Karen Morrison and Lucille Dunne Cambridge IGCSE® Mathematics Extended Practice Book cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107672727 © Cambridge University Press 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library ISBN-13 978-1-107-67272-7 Paperback Cover image: Seamus Ditmeyer/Alamy Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. IGCSE® is the registered trademark of Cambridge International Examinations. Contents Introduction v Unit 1 Chapter 1: Reviewing number concepts 1.1 Different types of numbers 1.2 Multiples and factors 1.3 Prime numbers 1.4 Powers and roots 1.5 Working with directed numbers 1.6 Order of operations 1.7 Rounding numbers 1 1 2 3 3 4 4 5 Chapter 2: Making sense of algebra 2.1 Using letters to represent unknown values 2.2 Substitution 2.3 Simplifying expressions 2.4 Working with brackets 2.5 Indices 7 7 8 8 9 9 Chapter 3: Lines, angles and shapes 3.1 Lines and angles 3.2 Triangles 3.3 Quadrilaterals 3.4 Polygons 3.5 Circles 3.6 Construction 12 12 15 16 18 19 19 Chapter 4: Collecting, organising and displaying data 4.1 Collecting and classifying data 4.2 Organising data 4.3 Using charts to display data 22 22 23 24 Chapter 7: Perimeter, area and volume 7.1 Perimeter and area in two dimensions 7.2 Three-dimensional objects 7.3 Surface areas and volumes of solids 39 39 43 44 Chapter 8: Introduction to probability 8.1 Basic probability 8.2 Theoretical probability 8.3 The probability that an event does not happen 8.4 Possibility diagrams 8.5 Combining independent and mutually exclusive events 48 48 49 Chapter 12: Averages and measures of spread 12.1 Different types of average 12.2 Making comparisons using averages and ranges 12.3 Calculating averages and ranges for frequency data 12.4 Calculating averages and ranges for grouped continuous data 12.5 Percentiles and quartiles 75 75 Unit 2 Chapter 5: Fractions 5.1 Equivalent fractions 5.2 Operations on fractions 5.3 Percentages 5.4 Standard form 5.5 Estimation 29 29 29 30 32 33 Chapter 6: Equations and transforming formulae 6.1 Further expansions of brackets 6.2 Solving linear equations 6.3 Factorising algebraic expressions 6.4 Transformation of a formula 35 35 35 36 37 50 51 52 Unit 3 Chapter 9: Sequences and sets 9.1 Sequences 9.2 Rational and irrational numbers 9.3 Sets 54 54 55 56 Chapter 10: Straight lines and quadratic equations 10.1 Straight lines 10.2 Quadratic expressions 59 59 61 Chapter 11: Pythagoras’ theorem and similar shapes 11.1 Pythagoras’ theorem 11.2 Understanding similar triangles 11.3 Understanding similar shapes 11.4 Understanding congruence 66 66 68 69 70 Contents 76 77 78 79 iii Unit 4 Chapter 13: Understanding measurement 13.1 Understanding units 13.2 Time 13.3 Upper and lower bounds 13.4 Conversion graphs 13.5 More money Chapter 14: Further solving of equations and inequalities 14.1 Simultaneous linear equations 14.2 Linear inequalities 14.3 Regions in a plane 14.4 Linear programming 14.5 Completing the square 14.6 Quadratic formula 14.7 Factorising quadratics where the coefficient of x2 is not 1 14.8 Algebraic fractions 81 81 82 84 85 86 89 89 91 92 93 94 95 Chapter 15: Scale drawings, bearings and trigonometry 15.1 Scale drawings 15.2 Bearings 15.3 Understanding the tangent, cosine and sine ratios 15.4 Solving problems using trigonometry 15.5 Angles between 0° and 180° 15.6 The sine and cosine rules 15.7 Area of a triangle 15.8 Trigonometry in three dimensions 99 99 100 101 105 105 105 108 108 Chapter 16: Scatter diagrams and correlation 16.1 Introduction to bivariate data 111 111 Chapter 19: Symmetry and loci 19.1 Symmetry in two dimensions 19.2 Symmetry in three dimensions 19.3 Symmetry properties of circles 19.4 Angle relationships in circles 19.5 Locus 126 126 127 128 129 131 96 96 Unit 5 Chapter 17: Managing money 17.1 Earning money 17.2 Borrowing and investing money 17.3 Buying and selling 114 114 115 116 Chapter 18: Curved graphs 18.1 Plotting quadratic graphs (the parabola) 18.2 Plotting reciprocal graphs (the hyperbola) 18.3 Using graphs to solve quadratic equations 18.4 Using graphs to solve simultaneous linear and non-linear equations 18.5 Other non-linear graphs 18.6 Finding the gradient of a curve 119 119 121 122 122 123 124 Chapter 20: Histograms and frequency distribution diagrams 135 20.1 Histograms 135 20.2 Cumulative frequency 137 Unit 6 Chapter 21: Ratio, rate and proportion 21.1 Working with ratio 21.2 Ratio and scale 21.3 Rates 21.4 Kinematic graphs 21.5 Proportion 21.6 Direct and inverse proportion in algebraic terms 21.7 Increasing and decreasing amounts by a given ratio Chapter 22: More equations, formulae and functions 22.1 Setting up equations to solve problems 22.2 Using and