AQA A-level Mathematics Year 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 eTextbook 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 find 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 specifications, 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 Year 2 Authors Sophie Goldie Approval message from AQA The core content of this digital textbook has been approved by AQA for use with our qualification This means that we have checked that it broadly covers the specification and that we are satisfied with the overall quality We have also approved the printed version of this book We not however check or approve any links or any functionality Full details of our approval process can be found on our website 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 publisher is ultimately responsible for the editorial control and quality of this digital book Please note that when teaching the AQA A-level Mathematics course, you must refer to AQA’s specification as your definitive source of information While this digital book has been written to match the specification, it cannot provide complete coverage of every aspect of the course A wide range of other useful resources can be found on the relevant subject pages of our website: aqa.org.uk Val Hanrahan Cath Moore Jean-Paul Muscat Susan Whitehouse Series editors Roger Porkess Catherine Berry Consultant Editor Heather Davis 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 (p319–320), Neil Sheldon (p410–413), Rose Jewell (p518–520) and MEI (p127–129 and p237–239) p35 Figure 3.1 data from United Nations Department of Economics and Social Affairs, Population Division World Population prospects: The 2015 Revisions, New York, 2015 p350 Table source: adapted from Table Q1.6(i),Executive summary tables: June 2013, Criminal justice statistics quarterly: June 2013 p354 Data from TableNTS0905 published by gov.uk, reproduced under the Open Government Licence www nationalarchives.gov.uk/doc/open-government-licence/version/3/ p403 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 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 Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser 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 education@bookpoint.co.uk Lines are open from a.m to 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 4718 52893 © 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 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 © Natinkabu/iStock/Thinkstock/Getty Images 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 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM Contents Getting the most from this book Prior knowledge Proof v Review: The sine and cosine rules 130 vii 1.1 Problem solving 1.2 Methods of proof Trigonometry 2.1 2.2 2.3 Radians Circular measure Small-angle approximations Review: Algebra R.1 Surds and indices R.2 Exponentials and logarithms Sequences and series 3.1 3.2 3.3 12 17 22 6.3 27 27 29 35 56 56 R.2 Polynomials 61 Review: Graphs and transformations 4.1 The language of functions 4.2 Composite functions 4.3 The modulus function Differentiation Review: Differentiation 5.1 The shape of curves 5.2 The chain rule 5.3 Connected rates of change 5.4 The product and quotient rules ractice questions: Pure P mathematics 6.1 6.2 R.1 Equations and inequalities Functions Trigonometric functions 12 Definitions and notation 36 Arithmetic sequences and series 43 Geometric sequences and series 47 Review: Algebra Working with triangles Problem solving: Triples Further algebra Trigonometric identities 8.1 8.2 8.3 64 71 80 90 9.2 127 Compound angle formulae Double angle formulae The forms rcos (q ± a), rsin (q ± a) Further differentiation 9.1 97 102 110 116 118 137 137 142 147 150 Review: Pascal’s triangle and 151 the binomial expansion 7.1 The general binomial expansion 153 7.2 Simplifying algebraic expressions 158 7.3 Partial fractions 163 64 96 Reciprocal trigonometric functions Working with trigonometric equations and identities Solving equations involving radians 130 134 9.3 169 170 174 177 184 Differentiating exponentials and logarithms 185 Differentiating trigonometric functions 189 Implicit differentiation 193 10 Integration Review: Integration 10.1 Finding areas 10.2 Integration by substitution 10.3 Integrating other functions 10.4 Integration involving the natural