Edex_Higher Math_00.qxd 17/03/06 09:14 Page i BRIAN SPEED KEITH GORDON KEVIN EVANS This high quality material is endorsed by Edexcel and has been through a rigorous quality assurance programme to ensure it is a suitable companion to the specification for both learners and teachers This does not mean that the contents will be used verbatim when setting examinations nor is it to be read as being the official specification – a copy of which is available at www.edexcel.org.uk This book provides indicators of the equivalent grade level of maths questions throughout The publishers wish to make clear that these grade indicators have been provided by Collins Education, and are not the responsibility of Edexcel Ltd Whilst every effort has been made to assure their accuracy, they should be regarded as indicators, and are not binding or definitive Edex_Higher Math_00.qxd 17/03/06 09:14 Page ii William Collins’ dream of knowledge for all began with the publication of his first book in 1819 A self-educated mill worker, he not only enriched millions of lives, but also founded a flourishing publishing house Today, staying true to this spirit, Collins books are packed with inspiration, innovation and a practical expertise They place you at the centre of a world of possibility and give you exactly what you need to explore it Collins Do more Published by Collins An imprint of HarperCollinsPublishers 77–85 Fulham Palace Road Hammersmith London W6 8JB Browse the complete Collins catalogue at www.collinseducation.com Acknowledgements With special thanks to Lynn and Greg Byrd The Publishers gratefully acknowledge the following for permission to reproduce copyright material Whilst every effort has been made to trace the copyright holders, in cases where this has been unsuccessful or if any have inadvertently been overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity © HarperCollinsPublishers Limited 2006 Edexcel material reproduced with permission of Edexcel Limited Edexcel Ltd accepts no responsibility whatsoever for the accuracy or method of working in the answers given 10 ISBN-13 978-0-00-721564-5 ISBN-10 0-00-721564-9 Grade bar photos © 2006 JupiterImages Corporation The author asserts his moral right to be identified as the author of this work © 2006 JupiterImages Corporation, p1, p23, p61, p83, p111, p149, p171, p209, p300, p315, p333, p363, p415, p509, p519, p531, p543, p559, p573 © Mr Woolman, p42 All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means – electronic, mechanical, photocopying, recording or otherwise – without the prior written consent of the Publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP British Library Cataloguing in Publication Data A Catalogue record for this publication is available from the British Library © Dave Roberts / Istock, p191 © Agence Images / Alamy, p279 © Sergeo Syd / Istock, p385 © PCL / Alamy, p437 © Michal Galazka / Istock, p479 © SuperStock / Alamy, p528 © Penny Fowler, p585 Commissioned by Marie Taylor, Vicky Butt and Michael Cotter Project managed by Penny Fowler Edited by Marian Bond and Paul Sterner Answer checker: Amanda Whyte Internal design by JPD Cover design by JPD Cover illustration by Andy Parker, JPD Page make-up and indexing by Gray Publishing Page make-up of Really Useful Maths! spreads by EMC Design Illustrations by Gray Publishing, EMC