AQA 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: ● Display interactive pages to their class ● Add notes and highlight areas ● Add double-page spreads into lesson plans Student eTextbooks are downloadable versions of the printed textbooks that teachers can assign to students so they can: ● Download and view on any device or browser ● Add, edit and synchronise notes across two devices ● Access their personal copy on the move 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. To fi nd out more and sign up for free trials visit: www.hoddereducation.co.uk/dynamiclearning 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 Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Email [email protected] Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: 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 reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo © Tim Gainey/Alamy Stock Photo 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/ Photo credits p.1 © Kittipong Faengsrikum/Demotix/Press association Images; p.19 © Randy Duchaine/Alamy Stock Photo; p.32 © Gaby Kooijman/123RF.com; p.53 © StockbrokerXtra/Alamy Stock Photo; p.65 © polifoto/123RF.com; p.99 (top) © Sakarin Sawasdinaka/123RF.com; p.99 (lower) © Rico Koedder/123RF.com; p.130 © Edward R. Pressman Film/The Kobal Collection; p.148 © ianwool/123RF. com; p.174 © ullsteinbild/TopFoto; p.180 © Emma Lee/Alamy Stock Photo; p.190 © Bastos/Fotolia; p.229 © NASA; p.247 © Graham Moore/123RF.com; p.264 © BioPhoto Associates/Science Photo Library; p.291 (top) © Jack Sullivan/Alamy Stock Photo; p.291 (lower) © Ludmila Smite/Fotolia; p.297 © arekmalang/Fotolia; p.350 (top) © molekuul/123.com; p.350 (lower) © Wavebreak Media Ltd/123RF. com; p.372 (top) © Tom Grundy/123RF.com; p.372 (lower) © George Dolgikh/Fotolia; p.383 © Fotoatelie/Shutterstock; p.403 ©Tan Kian Khoon/Fotolia; p.406 © Volodymyr Vytiahlovskyi/123RF.com; p.434 © Stocktrek Images, Inc./Alamy Stock Photo; p.437 © Dr Jeremy Burgess/Science Photo Library; p.438 © Herbert Kratky/123RF.com; p.440 © Matthew Ashmore/Stockimo/Alamy Stock Photo; p.445 © NASA; p.450 © V Kilian/Mauritius/Superstock; p.452 © scanrail/123RF.com; p.472 © FABRICE COFFRINI/AFP/GettyImages; p.484 © Peter Bernik/123RF.com. Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. viii