logarithmic function 10.5 Further integration by substitution 10.6 Integration by parts 197 198 202 211 216 220 227 229 iii 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM Running head verso Contents ractice questions: Pure P mathematics 237 Review: Coordinate geometry 240 R.1 Line segments R.2 Circles Problem solving: Eggs 11 Parametric equations 11.1 Graphs from parametric equations 11.2 Finding the equation by eliminating the parameter 11.3 Parametric differentiation 12 Vectors 240 243 248 250 252 255 261 266 12.1 Vectors 266 12.2 Using vectors to solve problems 273 13 Differential equations 278 13.1 First order differential equations 279 13.2 Solving differential equations by separating the variables 284 14 Numerical methods 14.1 Solving equations numerically 14.2 The Newton–Raphson method 14.3 Numerical integration Problem solving: Numerical integration 291 292 304 308 316 ractice questions: Pure P mathematics 319 Review: Working with data 321 R.1 Statistical problem solving Problem solving: Trains 15 Probability Review: Probability 15.1 The probability of events from two experiments 15.2 Conditional probability 321 332 16 Statistical distributions Review: The binomial distribution 16.1 Discrete random variables 16.2 The Normal distribution 353 353 355 362 17 Statistical hypothesis testing 378 Review 17.1 Interpreting sample data using the Normal distribution 17.2 Bivariate data: correlation and association 378 384 394 Practice questions: Statistics 410 18 Kinematics 414 Review: Motion in one dimension 414 18.1 Motion in two or three dimensions 423 19 Forces and motion Review: Forces and motion 19.1 Forces in equilibrium 19.2 Finding resultant forces 19.3 Newton’s second law in two dimensions 20 Moments of forces 20.1 Rigid bodies 21 Projectiles 21.1 Equations for projectile motion 21.2 Projectile problems 21.3 Further examples 21.4 The path of a projectile 21.5 General equations Problem solving: Fireworks and aeroplanes 438 438 446 456 462 470 471 482 483 486 490 498 499 504 334 22 A model for friction 506 334 22.1 A model for friction 507 339 344 Practice questions: Mechanics 518 Data set 521 Answers 523 Index 585 iv 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM 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 remaining content of A Level 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 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 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 T icon 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/AQAMathsYear2) 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 information Throughout the book the emphasis is on understanding and interpretation rather than mere routine calculations, but the various exercises nonetheless provide plenty of scope for practising basic techniques The exercise questions are split into three bands Band questions (indicated by a green bar) are designed to reinforce basic understanding Band 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 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 MP , use of ICT T and modelling M There are some multiple choice questions preceding each of these sets of practice questions to reflect those in the AQA papers This book follows on from A Level Mathematics for Year (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 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, v 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM Getting the most from this book including one or more exercises; in addition there are nine chapters that begin with a Review 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 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 the book At the end of each chapter there is a summary of the new knowledge that readers should have gained 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 Callout boxes and Note boxes are two other common features Callout boxes provide explanations for the current work Note boxes set the work in a broader or deeper context Another feature is a Caution icon , highlighting points where it is easy to go wrong The authors have taken considerable care to ensure that the mathematical vocabulary and notation are used correctly in this book, including those for variance and standard deviation, as defined in the AQA specification for A-level Mathematics In the paragraph