Design, Peters and Zabransky, Peter Cornwell, Bob Lea (Artists Partners), Martin Sanders (Beehive Illustration) and Laszlo Veres (Beehive Illustration) Production by Natasha Buckland Printed and bound in Italy by Eurografica SpA Edex_Higher Math_00.qxd 17/03/06 09:14 Page iii CONTENTS Chapter Number Chapter Fractions and percentages 23 Chapter Ratios and proportion 45 Chapter Shape 61 Chapter Algebra 83 Chapter Pythagoras and trigonometry 111 Chapter Geometry 149 Chapter Transformation geometry 171 Chapter Constructions 191 Chapter 10 Powers, standard form and surds 209 Chapter 11 Statistics 237 Chapter 12 Algebra 279 Chapter 13 Real-life graphs 303 Chapter 14 Similarity 315 Chapter 15 Trigonometry 333 Chapter 16 Linear graphs and equations 363 Chapter 17 More graphs and equations 385 Chapter 18 Statistics 415 Chapter 19 Probability 437 Chapter 20 Algebra 479 Chapter 21 Dimensional analysis 509 Chapter 22 Variation 519 Chapter 23 Number and limits of accuracy 531 Chapter 24 Inequalities and regions 543 Chapter 25 Vectors 559 Chapter 26 Transformation of graphs 573 Chapter 27 Proof 585 Answers 597 Index 635 iii Edex_Higher Math_00.qxd 16/03/06 15:45 Page iv Welcome to Collins GCSE Maths, the easiest way to learn and succeed in Mathematics This textbook uses a stimulating approach that really appeals to students Here are some of the key features of the textbook, to explain why Each chapter of the textbook begins with an Overview The Overview lists the Sections you will encounter in the chapter, the key ideas you will learn, and shows how these ideas relate to, and build upon, each other The Overview also highlights what you should already know, and if you’re not sure, there is a short Quick Check activity to test yourself and recap Maths can be useful to us every day of our lives, so look out for these Really Useful Maths! pages These double page spreads use big, bright illustrations to depict real-life situations, and present a short series of real-world problems for you to practice your latest mathematical skills on Each Section begins first by explaining what mathematical ideas you are aiming to learn, and then lists the key words you will meet and use The ideas are clearly explained, and this is followed by several examples showing how they can be applied to real problems Then it’s your turn to work through the exercises and improve your skills Notice the different coloured panels along the outside of the exercise pages These show the equivalent exam grade of the questions you are working on, so you can always tell how well you are doing iv Edex_Higher Math_00.qxd 16/03/06 15:45 Page v Every chapter in this textbook contains lots of Exam Questions These provide ideal preparation for your examinations Each exam question section also concludes with a fully worked example Compare this with your own work, and pay special attention to the examiner’s comments, which will ensure you understand how to score maximum marks Throughout the textbook you will find Activities – highlighted in the green panels – designed to challenge your thinking and improve your understanding Review the Grade Yourself pages at the very end of the chapter This will show what exam grade you are currently working at Doublecheck What you should now know to confirm that you have the knowledge you need to progress Working through these