transforming formulae 22.3 Functions and function notation 139 139 140 141 141 144 145 146 Chapter 23: Transformations and matrices 23.1 Simple plane transformations 23.2 Vectors 23.3 Further transformations 23.4 Matrices and matrix transformation 23.5 Matrices and transformations 153 153 158 161 163 164 Chapter 24: Probability using tree diagrams 24.1 Using tree diagrams to show outcomes 24.2 Calculating probability from tree diagrams 169 169 169 149 149 150 151 Answers Example practice papers can be found online, visit education.cambridge.org/extendedpracticebook iv Contents 171 Introduction This highly illustrated practice book has been written by experienced teachers to help students revise the Cambridge IGCSE Mathematics (0580) Extended syllabus. Packed full of exercises, the only narrative consists of helpful bulleted lists of key reminders and useful hints in the margins for students needing more support. There is plenty of practice offered via ‘drill’ exercises throughout each chapter. These consist of progressive and repetitive questions that allow the student to practise methods applicable to each subtopic. At the end of each chapter there are ‘Mixed exercises’ that bring together all the subtopics of a chapter in such a way that students have to decide for themselves what methods to use. The answers to all of these questions are supplied at the back of the book. This encourages students to assess their progress as they go along, choosing to do more or less practice as required. The book has been written with a clear progression from start to finish, with some later chapters requiring knowledge learned in earlier chapters. There are useful signposts throughout that link the content of the chapters, allowing the individual to follow their own course through the book: where the content in one chapter might require knowledge from a previous chapter, a comment is included in a ‘Rewind’ box; and where content will be practised in more detail later on, a comment is included in a ‘Fast forward’ box. Examples of both are included below: FAST FORWARD REWIND You learned how to plot lines from equations in chapter 10. Remember ‘coefficient’ is the number in the term. Tip It is essential that you remember to work out both unknowns. Every pair of simultaneous linear equations will have a pair of solutions. You will learn much more about sets in chapter 9. For now, just think of a set as a list of numbers or other items that are often placed inside curly brackets. Other helpful guides in the margin of the book are as follows: Hints: these are general comments to remind students of important or key information that is useful when tackling an exercise, or simply useful to know. They often provide extra information or support in potentially tricky topics. Tip: these are tips that relate to good practice in examinations, and also just generally in mathematics! They cover common pitfalls based on the authors’ experiences of their students, and give students things to be wary of or to remember in order to score marks in the exam. The Extended Practice Book mirrors the chapters and subtopics of the Cambridge IGCSE Mathematics Core and Extended Coursebook written by Karen Morrison and Nick Hamshaw (9781107606272). However, this book has been written such that it can be used without the coursebook; it can be used as a revision tool by any student regardless of what coursebook they are using. Various aspects of the Core syllabus are also revised for complete coverage. Also in the Cambridge IGCSE Mathematics series: Cambridge IGCSE Mathematics Core and Extended Coursebook (9781107606272) Cambridge IGCSE Mathematics Core Practice Book (9781107609884) Cambridge IGCSE Mathematics Teacher’s Resource CD-ROM (9781107627529) Introduction v 1 Reviewing number concepts 1.1 Different types of numbers a b Tip a b b≠ Exercise 1.1 1 Number Natural Integer Prime Fraction −57 03 − 14 2 7 3 512 2 (a) (b) (c) (d) > 1 3 < Unit 1: Number 1 1 Reviewing number concepts 1.2 Multiples and factors To find the LCM of a set of numbers, you can list the multiples of each number until you find the first multiple that is in the lists for all of the numbers in the set. FAST FORWARD You will use LCM again when you work with fractions to find the lowest common denominator of two or more fractions. See chapter 5. ! You need to work out whether to use LCM or HCF to find the answers. Problems involving LCM usually include repeating events. Problems involving HCF usually involve splitting things into smaller pieces or arranging things in equal groups or rows. Exercise 1.2 A 1 (a) (e) (b) (f) (c) (g) (d) (h) (a) (e) (b) (f) (c) (g) (d) (h) 2 Exercise 1.2 B 1 2 3 4 × 5 (a) (b) 2 Unit 1: Number × 1 Reviewing number concepts 1.3 Prime numbers not Exercise 1.3 You can use a tree diagram or division to find the prime factors of a composite whole number. 