on notation for sample variance and sample standard deviation (page 327), it explains that the meanings of ‘sample variance’, denoted by s2, 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 When answering questions, students are expected to match the level of accuracy of the given information However, there are times when this can be ambiguous For example ‘The mass of the block is kg’ could be taken to be an exact statement or to be true to just significant figure In many of the worked examples in this book such statements are taken to be exact A particular issue arises with the value of g, the acceleration due to gravity.This varies from place to place around the world Unless stated otherwise questions in this book are taken to be at a place where, to significant figures, it is 9.80 m s-2 So, providing that other information in the question is either exact or given to at least significant figures, answers based on this value are usually given to significant figures However, in the solutions to worked examples it is usually written as 9.8 rather than 9.80. Examination questions often include a statement of the value of g to be used and candidates should not give their answers to a greater number of significant figures; typically this will be figures for values of g of 9.80 m s-2 and 9.81 m s-2, and figures for 9.8 m s-2 and 10 m s-2 Answers to all exercise questions and practice questions are provided at the back of the book, and also online at www.hoddereducation.co.uk/AQAMathsYear2 Full step-by-step worked solutions to all of the practice questions are available online at www.hoddereducation.co.uk/AQAMathsYear2 All answers are also available on Hodder Education’s Dynamic Learning platform Finally a word of caution This book covers the content of Year 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 Catherine Berry and Roger Porkess *Please note that the marks stated on the example questions are to be used as a guideline only, AQA have not reviewed and approved the marks vi 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM Prior knowledge This book builds on work from AS/Year A level Mathematics AS work is reviewed either in sections at the start of chapters, or in separate review chapters in this Year A level Mathematics 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 of AS/Year Mathematics 2: Trigonometry builds on the trigonometry work in Chapter of AS/Year Mathematics ■ Chapter ■ Review: Algebra reviews the work on surds, indices, exponentials and logarithms from Chapters 2 and 13 of AS/Year Mathematics ■ Chapter 3: Sequences and series requires some use of logarithms, covered in Review: Algebra ■ Review: Algebra reviews the work on equations, inequalities and polynomials from Chapters 3, and of AS/Year Mathematics ■ hapter 4: Functions begins with a review of the work on transformations covered in Chapter C of AS/Year Mathematics ■ Chapter 5: Differentiation begins with a review of the work on differentiation covered in Chapter 10 of AS/Year Mathematics ■ Review: The sine and cosine rules reviews the work on triangles covered in part of Chapter of AS/Year Mathematics ■ Chapter 6: Trigonometric functions builds on the work in Chapter 2, and uses ideas about functions from Chapter ■ Chapter 7: Further algebra starts with a review of the work on the binomial expansion from Chapter of AS/Year Mathematics It also builds on work on the factor theorem and algebraic division, covered in Review: Algebra ■ Chapter 8: Trigonometric identities builds on the work in Chapter and Chapter ■ Chapter 9: Further differentiation builds on the work in Chapter It also requires the use of radians, covered in Chapter ■ Chapter 10: Integration starts with a review of the work on integration covered in Chapter 11 of AS/Year 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 ■ Review: Coordinate geometry reviews the work in Chapter of AS/Year Mathematics ■ Chapter 11: Parametric equations uses trigonometric identities covered in Chapter 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 and Chapter ■ Chapter 12: Vectors builds on the vectors work in Chapter 12 of AS/Year 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 852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM ■ Review: Working with data reviews the work in Chapters 14 and 15 of AS/Year Mathematics ■ Chapter 15: Probability starts with a review of the probability work in Chapter 16 of AS/Year Mathematics 16: Statistical distributions starts with a review of the work on the binomial distribution covered in Chapter 17 of AS/Year Mathematics It involves use of probability covered in Chapter 15 Prior knowledge ■ Chapter ■ Chapter 17: Statistical hypothesis testing starts with a review of the work on hypothesis testing covered in Chapter 18 of AS/Year 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 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 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 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) Photo credits p1 © Tatiana Popova – Shutterstock; p12 © Adam Majchrzak – Shutterstock; p35 © Dmitry Nikolaev – Fotolia; p44 © Granger Historical Picture Archive/Alamy Stock Photo; p64 © Gianfranco Bella – Fotolia; p96 © andreadonetti – 123RF; p117 © focal point – Shutterstock; p137 © Design Pics Inc – Getty Images; p150 © REUTERS/Alamy Stock Photo; p169 © EpicStockMedia – Fotolia; p184 © hanapon1002/Istock via Thinkstock; p197 © Alberto Loyo – Shutterstock; p248 © Ken Hewitt/iStockphoto.com; p250 © Sinibomb Images/Alamy Stock Photo; p266 © Matt Tilghman/123RF; p278 © volff – Fotolia; 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852893_FM_AQA_Maths_Y2_4E_i-viii.indd 16/11/17 10:00 PM Answers Multiple choice questions C B C D 38 − 43 × 51 18 60 0.8849 P(79.5 < X < 90.5) Practice questions: Statistics (page 410) Only countries show a value over 50 minutes, and 24 countries show a value below Of the two countries with the biggest populations, India is considerably above 50 minutes and China is just below It isn’t possible to be certain, but the figure may be plausible, particularly if it is rounded. [2] (ii) 50 minutes per day is about 304 hours per year This is roughly 39 days of 7.8 hours, so 39 working days. [2] It should say ‘in a Normal distribution (about) 95% of the distribution is within standard deviations of the mean’ (or about within standard deviation). [1] Replace 2s with 4s (or change the percentage) in the diagram. [1] (i) There may be fewer slugs to be seen in the winter months There may be fewer observers looking for slugs in the winter months [2] (ii) 93 (in month 8) is an anomalous dip It could be just randomness in the sampling Or, noting that the other observations in month are low, it could (i) be that there are fewer = 0.224 [3] 0.964 = 0.232 observers in August: (i) = + 1.0364s and perhaps they are on = − 0.8416s[2] holiday.[2] Solve: = 2.90 (iii) The rise and fall in the [2] figures is consistent with and s = 1.06 (ii) H : = 3.2, where μ is an annual life cycle the population mean However the low figures yield for variety B are also consistent with H1: < 3.2 [2] other hypotheses – for (iii) Sample mean is example that slugs 3.06 kg Critical ratio is hibernate in the winter – 3.06 − 3.2 = −1.556 so it isn’t possible [2] 0.9 ÷ 100 to say. [2] Compare with (i) Use X ~ B(10, 0.65) to z = −1.645[1] find P(X > 5) = 0.751.[2] Accept H0; insufficient (ii) H0: p = 0.65 where p reason to suppose his is the probability of yield will be below 3.2 choosing the quickest [1] check-out when there (iv) Assumption: the SD is are three available still 0.9.This is the best H1: p < 0.65 [2] information available, but (iii) P(X < 8) = 0.0060, the SD for the farmer’s P(X < 9) = 0.020, so crop could be different.[2] k = 0, 1, 2, …,7. [3] (v) Assumption: the farmer’s (iv) The first 20 visits might crop is a random sample be sufficiently random if, (of typical conditions on for example, they were his land) If, for example, at different times of day all the trial plants were However, patterns of grown in the same behaviour, by staff and location this might be customers, might be untypical of the land different immediately after as whole In that case opening a third checkthe mean yield is not a out Also, it is possible that reliable estimate. [2] the person asking the 1 1 question is still learning in (i) k ( + + + ) = 1, the first 20 trials. [2] 60 = 12 [2] so k = 125 25 (i) (a) 0.4 × 0.7 × 0.2 (ii) Vertical line chart. [1] + 0.4 × 0.3 × 0.8 + 0.6 × 0.7 × 0.8 (iii) P ( X = X ) = k ( + + + = 0.488 [2] 41 1 P X = X ) = k ( + + + 16 ) = 125 [2] (b) − 0.6 × 0.3 ×( 0.2 = 0.964 [2] 41 = 42 P ( X < X ) = 21 (1 − 125 ) 125 (ii) P(successful on all 3) 1 − 41 = 42 = 0.4 × 0.7 × 0.8 = P0.224 ( X < X ) = 2( 125 ) 125 [2] P(successful on all | successful on at least 1) P(successful on all ∩ successful on at least 1) = P(successful on at least 1) 16 41 ) = 125 574 852893_Ans_AQA_Maths_Y2_4E_568-584.indd 574 17/11/17 12:36 PM Review exercise (page 422) (i) v 25 10 O t 30 (ii) 30 s (iii) 2275 m (iv) v = 10 + 0.5t; v = 25; t > 30 (i) 3 m s−1 (ii) 2.25 m s−1 (i) 200 m (ii) 30 s (iii) 150 m (i) 1.875 m s−2 (ii) 14.6 m s−1 (i) 3 m s−1 (ii) 50 m (iii) 9.6 m s−2 (iv) 35 m (v) 5.25 m < t < 30 78.4 m (i) 4.11 s (ii) −20.2 m s−1 16 s (i) 3.5 m s−1, −3 m s−2 (ii) 4.47 m s−2; −153° (i) t i + t j 20 30 (ii) 5i + 33 13 j v = 2t i + ( 6t − t ) j; r = 23 t i + ( 3t − 13 t ) j 15.3° 8.11 m s−1 (i) initial velocity = initial velocity = 3.54 i − 3.54 j (ii) v = 8.54 i + 11.46 j ; r = 52.0i + 14.6 j 15 (i) v = ; 16 − 10t 10 21 13 m s = t3 − 4t2 + 6t + (ii) v = −10t + 12; s = −5t2 + 12t − (0, 0); (2, 8); (8, 16); (18, 24); (32, 32) (iii) 8j; 4i + 8j; 8i + 8j; 12i + 8j; 16i + 8j (iv) 21.5 m s−1 −4 0 v = ; a = −5 0 10 (i) v = 3t2 − 8t + 6; a = 0 −10 (ii) 1.6 s (iii) 22.8 m s−1 (iv) y = 2 + 16 x 15 (i) − x2 45 0 12 t (seconds) 16 (0, 2); (4, 2); (8, 1.6); (12, 0.8); (16, −0.4) y = when t = 2.4 and then the (ii) v = 20 i + (1 − 10t ) j;20 i − j projectile hits the ground v = 20i + (1 − 10t ) j;20i − j Exercise 18.1 (page 433) (iii) −10j (i) v = 4t i + 8 j (iv) y = 2 + x − x 20 80 (ii) Discussion point (page 426) vertical position (m) 11 (i) y 40 20 20 40 horizontal position (m) 60 x vertical position (m) Chapter 18 Answers Mean wing lengths differ by about SDs; mean weights differ by about of an SD So there is much more overlap of weights than there is of wing lengths So the comment is correct. [3] (ii) Females’ weights will vary according to whether or not they have recently laid eggs Males’ weights will not have this element of variation. [1] (iii) The vertical ‘striping’ in the data corresponds to the wing lengths being recorded to the nearest millimetre.[1] (iv) The data cloud looks broadly elliptical So it is appropriate to carry out the standard test on the pmcc.[2] (v) There is almost no chance (probability less than in 106) that so strong a correlation would be obtained in a sample of this size if there were no underlying correlation in the population. [2] (vi) It seems very likely that there would be strong evidence for a correlation in the male blackbirds too However, it is not certain. [1] It would not have been sensible to work with males and females as a single sample Combining two separate samples can often lead to spurious or misleading results. [1] (i) position (metres) y 30 20 10 x 10 20 30 horizontal position (m) 575 852893_Ans_AQA_Maths_Y2_4E_568-584.indd 575 17/11/17 12:36 PM (ii) 5 s (iii) 33.5 m s−1, 63.4° (or 116.6°) 13 (i) A: v sin 35°i + v cos 35°°j Chapter 19 B: −8.66i + j (ii) A: vt sin 35°i + vt cos 35°j; B: ( − 8.66t ) i + 5tj (iii) 6.1 km h−1 (iv) 0.4111… hours = 24.7 min 14 (i) 2t i + 4t j (ii) t i + 2t j (iii) t i + t j (iv) speed 9.61 m s−1; 56.3° to i direction 15 (i) 20i+10j (ii) T = s; 125 m 16 (i) 16 cm (ii) 20 cm (iii) 0 cm s−1, 0 cm s−1; 20 cm s−1, −16 cm s−1 (iv) t = 0, rA = rB = 16j t = 2, rA = rB = 20i (v) All components are zero, so model B is better 17 (i) Distance travelled (m) 750 18 (i) 2i + j + k R = 5g (ii) (2t − 5)i + (t (iii) 1.5 s; 3.5 m T T A B + 1) j + t k W = 5g 600 450 T1 4g There are a number of forces acting on the car which cancel each other out, resulting in no motion In order for that to be possible the cable must make small angles with the horizontal so that the vertical components of the tension cancel out the weight of the cable car In that case the tensions in the cable will be greater than the weight of the car 0.3 m s−2 500 m s−2 65 N (i) 3 m s−2; 2 s (ii) 1440 N (i) 29.5 N (ii) 19.5 N (iii) 24.5 N (i) (i) (ii) (ii) ( ) ) (i) R1 0.1g 420 1020 ( ( ) ( ) ) 0.5g A: R1 − 0.98 = 0.08; B: R2 − R1 − 4.9 = 0.4 (iii) R1 = 1.06 N, R2 = 6.36 N (ii) (i) caravan T2 25g T 250 N car B T2 R2 300 N Time (s) 50g R1 B A A No it doesn’t, maximum y = 9.48 when t = 2.67 120 5g (ii) 150 O 2g A: g − T1 = 8a; B: T1 − T2 − g = 2a C: T2 − g = 5a (iii) 0.65 m s−2, 73.2 N, 52.3 N T1 300 C T2 8g T2 B Review exercise (page 444) T1 A Opening activity (page 438) v = 8ti + − 4t + t j; a = 8i + −4 + t j;0