sections in the right way should mean you achieve your very best in GCSE Maths Remember though, if you get stuck, answers to all the questions are at the back of the book (except the exam question answers which your teacher has) We hope you enjoy using Collins GCSE Maths, and wish you every good luck in your studies! Brian Speed, Keith Gordon, Kevin Evans v Edex_Higher Math_00.qxd 16/03/06 15:45 Page vi ICONS You may use your calculator for this question You should not use your calculator for this question Indicates a Using and Applying Mathematics question Indicates a Proof question Edex_Higher Math_01.qxd 16/03/06 08:15 Page TO PAGE 217 Solving real problems This chapter will show you … Division by decimals Estimation Multiples, factors and prime numbers how to calculate with integers and decimals ● ● how to round off numbers to a given number of significant figures ● how to find prime factors, least common multiples (LCM) and highest common factors (HCF) What you should already know ● How to add, subtract, multiply and divide with integers ● What multiples, factors, square numbers and prime numbers are ● The BODMAS rule and how to substitute values into simple algebraic expressions Solving real problems Prime factors, LCM and HCF Negative numbers Rounding to significant figures Sensible rounding Division by decimals Estimation Approximation of calculations Multiplying and dividing by multiples of 10 Least common multiple Multiples, factors and prime numbers Prime factors Highest common factor Negative numbers Quick check Work out the following a 23 × 167 b 984 ÷ 24 c (16 + 9)2 Write down the following a a multiple of b a prime number between 10 and 20 c a square number under 80 d the factors of Work out the following a 2+3×5 b (2 + 3) × © HarperCollinsPublishers Limited 2007 c + 32 – Edex_Higher Math_01.qxd 1.1 16/03/06 08:15 Page Solving real problems This section will give you practice in using arithmetic to: ● solve more complex problems Key words long division long multiplication strategy In your GCSE examination, you will be given real problems that you have to read carefully, think about and then plan a strategy without using a calculator These will involve arithmetical skills such as long multiplication and long division There are several ways to these, so make sure you are familiar with and confident with at least one of them The box method for long multiplication is shown in the first example and the standard column method for long division is shown in the second example In this type of problem it is important to show your working as you will get marks for correct methods EXAMPLE A supermarket receives a delivery of 235 cases of tins of beans Each case contains 24 tins a How many tins of beans does the supermarket receive altogether? b 5% of the tins were damaged These were thrown away The supermarket knows that it sells, on average, 250 tins of beans a day How many days will the delivery of beans last before a new consignment is needed? a The problem is a long multiplication 235 × 24 The box method is shown × 200 30 20 4000 600 100 800 120 20 4000 600 00 + 800 20 20 5640 So the answer is 5640 tins b 10% of 5640 is 564, so 5% is 564 ÷ = 282 This leaves 5640 – 282 = 5358 tins to be sold There are 21 lots of 250 in 5358 (you should know that × 250 = 1000), so the beans will last for 21 