1 (a) (b) (c) 2 (a) (e) (b) (f) (c) (g) (d) (h) (a) (e) (b) (f) (c) (g) (d) (h) 3 1.4 Powers and roots n×n n n n×n×n n3 3 FAST FORWARD Powers greater than 3 are dealt with in chapter 2. See topic 2.5 indices. ! n Exercise 1.4 1 2 (a) 9 36 4 (e) (i) (m) 3 16 3 (b) (f) ( 25 ) 2 (g) 27 − 1 (j) 100 ÷ 4 × (n) 1 4 3 (c) 9 16 2 (o) (d) 64 + 36 9 (h) 169 − 144 (l) 16 × 3 27 16 (k) ( 13 ) + 64 + 36 1+ 3 9 16 1 − 3 −125 3 (a) (b) Unit 1: Number 3 1 Reviewing number concepts 1.5 Working with directed numbers − + Exercise 1.5 1 2 (a) Draw a number line to help you. (b) − (c) − − − − 3 (a) (b) (c) (d) (e) − − − − 1.6 Order of operations × Remember the order of operations using BODMAS: Exercise 1.6 Brackets Of Divide Multiply Add Subtract Tip FAST FORWARD The next section will remind you of the rules for rounding numbers. ! 4 Unit 1: Number 1 (a) + × (d) + (b) × (e) + × (c) × − (f) × − ÷ ÷ + 1 Reviewing number concepts (g) 1.453 + 5.34 3.315 4 03 11.5 (k) 2 9 − 1.43 12.6 1 98 − (n) 8 3 4 62 76 32 (h) 5 27 1.4 × 1.35 89 (m) 8 9 − 10.4 (j) (p) − (s) 4 072 8.2 − 4.09 (q) 3 (t) 6 8 + ( ) (v) 6 1 + 2.1 28 16 2 − 1. 4 1. 2 − 69 93 (w) 6.4 (1.2 .22 + 1 92 )2 (i) (l) (o) (r) 6 54 − 1 08 23 0.23 4.26 1.32 3.43 ( − ) 2 16.8 − 1 01 93 ( (u) 4.33 + 1.2 + 1 6 5 (x) ) 2 (4 8 − 916 ) × 4 3 1.7 Rounding numbers ≥ ≤ Exercise 1.7 FAST FORWARD 1 (i) (ii) (iii) Rounding is very useful when you have to estimate an answer. You will deal with this in more detail in chapter 5. ! (a) (d) (g) (b) (e) (h) (c) (f) (a) (b) (c) (d) (a) (e) (b) (f) (c) (g) (d) (h) 2 3 Unit 1: Number 5 1 Reviewing number concepts Mixed exercise 1 − 3 − 4 3 1 2 2 (a) (b) (c) (e) 3 (a) (b) (c) 4 5 (a) (b) (c) 30 + 10 30 (d) 6 (a) 100 ÷ 4 (b) 100 ÷ 4 (c) ( 64 ) 3 3 (d) 7 (a) 5.4 × 12.2 4. 1 2 (b) 12.2 3 92 (d) 3.8 × 12.6 4 35 (e) (a) (b) 2. 8 × 4. 2 2 3.32 × 6.22 12.65 (c) 2 04 + 1.7 × 4.3 ( (f) 2.55 − 3.1 + 05 5 ) 2 8 9 (a) (b) (c) 6 Unit 1: Number (c) (d) 3 + 2 Making sense of algebra 2.1 Using letters to represent unknown values x y + − Exercise 2.1 x, 1 Tip (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) x x 1 2 p 2 (a) (b) (c) H x 3 (a) (b) Unit 1: Algebra 7 2 Making sense of algebra 2.2 Substitution x=− x × − =− Exercise 2.2 1 A = bh A 1 (a) (b) (c) (d) REWIND Remember that the BODMAS rules always apply in these calculations. ! Take special care when substituting negative numbers. If you replace x with −3 in the expression 4x, you will obtain 4 × −3 = −12, but in the expression −4x, you will obtain −4 × −3 = 12. b 2 h 2 xy 3 a x x b m 4 y y c n a b c m3 n3 m2 mn n x 5 1 2 x 1 2 x (a) (b) 2.3 Simplifying expressions Remember, like terms must have exactly the same variables with exactly the same indices. So 3x and 2x are like terms but 3x2 and 2x are not like terms. Exercise 2.3 1 (a) 3x 2 + 6 x 8 Remember, multiplication can be done in any order so, although it is better to put variable letters in a term in alphabetical order, ab = ba. So, 3ab + 2ba can be simplified to 5ab. Remember, x × x = x2 y × y × y = y3 x÷x=1 8 Unit 1: Algebra 3 (b) x 2 y + 3x ac + 3ba 3x 2 y 2 yx (c) 2ab − 44ac (d) x + 2xx 4 + 3x 3x 2 y 3x − 1 (g) −2 xy × −3 y (h) −2 xy × 2 x 2 (k) 33abc 11ca y 2y (o) × x x x 2 (s) × 4 3y 2 (l) 45mn 20n xy y (p) × 2 x 3 x 9x (t) × 5 2 2 (e) −6m × 5n (f) 3xy 2 x (i) 12ab ÷ 3a (j) 12 x ÷ 48 xy 80 xy 2 (m) 12 x 2 y (q) 5a × 3a 4 3 (n) −36 x −12 xy −2 y (r) 7 × 5 2 Making sense of algebra 2.4 Working with brackets a b + c = ab + ac Exercise 2.4 Remember the rules for multiplying integers: 1 (a) (e) (h) +×+=+ −×−=+ +×−=− If the quantity in front of a bracket is negative, the signs of the terms inside the bracket will change when the brackets are expanded. (b) ( y 3) x x) (f) 2 ( x 1) (1 x ) − x ( − ) + 2( + ) − 4 2x ( x 2) (x )( )( (c) ( x ) 3x (g) x ( x 2 − 2x 2 x 1) (d) −2 x − ( − ) 2 (a) 2 x ( 12 x + 14 ) (b) −3 (d) ( x + y ) − ( 12 x 1 y) (e) ( − y ) − 2 x ( y − 2 x ) (c) −2 x − − (f) 2x (2x − 2) x ( x 2) (3 − 2x ) ( 5 ) (g) x (1 x ) x (2 x − 5) 2 x (1 3x ) 2.5 Indices x×x x 1 index laws: x m x n = x m+ n xm xn = xm n ( )n x mn x0 = 1 x −m = m y × y × y × y. 4 Exercise 2.5 A Tip xn y (a) x 4 y × y2 x6 x 4 y5 (b) 2 x 2 y 4 3x 3 y 2 xy 4 (c) 2 x 5 y 4 2 xy 3 2 x 2 y 5 3x 2 y 3 (d) x3 y7 x2 y8 × 3 xy 4 x y (e) 2 x 7 y 2 10 x 8 y 4 × 4 x 3 y 7 2x 3 y 2 (f) x9 y6 x3 y2 ÷ x 4 y2 x5 y (g) 10 x 5 y 2 3x 3 y ÷ 9 x 6 y 6 5x 7 y 4 (h) 7 y 3 x 2 5x 6 y 2 ÷ 5 y5 x 4 7x5 y3 (i) (j) (2 x y ) × ( x y ) ( y x ) 3(x y ) 1 x m (x ) = ( x ) 1 n m n m 4 3 2 2 3 3 4 4 2 2 2 3 2 5 (k) x 4 × x 2 y y 2 (l) (x y ) × (x y ) (x y ) (5x y ) ÷ 2xy 2 5 3 3 3 7 2 4x y 6 3 3 4 2 3 3 5x 2 y 4 Unit 1: Algebra 2 9 2 Making sense of algebra 2 Tip (a) x−4 x −m = 1 xm 5x −4 = y = x2 y x −2 x5 y 4 x −33 y 2 5 x4 (x ) ÷ −1 (d) x3 y 2 4 −10 y2 (e) x 2 ÷ 3 − x (y ) y −3 (2 x y ) (y x ) −33 x −4 y 3 x 7 y −5 × x 2 y −1 x −4 y 3 (b) (c) 2 1 1 1 (e) (64 x ) (i) ( x ) × xx (a) (x ) × xx (d) (x y ) 6 1 2 8 1 2 2 3 (b) x 3 × x 5 (c) (f) (8 x y ) (g) (j) (x y ) × (x (b) (x ) × xx (e) x2 y3 2 1 ÷ 3 4 x2 x y (b) −24 (−2)4 9 6 3 1 3 1 3 8 y 10 ) 1 2 ( 2 1 1 2 2 ) ( ) 2 x −2 y 6 x4 y 1 (f) 5 3 × x y 2 xy 3 −2 −4 3 (a) x 4 × x 4 3 1 x2 1 x3 (d) ( ) 1 1 2 × 1 3 1 3 x6 (h) 2 y xy 8 (k) xy 3 (x ) 1 x2 y2 xy 4 1 2 4 Tip 1 2 2 3 2 3 2 3 1 3 4 1 3 5 5 2 1 2 x3 y3 × xy 2 (c) 1 2 1 y3 2 1 ( x y ) × (x y ) 1 2 2 1 2 3 4 4 1 3 1 1 xy 4 4 4 (f) x 3 × 3 2 y2 x y 3 Exercise 2.5 B 1 (a) ( 4 (e) 256 (i) )2 − 14 ( ) 8 27 )( − 13 (f) 125 − 43 ( ) − 12 (j) 8 18 (c) 63 ( 3)4 (d) 8 3 (g) () (h) 5 1 −2 4 1 () 2 1 −3 8 2 (a) (b) (c) (d) 3 3 4 x 3 (a) (e) 10 Unit 1: Algebra 6 = x = x (b) 3x = x (f) = 1 25 (c) 3x = 811 (g) 3 x − = (d) (h) x x+ = 2 Making sense of algebra Mixed exercise x 1 (a) (b) (c) (d) (e) (f) (g) (h) x 2 (a) x = x (b) x = − (c) x = a=− b= b (c − a ) (b) b a 3 −2a + 3b (a) 2ab 3 2 ( − 1) (d) c − a (b − 1) 1 3 c= a b2 c a2 (c) (e) a 3b2 − 2a 2 + a 4b2 ac 3 4 (a) 3a 4b + 6a 3b (b) x 2 + 4x 4x x 2 (d) 2 ( (e) 16 x 2 y ÷ 4 y 2 x 3) − ( x 4 ) 2 x 2 (c) −2a 2b(2a 2 − 3b2 ) 10 x 2 5xy (f) 2x 5 (a) 2 ( 4 x 3) 3( x + 1) (c) x ( x + ) + 3x − 3( x − (b) 3 (2 x + 3) 2 ( 4 3x ) (d) x 2 ( x + 3) 2 x 3 ( x − 5) ) 6 (b) 5x 2 × 3x7 x (d) (2 xy 2 ) 4x3 (e) 5 y 4 (2xy ) (3x y ) (h) (x y ) 2 ( xy ) (a) (125x y ) (b) (x y ) −2 2 3 (x ) (x ) (f) (x y ) × (c) (x ) × (x y) 3 (g) 3 (c) 3 5 7 (a) 15x 2 18 x − 4 −3 x −3 y 3 ÷ 2 −1 x y 2 4 8 2 3 (x y ) (xy ) 2 4 2 3 3 2 7 3 2 1 3 ( ) 1 4 6 3 1 2 1 × 1 x2 y2 x3 y2 2 y 3 1 2 −4 5 2 1 2 x2 y x3 y3 ÷ 3 5 (d) 1 3 x y 2 xy Unit 1: Algebra 11 3 Lines, angles and shapes 3.1 Lines and angles < ° ° > > ° ° < < ° °. ° ° °. °. °. °, Exercise 3.1 A 1 10 11 12 1 9 (a) (i) 2 3 8 7 (i) 2 4 6 (ii) (iii) (b) (c) 5 1 4 (ii) (d) ° 2 3 4 (a) 12 Unit 1: Shape, space and measures ° (b) x° (c) x° 3 Lines, angles and shapes 5 (a) (d) Tip (b) (e) ° x° (c) x° (f) +x ° ° x° Exercise 3.1 B MN 1 PQ a. Q M Remember, give reasons for statements. Use these abbreviations: a O N 37° P Supp ∠s ∠s on line ˆ = 85° QOR R Comp ∠s x 2 (a) ∠s round point (b) A Vertically opposite ∠s E 27° D 3x + 25° O x 112° x 6x C 5x + 35° P B Exercise 3.1 C x 1 Remember, give reasons for statements. Use these abbreviations to refer to types of angles: Alt ∠s M x Corr ∠s N O y P R M E y B 57° 72° S 95° Q Co-int ∠s y (b) (a) P T C x F 81° G N Q H (c) A K Bx I C 65° D (d) E F 108° G y J x xx x x x x x H Unit 1: Shape, space and measures 13 3 Lines, angles and shapes x 2 (a) (b) E A F 112° B M x G C S N 57° Q T D H (c) A 30° (d) A B C x C D D E 60° (e) F A x C B 14 Unit 1: Shape, space and measures D 50° G 45° E F B E 126° 108° x F 3 Lines, angles and shapes 3.2 Triangles ° °. Exercise 3.2 REWIND You may also need to apply the angle relationships for points, lines and parallel lines to find the missing angles in triangles. ! 1 (a) 29° 48° (b) (c) 113° 37° 31° a b 62° (d) c d 53° (e) e (f) 32° x 61° (g) 21° 54° (h) (i) x x y x 30° 102° 57° y Unit 1: Shape, space and measures 15 3 Lines, angles and shapes x 2 (a) (b) A A x B B 2x C C D (c) (d) M O M O N x x N 18° P P Tip 3 ABC ∠A ∠B x ∠C B C x. 3.3 Quadrilaterals ° ° ° ° °. 16 Unit 1: Shape, space and measures 3 Lines, angles and shapes Exercise 3.3 1 REWIND The angle relationships for parallel lines will apply when aquadrilateral has parallel sides. ! (a) (b) (c) (d) (e) (f) (g) (h) (i) ° 2 a f e d g f b 63° a b c ed c 3 (a) A 79° (b) A B (c) B 32° 38° x D D D 58° (d) M B C E C N x A O (e) (f) 3x F C N M c d 68° O x Q 65° R e 2x P PQRS 4 (a) ∠R 5 ∠P ∠S ∠Q (b) ∠R PMNO PN (a) ∠MNP (b) ∠MNO ∠R Q a b P ∠Q. (c) ∠Q. MO Q ∠QMN ∠PNO (c) ∠PON. Unit 1: Shape, space and measures 17 3 Lines, angles and shapes 3.4 Polygons n− × ° n ° Exercise 3.4 1 Tip (i) (ii) (a) (b) (c) ° 2 ° 3 °. 