 Modelling curves 280
of motion 434
Practice questions: Pure
20.1 Force diagrams 434
mathematics 3 288
20.2 Force and motion 440
14 Data collection 291 20.3 Types of forces 442
20.4 Pulleys 447
14.1 Using statistics to solve problems 292
20.5 Applying Newton’s second law
14.2 Sampling 297
along a line 450
15 Data processing, presentation 20.6 Newton’s second law applied
and interpretation 306 to connected objects 457
Problem solving: Reviewing models
15.1 Presenting different types of data 308
for air resistance 468
15.2 Ranked data 312
15.3 Discrete numerical data 317 21 Variable acceleration 472
15.4 Continuous numerical data 324
21.1 Using differentiation 473
15.5 Bivariate data 335
21.2 Finding displacement
15.6 Standard deviation 342 from velocity 475
21.3 The constant acceleration
16 Probability 350
formulae revisited 479
16.1 Working with probability 351
Problem solving: Human acceleration 484
Problem solving: Alphabet puzzle 368
Practice questions: Mechanics 487
Problem solving: Estimating minnows 370
17 The binomial distribution 372 Dataset 490
17.1 Introduction to binomial
Answers* 492
distribution 373
Index 561
17.2 Using the binomial distribution 377
18 Statistical hypothesis testing
*Please note that the marks stated on
using the binomial distribution 383
the example questions are to be used as a
18.1 The principles and language of guideline only, AQA have not reviewed and
hypothesis testing 385 approved the marks.
18.2 Extending the language of
hypothesis testing 391
Practice questions: Statistics 399
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 AS Mathematics and so provides a complete course for the first of the
two years of Advanced Level study. The requirements of the second year are met in a second 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
(cid:81)
Proof
(cid:81)
Use of ICT
(cid:81)
Modelling
(cid:81)
Working with large data sets in statistics.
(cid:81)
This book embraces these ideas. 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 PS icon 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. Questions
which involve an element of modelling are identified by the M icon.
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. A large data
set is provided at the end of the book but this is essentially only for reference. It is also available online
as a spreadsheet (www.hoddereducation.co.uk/AQAMathsYear1) 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. Places where ICT can be used are highlighted by a T icon.
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, while most exercises precede these with one or two
questions designed to help students bridge the gap between GCSE and AS Mathematics; these questions
are signposted by a icon. These include a 'thinking' question which addresses a key stumbling block
in the topic and a multiple choice question to test key misconceptions. Band 2 questions (yellow bar) are
broadly typical of what might be expected in an examination: some of them cover routine techniques;
others are design 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. In addition, extensive online support,
including further questions, is available by subscription to MEI’s Integral website, http://integralmaths.org.
(Please note that these external links are not being entered in an AQA approval process.)
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 MP , use of ICT T and modelling M .
The book is written on the assumption that readers have been successful in GCSE Mathematics, or its equivalent,
and are reasonably confident and competent with that level of mathematics. There are places where the work
depends on knowledge from earlier in the book and this is flagged up in the margin in Prior knowledge boxes.
This should be seen as an invitation to those who have problems with the particular topic to revisit it earlier in
book. At the end of each chapter there is a summary of the new knowledge that readers should have gained.
v
Two common features of the book are Activities and Discussion points. These serve rather different purposes.
The Activities are designed to help readers get into the thought processes of the new work that they are
about to meet; having done an Activity, what follows will seem much easier. The Discussion points invite
readers to talk about particular points with their fellow students and their teacher and so enhance their
understanding. Another feature is a Caution icon , highlighting points where it is easy to go wrong.
ok
so
ro The authors have taken considerable care to ensure that the mathematical vocabulary and notation are
eb
d vis used correctly in this book, including those for variance and standard deviation, as defined in the AQA
ah specification for AS Level in Mathematics. In the paragraph on notation for sample variance and sample
et
hm standard deviation (page 344), it explains that the meanings of ‘sample variance’, denoted by s2, and ‘sample
g o
nr standard deviation’, denoted by s, are defined to be calculated with divisor (n – 1). In early work in statistics
nit f it is common practice to introduce these concepts with divisor n rather than (n – 1). However there is
ns
uo no recognised notation to denote the quantities so derived. Students should be aware of the variations
Rm
e in notation used by manufacturers on calculators and know what the symbols on their particular models
h represent.
t
g
n Answers to all exercise questions and practice questions are provided at the back of the book, and also online
i
t
t at www.hoddereducation.co.uk/AQAMathsYear1. Full step-by-step worked solutions to all of the practice
e
G questions are available online at www.hoddereducation.co.uk/AQAMathsYear1. All answers are also available
on Hodder Education’s Dynamic Learning platform. (Please note that these additional links have not been
entered into the AQA approval process.)