days before another delivery is needed © HarperCollinsPublishers Limited 2007 Edex_Higher Math_01.qxd 16/03/06 08:15 Page CHAPTER 1: NUMBER EXAMPLE A party of 613 children and 59 adults are going on a day out to a theme park a How many coaches, each holding 53 people, will be needed? b One adult gets into the theme park free for every 15 children How many adults will have to pay to get in? a We read the problem and realise that we have to a division sum: the number of seats on a coach into the number of people This is (613 + 59) ÷ 53 = 672 ÷ 53 53 | 672 530 142 06 36 The answer is 12 remainder 36 So, there will be 12 full coaches and one coach with 36 people on So, they would have to book 13 coaches b This is also a division, 613 ÷ 15 This can be done quite easily if you know the 15 times table as × 15 = 60, so 40 × 15 = 600 This leaves a remainder of 13 So 40 adults get in free and 59 – 40 = 19 adults will have to pay EXERCISE 1A There are 48 cans of soup in a crate A supermarket had a delivery of 125 crates of soup a How many cans of soup were received? b The supermarket is having a promotion on soup If you buy five cans you get one free Each can costs 39p How much will it cost to get 32 cans of soup? Greystones Primary School has 12 classes, each of which has 24 pupils a How many pupils are there at Greystones Primary School? b The pupil–teacher ratio is 18 to That means there is one teacher for every 18 pupils How many teachers are at the school? Barnsley Football Club is organising travel for an away game 1300 adults and 500 juniors want to go Each coach holds 48 people and costs £320 to hire Tickets to the match are £18 for adults and £10 for juniors a How many coaches will be needed? b The club is charging adults £26 and juniors £14 for travel and a ticket How much profit does the club make out of the trip? First-class letters cost 30p to post Second-class letters cost 21p to post How much will it cost to send 75 first-class and 220 second-class letters? Kirsty collects small models of animals Each one costs 45p She saves enough to buy 23 models but when she goes to the shop she finds that the price has gone up to 55p How many can she buy now? Eunice wanted to save up for a mountain bike that costs £250 She baby-sits each week for hours for £2.75 an hour, and does a Saturday job that pays £27.50 She saves three-quarters of her weekly earnings How many weeks will it take her to save enough to buy the bike? © HarperCollinsPublishers Limited 2007 Edex_Higher Math_01.qxd 16/03/06 08:15 Page CHAPTER 1: NUMBER The magazine Teen Dance comes out every month In a newsagent the magazine costs £2.45 The annual (yearly) subscription for the magazine is £21 How much cheaper is each magazine bought on subscription? Paula buys a music centre She pays a deposit of 10% of the cash price and then 36 monthly payments of £12.50 In total she pays £495 How much was the cash price of the music centre? 