4 (a) (b) ° 5 (a) (b) 120° 125° C 50 ° 98° 100° x – 10° B H 56° 100° y 17° D x– x 130° 84° x 161° 110° (c) A x E 44° x 100° F 18 Unit 1: Shape, space and measures C 125° G 3 Lines, angles and shapes 3.5 Circles Exercise 3.5 O 1 (a) (b) DE (c) MON (d) (e) M MP. 3.6 Construction A How to bisect an angle. Tip B How to draw the perpendicular bisector of a line. Exercise 3.6 1 ! = ABC 2 ΔABC 3 ΔMNO ° AC = MN = NO CB = NO = NO X AB = MO = MO Y XY Unit 1: Shape, space and measures 19 3 Lines, angles and shapes ΔDEF 4 DE = (a) (b) FE = DF = DEF 5 Mixed exercise (a) 1 (b) G A 23° C x E F A x (c) B C D 71° C D x D (e) y 104° 32° (f) y 110° x 30° 40° 68° B (d) (g) E A 111° 98° 16° (h) y x y 110° x z x 2 (a) (b) x (c) A x x 50° C 70° 120° E 60° 20 Unit 1: Shape, space and measures B 50° D 110° B 3 Lines, angles and shapes O 3 A (a) (i) DO (b) (c) (d) O (ii) AB OB (iii) AC AC B. B AB 4 D AB C 5 ΔABC X AB XY BX̂Y. AB = BC = AC = AB Unit 1: Shape, space and measures 21 4 Collecting, organising and displaying data 4.1 Collecting and classifying data 2 12 Exercise 4.1 Student 1 2 Gender 3 4 5 M M M 6 7 8 M 7 7 8 7 Mass (kg) Eye colour Hair colour No. of brothers/ sisters 1 (a) (b) (c) (d) (e) 22 Unit 1: Data handling 10 M Height (m) Shoe size 9 1 4 Collecting, organising and displaying data 4.2 Organising data Exercise 4.2 In data handling, the word frequency means the number of times a score or observation occurs. 1 Phone calls Tally Frequency 1 2 (a) Number of mosquitoes 0 1 2 3 4 5 6 Frequency (b) Unit 1: Data handling 23 4 Collecting, organising and displaying data Score Frequency 3 (a) (b) (c) (d) (e) 4 Student 1 Gender 2 3 4 5 M M M 6 7 8 M Hair colour 1 (a) Eye colour Male Female (b) (c) 4.3 Using charts to display data 24 Unit 1: Data handling Brown Blue Green 10 M Eye colour No. of siblings (brothers/sisters) 9 1 4 Collecting, organising and displaying data Exercise 4.3 1 Number of students in each year Year 8 Key Year 9 Year 10 (a) (b) = 30 students (c) (d) (e) (f) Year 11 (g) Year 12 Tip 2 City Tokyo Seoul Mexico City New York Mumbai Population (millions) 3 (a) (b) (c) (e) (f) (g) (d) A. Number of students in 10A 20 B. Favourite sport of students in 10A 18 Frequency 16 14 Frequency Tip 12 10 8 6 4 14 12 10 8 6 4 2 0 Key boys girls soccer basketball athletics Sport 2 0 boys girls Gender Unit 1: Data handling 25 4 Collecting, organising and displaying data 4 Cereal Girls Hot porridge Bread 8 Boys (a) (b) 5 Tip Traffic passing my home n × 100 360 (a) (b) (c) (d) handcarts bicycles taxis motorcycles n cars trucks buses tuk-tuks 6 (a) (b) (c) (d) 7 20 140 10 120 Rainfall (mm) Temperature (°C) 30 0 –10 –20 –30 –40 Unit 1: Data handling 80 60 40 J F (a) (b) 26 100 M A M J J Month A S O N D 20 J F M A M J J Month A S O N D 4 Collecting, organising and displaying data (c) (d) (e) (f) (g) Mixed exercise 1 (a) (b) (c) (d) (e) (f) 2 3 Mobile phones and land lines, per 100 people Number of phones per 100 people 100 90 80 70 (a) (b) (c) (d) 60 50 (e) 40 30 (f) 20 (g) 10 0 Canada USA Germany Denmark UK Key Mobile phones Sweden Italy Land lines Unit 1: Data handling 27 4 Collecting, organising and displaying data 4 Student 1 Gender 7 M M M 8 M M Height (m) Shoe size 7 7 8 7 Mass (kg) Eye colour Hair colour No. of brothers/sisters (a) (b) (c) 5 Year FAST FORWARD In part (b), the ‘percentage depreciation’ requires you to first calculate how much the car’s value decreased in the year she had it, and then calculate this as a percentage of the original value. You will see more about percentage decrease in chapter 5. ! 28 Unit 1: Data handling (a) (b) (c) Value of car 1 1 5 Fractions 5.1 Equivalent fractions You can cross multiply to make an equation and then solve it. For example: Exercise 5.1 1 1 x = 2 28 2 x = 28 x = 14 (a) 2 26 = 5 x (b) 5 120 = 7 x (c) 6 66 = 5 x (d) 11 143 = 9 x (e) 5 80 = 3 x (f) 8 x = 12 156 5.2 Operations on fractions ÷ Tip × Exercise 5.2 1 (a) 3 5 40 (e) 14 (b) 1 12 22 3 (f) 2 4 (c) 11 24 30 (d) 3 75 100 35 45 2 Remember: you can cancel to simplify when you are multiplying fractions; and the word ‘of’ means ×. (a) 1 4 5 12 1 1 2 3 (e) 2 4 × (i) 8 9 of 81 (b) 9 ×7 13 (c) 3 (f) 1 12 1 × ×2 5 19 2 2 1 (g) (j) 3 of 4 2 (k) 1 3 1 2 1 2 (d) 2 4 of 360 of 9 16 50 (h) (l) 3 4 3 4 1 3 of 2 2 5 2 7 of 2 1 3 Unit 2: Number 29 5 Fractions Tip 3 1 6 + (e) 2 1 8 1 1 7 (f) 4 (i) 5 7 1 1 3 (j) 1 − (a) 2 3 8 (b) 9 10 7 12 − 3 10 (c) 3 1 2 3 4 4 1 + 7 3 (g) 1 7 3 1 13 − 1 3 (k) 2 − 4 5 17 3 (d) 2 1 2 (h) 3 9 10 2 (l) 1 − 4 9 13 3 2 3 5× 3 1 3 7 8 4 (a) 8 ÷ (d) (b) 12 ÷ 3 2 18 ÷ 9 30 (g) 1 REWIND 1 14 26 ÷ (e) 10 13 7 8 (c) 8 4 ÷ 9 5 (h) 3 6 15 7 8 ÷ 12 3 7 2 (f) 1 5 2 3 (i) 5 1 5 2 9 1 3 10 5 The order of operations rules (BODMAS) that were covered in chapter 1 apply here too. ! 2 3 (a) 4 + × (d) 2 7 8 1 3 (b) 2 8 1 − 6 3 4 8 (e) 5 15 5 1 (g) ÷ − × 8 4 6 5 1 8 5 1 × 6 4 2 1 − 7 5 8 5 8 + × (c) 3 2 × +6 7 3 3 5 1 (f) 5 ÷ − × 11 12 6 1 3 2 3 3 (h) 2 4− × 3 10 17 (i) 2 1 2 7÷ − × 9 3 3 7 12 6 (a) (b) 3 4 7 (a) 4 1 2 (b) 5.3 Percentages Exercise 5.3 A 1 (a) (e) 30 Unit 2: Number 1 6 (b) (f) 5 8 (c) (g) 93 312 (d) (h) 2 7 5 Fractions 2 (a) (b) (c) (d) (e) 3 Tip (a) (e) (b) (f) (c) (d) (a) (e) (i) (b) (f) (j) (c) (g) (d) (h) 4 5 Original amount New amount Percentage increase or decrease (a) (b) (c) (d) (e) (f) (g) 6 (a) (c) (e) (b) (d) (f) (a) (c) (e) (b) (d) (f) 7 Exercise 5.3 B 1 2 3 3 1 2 4 Unit 2: Number 31 5 Fractions 5 6 7 8 Tip Exercise 5.3 C 1 2 (a) (b) 3 4 (a) (b) (c) 5.4 Standard form a× looks Tip Exercise 5.4 A 1 (a) (e) (i) 32 Unit 2: Number (b) (f) (j) (c) (g) (k) (d) (h) (l) k 5 Fractions If the number part of your standard form answer is a whole number, there is no need to add a decimal point. REWIND Remember, the first significant figure is the first non-zero digit from the left. ! 