Finally a word of caution. This book covers the content of AS Level Mathematics and is designed to
help provide readers with the skills and knowledge 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 a data set about cycling accidents to give readers experience of working with a large data
set. Examination questions will test similar ideas but they will be based on different data sets; for more
information about these sets readers should consult the specification. 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.
Catherine Berry
Roger Porkess
vi
Prior knowledge
This book builds on GCSE work, much of which is assumed knowledge.
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: Problem solving pervades the whole book
(cid:81)
Chapter 3: Quadratic equations and graphs requires some manipulation of surds (chapter 2)
(cid:81)
Chapter 4: Equations and inequalities uses work on solving quadratic equations (chapter 3)
(cid:81)
Chapter 5: Coordinate geometry requires the use of quadratic equations (chapter 3) and
(cid:81)
simultaneous equations (chapter 4)
Chapter 6: Trigonometry requires some use of surds (chapter 2) and quadratic equations (chapter 3)
(cid:81)
Chapter 7: Polynomials builds on the work on quadratic equations (chapter 3)
(cid:81)
Chapter 8: Graphs and transformations brings together work on quadratic graphs (chapter 3),
(cid:81)
trigonometric graphs (chapter 6) and polynomial graphs (chapter 7)
Chapter 9: The binomial expansion builds on polynomials (chapter 7)
(cid:81)
Chapter 10: Differentiation draws on a number of techniques, including work on indices (chapter 2),
(cid:81)
quadratic equations (chapter 3), coordinate geometry (chapter 5) and polynomial graphs (chapter 8)
Chapter 11: Integration follows on from differentiation (chapter 10)
(cid:81)
Chapter 12: Vectors builds on coordinate geometry (chapter 5)
(cid:81)
Chapter 13: Logarithms and exponentials builds on work on indices (chapter 2)
(cid:81)
Chapter 15: Data processing, presentation and interpretation follows on from data collection
(cid:81)
(chapter 14)
Chapter 17: The binomial distribution draws on ideas from probability (chapter 16) and the
(cid:81)
binomial expansion (chapter 9)
Chapter 18: Hypothesis testing uses ideas from probability (chapter 16) and the binomial
(cid:81)
distribution (chapter 17)
Chapter 19: Kinematics requires fluency with quadratic equations (chapter 3) and simultaneous
(cid:81)
equations (chapter 4)
Chapter 20: Forces ties in with work on vectors (chapter 12), although these two chapters could be
(cid:81)
covered in either order
Chapter 21: Variable acceleration uses differentiation (chapter 10) and integration (chapter 11).
(cid:81)
vii
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.
Question 5 on page 322 is taken from OCR, Core Mathematics Specimen Paper H867/02, 2015. The
answer on page 541 is also reproduced by permission of OCR.
Practice questions have been provided by Chris Little (p288–290), Neil Sheldon (p399–402), Rose Jewell
(p487–489), and MEI (p96–98, p186–189).
p.309 The smoking epidemic-counting the cost, HEA, 1991: Health Education Authority, reproduced
under the NICE Open Content Licence: www.nice.org.uk/Media/Default/About/Reusing-our-
content/Open-content-licence/NICE-UK-Open-Content-Licence-.pdf; p.310 Young People Not in
Education, Employment or Training (NEET): February 2016, reproduced under the Open Government
Licence www.nationalarchives.gov.uk/doc/open-government-licence/version/3/; p.334 The World Bank:
Mobile cellular subscriptions (per 100 people): http://data.worldbank.org/indicator/IT.CEL.SETS.P2;
p.340 Historical monthly data for meteorological stations: https://data.gov.uk/dataset/historic-monthly-
meteorological-station-data, reproduced under the Open Government Licence www.nationalarchives.
gov.uk/doc/open-government-licence/version/3/; p.341 Table 15.26 (no.s of homicides in England &
Wales at the start and end of C20th): https://www.gov.uk/government/statistics/historical-crime-data,
reproduced under the Open Government Licence www.nationalarchives.gov.uk/doc/open-government-
licence/version/3/; p.362 Environment Agency: Risk of flooding from rivers and the sea, https://flood-
warning-information.service.gov.uk/long-term-flood-risk/map?map=RiversOrSea, reproduced under the
Open Government Licence www.nationalarchives.gov.uk/doc/open-government-licence/version/3/
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viii