1.2 Division by decimals This section will show you how to: ● divide by decimals by changing the problem so you divide by an integer Key words decimal places decimal point integer It is advisable to change the problem so that you divide by an integer rather than a decimal This is done by multiplying both numbers by 10 or 100, etc This will depend on the number of decimal places after the decimal point EXAMPLE Evaluate the following a 42 ÷ 0.2 b 19.8 ÷ 0.55 a The calculation is 42 ÷ 0.2 which can be rewritten as 420 ÷ In this case both values have been multiplied by 10 to make the divisor into a whole number This is then a straightforward division to which the answer is 210 Another way to view this is as a fraction problem 42 42 10 420 210 = × = = = 210 0.2 0.2 10 b 19.8 ÷ 0.55 = 198 ÷ 5.5 = 1980 ÷ 55 This then becomes a long division problem This has been solved by the method of repeated subtraction – – – 1980 1 00 880 440 440 440 20 × 55 × 55 × 55 36 × 55 EXERCISE 1B Evaluate each of these a 3.6 ÷ 0.2 b 56 ÷ 0.4 c 0.42 ÷ 0.3 d 8.4 ÷ 0.7 f 3.45 ÷ 0.5 g 83.7 ÷ 0.03 h 0.968 ÷ 0.08 i e 4.26 ÷ 0.2 7.56 ÷ 0.4 e 2.17 ÷ 3.5 Evaluate each of these a b 6.36 ÷ 0.53 c 0.936 ÷ 5.2 d 162 ÷ 0.36 f 67.2 ÷ 0.24 98.8 ÷ 0.26 g 0.468 ÷ 1.8 h 132 ữ 0.55 i 0.984 ữ 0.082 â HarperCollinsPublishers Limited 2007 Edex_Higher_ANSWERS.qxd 17/03/06 09:26 Page 622 ANSWERS: CHAPTER 18 Exercise 18C Exercise 18E a Positive correlation, reaction time increases with amount of alcohol drunk b Negative correlation, you drink less alcohol as you get older c No correlation, speed of cars on MI is not related to the temperature d Weak, positive correlation, older people generally have more money saved in the bank c ≈ 19 cm/s d ≈ 34 cm c Greta d ≈ 67 e ≈ 72 b Yes, usually (good correlation) b No correlation, so cannot draw a line of best fit a Exercise 18D a c a c a c a c a c a c a c cumulative frequency 1, 4, 10, 22, 25, 28, 30 54 secs, 16 secs cumulative frequency 1, 3, 5, 14, 31, 44, 47, 49, 50 56 secs, 17 secs d Pensioners, median closer to 60 secs cumulative frequency 12, 30, 63, 113, 176, 250, 314, 349, 360 605 pupils, 280 pupils d 46–47 schools cumulative frequency 2, 5, 10, 16, 22, 31, 39, 45, 50 20.5°C, 10°C cumulative frequency 9, 22, 45, 60, 71, 78, 80 56, 43 d 17.5% cumulative frequency 6, 16, 36, 64, 82, 93, 98, 100 225p, 110p cumulative frequency 8, 22, 47, 82, 96, 100 £1605, £85 d 13% Quick check a Perhaps around 0.6 c Very close to d b Very close to e Exercise 19A 21 37 163 329 –, ––, 10, –––, –––, ––––, –––– b c 25 –– 200 250 1000 2000 – e 1000 19 27 53 69 –––, –––, ––, –––, ––– b 40 200 200 25 200 200 No, it is weighted towards the side with numbers and 32 is too high, 20 of the 50 throws between 50 and 100 unlikely to be b Yes 1a d 2a ` c 3a 622 Students Pensioners Time (minutes) 10 b The students are much slower than the pensioners Although both distributions have the same inter-quartile range, the students’ median and upper quartile are minute, 35 seconds higher The fastest person to compete the calculations was a student, but so was the slowest a The resorts have similar median temperatures, but Resort B has a much wider temperature range The greatest extremes of temperature are recorded in Resort B b Resort A is probably a better choice as the weather seems more consistent a Men Women 10 15 20 25 30 35 Salary (£1000s) 40 45 b Both distributions have a similar inter-quartile range, and there is little difference between the upper quartile values Men have a wider range of salaries, but the higher men’s median and the fact that the men’s distribution is negatively skewed and the women’s distribution is positively skewed indicates that men are better paid than women b £1605, £85 c ii symmetric a Symmetric b Negatively skewed c