2 (a) (e) (i) × × × (b) (f) − (c) (g) × × − (d) (h) × × − × − × − Exercise 5.4 B 1 (a) (b) (d) 9500 0.00054 (g) (h) (c) ÷ (f) 4525 × 8760 0.00002 ÷ (e) × × 5 25 108 (i) 3 9 1 × 10 −8 2 (a) (d) (g) × × × × − (b) (e) (h) × × ÷ × × − × × × − (c) (f) (i) × × ÷ 3 9 1 × 10 −8 × × S 3 × × × × ÷ × − × (a) (b) × 4 × × (a) (b) 5.5 Estimation Exercise 5.5 Remember, the symbol ≈ means ‘is approximately equal to’. 1 (a) (c) × × (a) + (b) (d) ≈ ≈ × ≈ ÷ ≈ 2 (c) 9 3 × 7.6 5 9 × 0 95 + (b) + ÷ (d) 8 9 × 8 98 2 Unit 2: Number 33 5 Fractions Mixed exercise 1 (a) (b) × (c) 36.4 6.32 9.987 ÷ 2 (a) 160 200 (a) 4 9 48 72 (b) (c) (d) 64.25 × 3.0982 36 54 3 (f) 2 × 1 3 3 8 (b) 84 × 9 1 2 3 4 3 (g) 4 4 2 (c) 5 1 ÷ 9 3 (h) 9 1 1 7 5 9 4 (a) (b) 5 6 (a) (b) 7 × 8 (a) (b) × S 9 (a) (b) (c) 34 Unit 2: Number (d) 9 3 − 11 4 (i) 9 1 5 7 5 9 8 1 ×2 (e) 5 24 + 7 16 (j) 4 ÷ 18 5 25 + 2 3 2 6 Equations and transforming formulae 6.1 Further expansions of brackets Remember: +×−=− −×+=− Exercise 6.1 1 (a) − x + y (d) x − y (g) − a (j) − x + +×+=+ −+−=+ (b) − a − b (e) − x x + y (h) − x + y (k) x x − y (c) − − x + y) (f) − x − (i) x − x − (l) − x x − y) 2 (a) (b) − x y − x−y + x − (d) − x ( − ) − 2y( (g) − x ( )+2−( 1 2 1 4 − + 2 ) (e) − 3) xy − − x + xy +y − −x (h) − x − y + x x − y (c) − x + y − x y − (f) x (i) 1 − 2 − y −y − x ) ( − )+3−( + 7) 6.2 Solving linear equations Exercise 6.2 In this exercise leave answers as fractions rather than decimals, where necessary. x 1 (a) (e) (i) x+ = x+ x− =x+ x+ = x− (b) (f) (j) x+ =x+ x− = −x x− = x− (c) (g) (k) x− = x+ − x= x+ x+ = x− (d) x − = x + (h) + x = x − (l) x + = x − Unit 2: Algebra 35 6 Equations and transforming formulae When an equation has brackets it is usually best to expand them first. To remove the denominators of fractions in an equation, multiply each term on both sides by the common denominator. x 2 (a) x− = (e) − x − = − (i) x + = x − (l) x− − x− (b) x + = (f) − x = (j) x − = x + = x− (c) (g) (k) x+ = x+ =x x+ − x− (d) (h) x− = x+ = x = x 3 (a) x − 3 6 2 (b) x + 2 = 11 3 (e) x − 2 = 5 3 2x 1 (i) =x 5 (f) 12 x − 1 = 9 5 2 3 = x −6 (j) 5 (m) 2x x − =7 3 2 (n) −2 (d) 28 − x = 12 6 5 2x (h) = −1 4 (c) 4 x = 16 6 5 2 = −1 (g) 3 (k) 10 x + 2 =6−x 3 (l) x x − =3 2 5 x − xy ab − bc ( + 4) = x+7 2 6.3 Factorising algebraic expressions Exercise 6.3 Remember, x2 means x × x, so x is a factor of x2 Find the HCF of the numbers first. Then find the HCF of the variables, if there is one, in alphabetical order. 1 (a) x (d) a ab (g) xy xyz (j) x y z xy z (b) (e) (h) (k) xy pq ab (c) (f) (i) (l) a ab abc xy (a) x+ (e) ab + a (i) xy − yz (b) (f) (j) + y x− y x − xy (c) (g) a− xyz − xz (d) (h) (a) x + x (e) ab + b (i) abc − a b c (b) (f) (j) a−a xy − x y x − xy (c) (g) (k) x + x x− x ab − b (d) (h) (l) x yz pq ab b ab ab xy 2 3 Remember, if one of the terms is exactly the same as the common factor, you must put a 1 where the term would appear in the bracket. 4 (a) x + y + +y (c) a+ b − a a+ b (e) x − y + − y (g) +y −x y+ (i) x x − − x − (k) x x + + y + x 36 Unit 2: Algebra (b) x y − + y − (d) a a − b − a − b) (f) x x − + x − (h) a b − c − c − b) (j) x x − y − x − y) (l) x − y − x x − y) x− x x y − xy a b − ab 6 Equations and transforming formulae 6.4 Transformation of a formula A = bh. Exercise 6.4 A Tip 1 m D = km 2 c y = mx + c 3 P = ab − c b 4 a = bx + c b a 5 (a) a + b = c (b) a − b = c (e) bc − a = d (f) bc − a = −d (i) abc − d = e (j) cab + d = ef (m) c a − b = d (n) d a + b = c (c) ab − c = d 2a b =d (g) c (k) (d) ab + c = d c ba (h) =e d ab + de = f c (l) c + ab =e d Exercise 6.4 B P= l+b 1 P l b (a) (b) Tip b b C= 2 (a) (b) (c) Tip r r r = = A= 3 a h (a b) 2 b h b a= h= T 4 P B T P B (a) (i) (ii) Unit 2: Algebra 37 6 Equations and transforming formulae (b) (c) (d) B m 5 t (a) (b) Mixed exercise m t t x 1 (a) x− =− (e) x− = − x (b) x+ =− (f) x− = (c) x+ x 2 (a) m = nxp − r (b) m = 2x 4 =2 7 (d) 5 = (g) 3x 7 = 1 4 x 4 8 1 4x 5 (h) 3(2 x − 5) = x + 1 5 2 nx + p q 3 Remember to inspect your answer to see if there are any like terms. If there are, add and/or subtract them to simplify the expression. (a) x− + (d) − y − y − y (g) − x x − + x (b) − x x − (e) x− + x+ (h) x x + − x x − (c) − x − y + (f) x x − + x − (a) x − (d) xy − x (g) x + x − (b) (e) (h) (c) − x − (f) x−y +x x−y 4 x+ = 5 (a) (b) x−7 x− y x y + xy x x + y − x x + y) × (c) 2x (d) 19x 4 x+9 REWIND 6 Use geometric properties to make the equations (see chapter 3). ! 