Negatively skewed d Symmetric e Negatively skewed f Positively skewed g Negatively skewed h Positively skewed i Positively skewed j Symmetric 1 38 21 77 1987 a –, –, –––, ––, –––, –––– b 100 50 200 5000 a 0.346, 0.326, 0.294, 0.305, 0.303, 0.306 b 0.231, 0.168, 0.190, 0.16, 0.202, 0.201 c Red 0.5, white 0.3, blue 0.2 d e Red 10, white 6, blue b 20 7a 107 169 91 38 a Caryl, most throws b –––, –––, –––, ––– c Yes 275 550 550 275 a Method B b B c C d A e B f A g B h B 10 a Not likely b Impossible c Not likely d Certain e Impossible f 50–50 chance g 50–50 chance h Certain i Quite likely © HarperCollinsPublishers Limited 2007 Edex_Higher_ANSWERS.qxd 17/03/06 09:26 Page 623 ANSWERS: CHAPTER 19 Exercise 19B Exercise 19F a b 2, 12 1 1 –– –– c P(2) = ––, P(3) = 18, P(4) = 12, P(5) = –, P(6) = ––, 36 36 1 –– –– P(7) = –, P(8) = ––, P(9) = –, P(10) = 12, P(11) = 18, 36 P(12) = –– 36 1 5 –– –– –– d i 12 ii – iii – iv –– v 12 vi 18 36 11 –– a 12 b –– c – d – 36 –– –– a –– b 11 c 18 36 36 1 –– a 18 b – c – d e – 1 a – b – c – d – 4 4 –– a b i –– ii 13 iii – iv – 25 25 5 7 a – b – c – d – 8 8 a 16 b 32 c 1024 d 2n 1 –– a 12 b – c – a Yes b Yes c No d Yes e Yes f Yes Events a and f – 3 –– –– –– a i 10 ii 10 iii 10 b All except iii c Event iv 3 –– –– –– –– b i 10 ii 10 iii 10 iv 10 c All except iii d Event ii a – b – c All except ii d Outcomes overlap 8 –– 20 Not mutually exclusive events a i 0.25 ii 0.4 iii 0.7 b Events not mutually exclusive c Man/woman, American man/American woman d Man/woman Exercise 19C 25 1000 a 260 b 40 c 130 d 10 a 150 b 100 c 250 d 167 b 833 1050 a Each score expected 10 times b 3.5 c Find the average of the scores, which is 21 (1 + + + + + 6) divided by 400 Exercise 19G a a a a a a a a 1 – b – c – 4 12 25 –– –– b 13 c i ––– ii ––– 13 169 169 1 – b – d i – ii – iii – 6 12 – b i –– ii –– 25 25 – b – c – 8 0.14 b 0.41 c 0.09 – –– –– c i – ii 15 iii 15 b c 1 23 b 20% c –– d 480 25 10 b c 14% d 15% –– c i – ii 16 iii – 4 51 16 b 16 c 73 d –– 73 c – The greenhouse sunflowers are bigger on average The garden sunflowers have a more consistent size (smaller range) a 40% b 45% c No as you don’t know how much the people who get over £350 actually earn a a b a b a b × × 10 3 Exercise 19D e 15 days Exercise 19H 4 a – b – 9 a ––– b ––– 169 169 1 a – b – ––– 216 –– a –– b 12 25 25 a 0.08 b 0.32 a 0.336 b 0.452 c 0.48 c 0.024 Exercise 19I Exercise 19E a a a a a a a a e – – –– 13 –– 11 – – – – – b c b c –– –– b 13 c 13 b –– c –– 11 11 11 11 –– –– b – c 15 d 15 e – 0.6 b 120 0.8 b 0.2 0.75 b 0.6 c 0.5 d 0.6 i Cannot add P(red) and P(1) as events are not mutually exclusive ii 0.75 17 –– a 20 b – c – 10 Probability cannot exceed 1, and probabilities cannot be summed in this way as events are not mutually exclusive a a a a a b c 125 ––– 216 –– 16 91 (0.579) b ––– (0.421) 216 15 –– b 16 0.378 b 0.162 c 0.012 d 0.988 –– –– b –– c 16 25 25 25 91 ––– i ––– (0.005) ii 125 (0.579) iii ––– (0.421) 216 216 216 625 671 ii –––– (0.482) iii –––– (0.518) i –––– (0.00077) 1296 1296 1296 3125 4651 ii –––– (0.402) iii –––– (0.598) i –––– (0.00013) 7776 7776 7776 5n 5n ii n iii – n n 6 32 b ––– (0.004) a ––– (0.132) 243 243 119 b ––– c ––– a – 120 120 d i 242 c ––– (0.996) 243 Exercise 19J 27 189 a –––– b –––– 1000 1000 a –––– (0.00077) 1296 © HarperCollinsPublishers Limited 2007 441 343 c –––– d –––– 1000 1000 625 ––– b –––– (0.482) c 125 1296 324 623 Edex_Higher_ANSWERS.qxd 17/03/06 09:26 Page 624 ANSWERS: CHAPTER 19 a a a a a a a 7 – b –– c –– d – e – 18 18 9 0.154 b 0.456 0.3024 b 0.4404 c 0.7428 0.9 b 0.6 c 0.54 d 0.216 0.6 b 0.6 c 0.432 d Independent events 1 – b – c –– d –– 9 27 27 0.126 b 0.4 c 0.42 d 0.054 Exercise 19K 1 a –– 60 2a – b 50 b c i – ii – iii Quick check a x a 17, 20, 23b 28, 36, 45 d 49, 64, 81 a b c 15, 10, e c a 2(x + 3) b x(x – 1) 2 b 2x – 5x – a x = – 2y b x = + 3y c x – 4x + c x = 4y – Exercise 20A f 19x b 20 5x + g 23x c 20 7x – 3x + 2y d 13x – 15 h i x2y + e 4x 3x – x b 11x 20 7x 20 3x – 2y –7x – h 15 c e 6x2 + 5x + g f h a 2x2 + x 15 b 2x – x–3 i x y j 2x d 2xy g c h f i 624 2x – 12x + 18 75 x2 – 5x + 48 j 2x e 5x 2x – 4x + 17x + 10 f 13x + 10 i x+3 a 3, –1.5 x–1 i x–1 a 2x + x+1 d x–1 c x j b 4, –1.25 –8x + j 10 2x + b x+3 2x + e 4x – 3x2 16 g 3x2 – 5x – 10 d 2x2 – 6y2 c 3, –2.5 c d 0, 2x – 3x – Exercise 20B xy2 – 4y a (5, –1) b (4, 1) c (8, –1) a (1, 2) and (–2, –1) b x = –4, y = 1; x = –2, y = a (3, 4) and (4, 3) b (0, 3) and (–3, 0) c (3, 2) and (–2, 3) a (2, 5) and (–2, –3) b (–1, –2) and (4, 3) c (3, 3) and (1, –1) a (2, 4) b (1, 0) c The line is a tangent to the curve x–1 g –8x + x–1 i j 10 a b c d e 0.75 f x x2 – 2x 3xy 2xy a b c d e 10 14 f x–1 d b 12x – 23 j 10 a h c 2x(5x + 1) a x + 8x + 12 5x a 5 ii – b i –– b ii –– 3a i – 8 12 12 15 4 a i –– ii –– b i –– ii –– 13 13 91 13 – –– –– – 5a i ii 15 b 15 c d Both events are independent 21 7 a ––– b –– c –– d –– 120 40 40 24 2 – – – – 8a b c d 9 a 0.000495 b 0.00198 c 0.000018 d 0.00024 10 a 0.54 b 0.38 c 0.08 d 1 1 11 a RFC, FRC, CFC, CRC b – c – d – e 3 Probability is the same regardless of which day he chooses Exercise 20C a 21, 34: add previous terms b 49, 64: next square number c 47, 76: add previous terms 15, 21, 28, 36 61, 91, 127 –, –, –, –, – a 6, 10, 15, 21, 28 b It is the sums of the natural numbers, or the numbers in Pascal’s Triangle a 2, 6, 24, 720 b 69! Exercise 20D a c e g i k 13, 15, 2n + 33, 38, 5n + 20, 23, 3n + 21, 25, 4n – 17, 20, 3n – 24, 28, 4n + © HarperCollinsPublishers Limited 2007 b d f h j l 25, 29, 4n + 32, 38, 6n – 37, 44, 7n – 23, 27, 4n – 42, 52, 10n – 29, 34, 5n – Edex_Higher_ANSWERS.qxd 17/03/06 09:26 Page 625 ANSWERS: CHAPTER 20 a 3n + 1, 151 b 2n + 5, 105 c 5n – 2, 248 d 4n – 3, 197 e 8n – 6, 394 f n + 4, 54 g 5n + 1, 251 h 8n – 5, 395 i 3n – 2, 148 j 3n + 18, 168 k 7n + 5, 355 l 8n – 7, 393 a 33rd b 30th c 100th = 499 a i 4n + ii 401 iii 101, 25th b i 2n + ii 201 iii 99 or 101, 49th and 50th c i 3n + ii 301 iii 100, 33rd d i 2n + ii 206 iii 100, 47th e i 4n + ii 405 iii 101, 24th f i 5n + ii 501 iii 101, 20th g i 3n – ii 297 iii 99, 34th h i 6n – ii 596 iii 98, 17th i i 8n – ii 799 iii 103, 13th j i 2n + 23 ii 223 iii 99 or 101, 38th and 39th 2n + · 5a b Getting closer to – (0.6) 3n + c i 0.667 774 (6dp) ii 0.666 778 (6dp) d 0.666 678 (6dp), 0.666 667 (6dp) 4n – 6a b Getting closer to – (0.8) 5n + c i 0.796 407 (6dp) ii 0.799 640 (6dp) d 7a 8a c 9a 0.799 964 (6dp), 0.799 9996 (7dp) 64, 128, 256, 512, 1024 b i 2n – ii 2n + iii × 2n The number of zeros equals the power b i 10n – ii × 10n Even, b Odd, + Odd Even × Odd Even Odd Even Odd Odd Odd Odd Even Even Even Even ii 2n2 iii n2 – 10 a 36, 49, 64, 81, 100 b i n2 + 11 + + + = 16 = , + + + + = 25 = 52 12 a 28, 36, 45, 55, 66 b i 210 ii 5050 c You get the square numbers 13 a Even b Odd c Odd d Odd e Odd f Odd g Even h Odd i Odd 14 a Odd or even b Odd or even c Odd or even d Odd e Odd or even f Even c 97 c 121 a i 35, 48 ii n2 – b i 38, 51 ii n2 + c i 2 39, 52 ii n + d i 34, 47 ii n – e i 35, 46 ii n2 + 10 a i 37, 50 ii (n + 1)2 + b i 35, 48 ii (n + 1)2 –1 c i 41, 54 ii (n + 1) + d i 50, 65 ii (n + 2)2 + e i 48, 63 ii (n + 2)2 – a i n +4 ii 2504 b i 3n + ii 152 c i (n + 1)2 –1 ii 2600 d i n(n + 4) ii 2700 e i n2 + ii 2502 f i 5n – ii 246 a 2n2 – 3n + b 3n2 + 2n – c – n2 + – n + 2 Exercise 20G b + 5c –8y b(q + p) q–p u √« – a) (1 «««« –x a(q – p) q+p π(2h + k) + st 2+s P 10 2R – R–1 11 a 12 100A 100 + RY 13 a b = π + 2k A b Ra a–R + st 2+s √ 2A π+ √« – 1) (««««« k b a= Rb b–R + 2y 2W + 2zy c x= y–1 z+y d Same formula as in a 15 a Cannot factorise the expression 3V 3V b c Yes, 5π r (2r + 3h) 14 a ––– √ 2W – 2zy z+y Exercise 21B P = 4l C = πd A = – bh A = πr2 V = l3 V = πr2h Exercise 21A P = 2a + 2b P = 4x P = 4x + 4y P = 5x + 2y + 2z P = 2h + (2 + π)r b 332 d 50th diagram d 49th set Quick check 18 b 4n + c 12 i 24 ii 5n – iii 224 b 25 i 20 cm ii (3n + 2) cm iii 152 cm i 20 ii 162 b 79.8 km i 14 ii 3n + iii 41 b 66 i ii n iii 18 b 20 2n b i 100 × 2n – ml ii 1600 ml 16 x = Exercise 20E b 4n – b 2n + a a a a a a a Exercise 20F Even Even 10 P=a+b+c+d P = p + 2q P = a + 3b P = 2πr P = 2l + πd A = a2 + ab A = bh A = πr2 1 A = – bh + – bw 2 A = – πd + – dh 2 10 A = – bh A = – (a + b)h A = 2ad – a2 A = 2rh + πr2 1 A = –πD2 + – (b + D)w Exercise 21C V = 6p3 V = – bhw V = aqt + bpt – apt © HarperCollinsPublishers Limited 2007 V = πr2h V = – bhl V = abl + adl + 2cdl 625 Edex_Higher_ANSWERS.qxd 17/03/06 09:26 Page 626 ANSWERS: CHAPTER 21 Exercise 21D a h o v a h o v A A V A C I C C b i p w b i p w L L A A I C I C c j q x c j q x L V V A C I C C d k r y d k r y A A A V I C C C e V l L s A z V e C l C s I f V m V t A f I m C t C a C, L b I c C, V d C, L e I f I g C, V h C, V i C, V j C, V k C, L l I m C, V a b 2, c d 2, Inconsistent a A is F2, B is F4, C is F3 b F1 is the total length of the curved edges g V n A u L g C n C u C Quick check a 25 b c 27 d a 48 b – Exercise 22A a a a a a a a 15 75 150 22.5 175 miles £66.50 44 b b b b b b b 6 12 hours 175 kg 84 m2 a a a a a 100 27 56 192 25.6 b b b b b 10 10 1.69 2.25 Quick check a 6370 b 6400 a 2.4 b 47.3 c 6000 b d f h j b d f h j l n p r t 115 р 120 Ͻ 125 49.5 р 50 Ͻ 50.5 16.75 р 16.8 Ͻ 16.85 14 450 р 14 500 Ͻ 14 549 52.5 р 55 Ͻ 57.5 16.5 р 17 Ͻ 17.5 237.5 р 238 Ͻ 238.5 25.75 р 25.8 Ͻ 25.85 86.5 р 87 Ͻ 87.5 2.185 р 2.19 Ͻ 2.195 24.5 р 25 Ͻ 25.5 595 р 600 Ͻ 605 995 р 1000 Ͻ 1050 7.035 р 7.04 Ͻ 7.045 6.995 р 7.00 Ͻ 7.005 Exercise 23B a 7.5, 8.5 d 84.5, 85.5 626 b b b b b 225 10 atm mm 40 m/s 400 miles Tm = 12 Wx = 60 Q(5 – t) = 16 Mt = 36 W√T = 24 gp = 1800 td = 24 ds2 = 432 p√h = 7.2 W√F = 0.5 a a a a a a a a c a 20 –3.2 4.8 £15 °C 1.92 km 2.4 atm t/h b b b b b b b b b b 2.5 100 36 12 km m/s 100 m 0.58 t/h Exercise 23C Exercise 23A a 6.5 р Ͻ 7.5 c 3350 р 3400 Ͻ 3450 e 5.50 р Ͻ 6.49 g 15.5 р 16 Ͻ 16.5 i 54.5 р 55 Ͻ 55.5 a 5.5 р Ͻ 6.5 c 31.5 р 32 Ͻ 32.5 e 7.25 р 7.3 Ͻ 7.35 g 3.35 р 3.4 Ͻ 3.45 i 4.225 р 4.23 Ͻ 4.235 k 12.665 р 12.67 Ͻ 12.675 m 35 р 40 Ͻ 45 o 25 р 30 Ͻ 35 q 3.95 р 4.0 Ͻ 4.05 s 11.95 р 12.0 Ͻ 12.05 £50 3.2 °C 388.8 g 2J £78 g 0.055, 0.065 h 250 g, 350 g i 0.65, 0.75 j 365.5, 366.5 k 165, 175 l 205, 215 a