2x + 8 12x – 45 180 – 3x 38 Unit 2: Algebra 4x – 10 x 4x + 15 6x – 45 7 Perimeter, area and volume 7.1 Perimeter and area in two dimensions C = πd C = 2πr bh A= 2 A = s2 A = bh A = bh A = bh 1 A = (produc f diagonals) 2 A= ( fp A = πr2 2 ll l d )h Exercise 7.1 A 1 (a) (b) 32 mm (c) 11.25 cm 19 mm 28 mm 45 mm (d) (e) 21 mm 14 mm (f) 1.5 cm 6.8 cm 5.3 cm 3.4 cm 4.9 cm m m 92 7.2 cm 69 mm Unit 2: Shape, space and measures 39 7 Perimeter, area and volume π= 2 Tip (a) (b) (c) 7 cm 5m π (e) π (f) (d) 21 mm 4.5 m 4m (g) 3m 8 mm 60° 8 cm 3 4 5 Tip 6 7 (a) (b) Remember, give your answer in square units. Exercise 7.1 B 1 (a) 12.5 cm 17 cm (b) 1.7 m (c) 90 cm 14 cm 19 cm (d) 21 cm 25 cm 15 cm 19 cm 20 cm 35 cm 11 cm 21 cm 12 cm 5 cm 40 (g) 112 mm 72 mm 13 cm Unit 2: Shape, space and measures 41 mm 67 mm (h) (i) 8 cm 15 cm 10 cm (f) 12 cm 7 cm cm 10 (e) 11 cm 7 cm 17 cm 7 Perimeter, area and volume π= 2 (a) (b) (c) (d) (e) 300° 100 mm 20° 10 cm 140 mm 8 cm 14 mm 3 Tip (a) (b) 2 6 6 (c) 6 3 (d) 22 13 2 12 9 18 20 6 7 8 35 20 5 21 5 12 (f) 9 1 13 6 14 14 36 (g) (h) 14 25 3 45 30 30 90 90 40 2 95 15 Tip (i) 20 4 (f) π= (a) 2.5 cm m 6c (b) (c) 9 cm 4.5 cm 11 cm 2 cm 7 cm 4.8 cm 5 cm (d) (e) 4 cm 6.5 cm 3 cm 5.5 cm cm 6.3 5 6 (f) 15 cm 32.4 cm 40 mm (e) 40 cm 50 mm 30 mm 8 cm 20 cm × × 2 Unit 2: Shape, space and measures 41 7 Perimeter, area and volume Exercise 7.1 C AB 1 (a) (b) A (c) A O 21 mm O 120° O 10 cm B 40° A B X 2 12 mm B Y XY 637 1k 60° m Y X 3 (a) (b) 12 cm 60° (c) 20 cm 18 mm 150° 4 2 42 Unit 2: Shape, space and measures 7 Perimeter, area and volume 7.2 Three-dimensional objects Exercise 7.2 1 (a) (b) ×2 ×6 (c) ×2 ×2 (d) ×4 ×1 ×8 2 (a) (b) (c) 3 (a) (b) (c) (d) Unit 2: Shape, space and measures 43 7 Perimeter, area and volume 7.3 Surface areas and volumes of solids h 1 V = πr 2 h πrl 3 V=l×b×h l V = al a b l h l area of base × h V= 3 h 4 V = πr 3 3 πr2 Exercise 7.3 A π= Tip 1 (a) (b) 0.5 mm 18.4 m 1.2 mm 0.4 mm 12 m 14 m (c) (d) 8m 4m 12 mm 1.5 cm 2 2 (a) (b) Remember, 1 m2 = 10 000 cm2 44 3 × (a) (b) (c) Unit 2: Shape, space and measures × 2 7 Perimeter, area and volume 4 Tip (a) (b) 45 mm 4 cm (c) (d) 65 mm 3 cm 80 mm 50 mm 2 cm m 40 c A = 28 cm2 8 cm 20 mm (e) 8 cm 10 cm (f) (g) (h) 1.25 m 4m 12 cm 1.2 m 1.2 m 25 12 cm cm 1.25 m 5m 1.2 5 6 (a) (b) 3 7 8 3 Volume (mm3) 64 000 64 000 64 000 64 000 Length (mm) Breadth (mm) Height (mm) Exercise 7.3 B 1 1.2 cm 3.5 cm (a) (b) (c) 20 m (d) (e) 3 cm 3.5 cm 2.7 cm 8 cm 36 mm 40 mm 12 cm Unit 2: Shape, space and measures 45 7 Perimeter, area and volume 2 (a) (i) (ii) (b) Mixed exercise 1 (a) (b) 2 2 3 170 mm 40 mm (c) 120 mm 150 mm 2 cm (b) 50 mm 192 mm 5 cm 6 cm (a) 320 mm (d) 5 cm (e) 8 cm 4 cm 6 cm 6 cm 12 cm (g) 12 cm 30° 10° 46 Unit 2: Shape, space and measures (f) 5 cm 1 cm 7 cm 5 cm 2 cm 3 cm 6 cm 3 cm 5 cm 7 Perimeter, area and volume 4 MNOP NO 2 M 12 m N 10 m P O 5 20 mm A 40 mm 120 mm B 20 mm 15 mm (a) (b) (c) (d) 6 7 8 Unit 2: Shape, space and measures 47 8 Introduction to probability 8.1 Basic probability 1 2 Exercise 8.1 1 Red //// //// //// White //// //// //// /// Green //// //// //// // (a) (b) (c) (d) 2 Result Frequency 44 48 Unit 2: Data handling 8 Introduction to probability (a) (b) (i) (ii) (iii) 8.2 Theoretical probability P(outcome) = number of favourable outcomes number of possible outcomes P(H) = Tip 1 2 all Exercise 8.2 1 (a) (b) (i) (iii) (v) (vii) (ix) (ii) (iv) (vi) (viii) < < < < > 2 (a) (b) 3 (a) (b) (c) 4 (a) (b) W (c) Unit 2: Data handling 49 8 Introduction to probability 5 (a) (d) (b) (e) < (c) 6 8.3 The probability that an event does not happen ′ ′ = A Exercise 8.3 1 3 8 2 3 Flavour Strawberry Lime Computers Sewing Lemon Blackberry Apple P(flavour) (a) (b) (c) (d) 4 Club P(Club) (a) (b) (c) (d) 50 Unit 2: Data handling Woodwork Choir Chess 8 Introduction to probability 8.4 Possibility diagrams Tip Exercise 8.4 1 (a) (b) 2 FAST FORWARD Tree diagrams are also probability space diagrams. These are dealt with in detail in chapter 24. ! (a) (b) (c) (d) 3 Drinks: Snacks: (a) (b) (c) Unit 2: Data handling 51 8 Introduction to probability 8.5 Combining independent and mutually exclusive events – are – are – Exercise 8.5 1 (a) (b) (c) (d) 2 (a) (b) (c) (d) (e) (f) (g) 3 (a) (b) (c) (d) (e) 52 Unit 2: Data handling 8 Introduction to probability Mixed exercise 1 (a) (b) (c) (d) 2 (a) (b) (c) (d) (e) (f) (g) 3 (a) (b) (c) (d) 4 (a) (b) (c) (d) 5 6 Unit 2: Data handling 53 9 Sequences and sets 9.1 Sequences 1 10 n You should recognise these sequences of numbers: square numbers: 1, 4, 9, 16 . . . = a+ n a (a) (e) (b) (f) (c) (g) 1 1 4 2 2 (a) (b) (c) (d) (e) 1 2 3 (a) (b) (c) (d) (e) (f) = n+ =n = n n =n n =n n n = n n n n 4 (a) (b) (c) (d) Unit 3: Number n 1 triangular numbers: 1, 3, 6, 10 . . . 54 n n Exercise 9.1 cube numbers: 1, 8, 27, 64 . . . Fibonacci numbers: 1, 1, 2, 3, 5, 8 . . . d d n (d) (h) 9 Sequences and sets 5 (a) (b) − (c) (d) (e) 6 n=1 n=2 (a) (b) (c) (d) n=3 n 9.2 Rational and irrational numbers a b b b a b a b In 1.2! , the dot above the two in the decimal part means it is recurring (the ‘2’ repeats forever). If a set of numbers recurs, e.g. 0.273273273..., there will be a dot at the start and end of the recurring ! !. set: 0.273 a Exercise 9.2 1 (a) 3 , 8 16 , 3 16 , (b) 23, 45 , 0.66! , 22 , 7 3 4 12 , 0.090090009. . ., 31 , 3 0.020202. . ., 1 , 3 90 , π, 5 , 8 , 0.834 2 2 (a) 0 4! (d) 0.114! !! (b) 0 74 ! ! (e) 0.943 (c) 0 87! ! ! (f) 0.1857 Unit 3: Number 55 9 Sequences and sets 9.3 Sets ∈ ∅ ℰ ⊂ ∪ A A A x x x Exercise 9.3 A Tip 1 (a) ∈ (b) ∈ (c) (d) ⊂ (e) (f) (g) A (h) ℰ ⊂ ∩ ∪ B A B A A 2 A (a) (b) (c) (d) (e) (f) Sometimes listing the elements of each set will make it easier to answer the questions. Unit 3: Number B C B∩C C A A A 3 ℰ C (a) (b) (c) (d) (e) (f) 56 A A A∩B B∪C A ∩B B ∩ C)΄ A ∩ B΄ A∪B∪C B 9 Sequences and sets 4 (a) x x ∈ (b) x x ∈ x x 5 (a) (b) Exercise 9.3 B Tip 1 ℰ P C 2 (a) (b) (c) (d) (e) (f) REWIND C) P΄) C∩P P∪C P ∪ C)΄ P⊂C 3 Exam questions often combine probability with Venn diagrams. Revise chapter 8 if you’ve forgotten how to work this out. ! (a) (b) (c) Mixed exercise n 1 (a) (b) (c) 2 n n (a) (b) (c) 3 5 7 8 17 1 , 0.213231234. . ., 25 , , 0.1, − 0.654, 2 , 22 5 , 4π Unit 3: Number 57 9 Sequences and sets 4 (a) ! 3! ! ! (b) 0.286 5 B C (a) (b) (c) (d) n (i) (ii) (iii) (iv) (v) 58 Unit 3: Number ∩ B = biology C = chemistry 10 Straight lines and quadratic equations 10.1 Straight lines x y x y x m= x y y y change in y change in x x y m c x+c y y x y y m y c x x y y Exercise 10.1 Tip x 1 − − x (a) y = x + 5 (b) y (e) x = 4 (i) 0 x 2 y − 1 (f) y = −2 1 (j) x + y = − 2 a 2 (c) y 7 2x (g) y 2x − (d) y 1 2 x −2 (h) 4 2x 2 − 5y e f j a 3 Remember, parallel lines have the same gradient. 2x − 1 − 4 (a) y 3x + 3 (c) y 3x (e) 2 y (g) y = 8 y = x+3 (b) y 1 x−4 2 3x + 7 (d) y 0 8x − 7 y 3x + 2 y = −9 3 2 y = x+2 1 x −8 2 y y (f) 2 y 3x = 2 y (h) x = −3 1 2 x= 8x + 2 1 5x + 2 Unit 3: Algebra 59 10 Straight lines and quadratic equations 5 (a) (b) y 6 (4, 6) –1 4 1 4 4 –1 –2 0 –1 (g) y 1 x –4 2 (1, –2) 0 1 2 3 4 5 6 7 (h) y –3 0 3 y x –8 x m y (a) 3x − 4 (b) y (e) y = x 1 + 2 4 (f) y = (i) x + 3 y = 14 y (m) x + = −10 2 x −1 4x −2 5 (j) x + y + 4 = 0 x 4 3 2 1 –3 6 y = mx + c x 2 3 0 Tip 4 6 7 6 5 4 3 2 1 2 2 6 –2 (f) y (–2, 1) x y 0 –2 3 (3, –3) 4 –2 2 –3 0 (e) 0 (d) y x –2 2 –4 (c) y c (c) y 1 x+5 2 (g) y = 7 (k) x 4 y –4 –1 –2 –3 0 (d) y 4 x (h) y 3x (l) 2 x 5 y 7 y 9 8 7 6 5 4 3 2 1 x 0 –9 –8 –7 –6–5 –4–3 –2–1 1 2 3 4 5 6 –2 –3 (j) –4 –5 –6 –7 (l) (k) –8 –9 (i) y (d) (a) (c) 3 2 1 0 –3 –2 –1 –2 (b) –3 (a) y x (d) x + y 2 60 Unit 3: Algebra 3 2 1 x 0 –3 –2 –1 –2 (e) –3 (f) 1 2 3 x 8 y (h) y (b) y (e) x x 1 2 3 (g) 1x 2 y (c) y x 8 10 Straight lines and quadratic equations 9 (a) (b) (c) (d) (e) (f) 10 10.2 Quadratic expressions x x x y) y) x x xy xy y y x x x x x x x x x x b a x x x x x a b x x x Exercise 10.2 A Tip 1 (e) ( x + 2))(( x + 3) ( x + 5))(( x + 7 ) ( x 1))(( x − 3) (g) (y (a) (c) (x 7 ))(( y − 2) )( (b) ( x + 2))(( x − 3) (d) (f) ) )( x + 7) ( x 5)( (2x 1)( x + 1) (h) (2 x y )(3x 2 y ) (j) (x 11) ( x + 12) 1 1 (k) x + 1 1 x (l) 2 2 (x 3)(2 3x ) (i) 2 2 + 1 2x − 3 (m) (3x 2)(2 4 x ) Unit 3: Algebra 61 10 Straight lines and quadratic equations Remember the expansions for the square of a sum or a difference. The difference of two squares always gives you two brackets that are identical except for the signs, so you may be able to write down the answer to an expansion just by inspection. 2 (a) ( x + 4 ) 2 (d) ( y − 2) 2 (g) (3 2) 2 (j) ( 4 6) 2 (m) (6 3 y ) 2 (b) (e) (h) (k) ( x − 3)2 (c) ( x + 5) 2 (f) (2 x y ) 2 (i) (2 5) 2 (l) ( 4 2 x ) 2 (x + y) (2x 3 y )2 (3 − x )2 2 3 (a) (c) (e) 5))(( x + 5) (x (i) (j) ( (x (d) x 2 (4 3x )(3x 4 ) (f) 3 2z )( 4 x y 2z ) ( 2 2y) (2x 2 y )(2x 4 xy 5 y ) (5 y 4 xy )(4x (8 x y 7 z )(8 x y 7 z ) (g) 4 x 2 y 2 (h) (b) (2 x 5)(2 x 5) (3 7 y )(7 y 3) 2 4 2 2 3 2 2 y2 x3 ) y2 ) 4 2 3 2 )( )( y2 x2 + y2 2 2 2 Exercise 10.2 B 1 Tip (a) x 2 + 4x 4x 4 (c) x 2 + 6x 6x 9 (e) 15 + 8x 8 + x2 (g) x 2 8x 8 x 15 (i) 26 − 27x 27 + x 2 (k) x 2 + 3x 3 x 10 (m) 12 − 7x 7 + x2 (o) −54 + 3x + x 2 (b) (d) (f) (h) (j) (l) (n) x 2 + 7x 7 x 12 4 5x 5 + x2 x 2 9x 9x 8 3 4x 4 + x2 x2 7x 8 x 2 4 x 32 −12 + x + x 2 (a) (c) (e) (g) (i) 5x 2 15x 10 3x 3 122 2 9 x x3 x 2 + 20 x x 3 5 x 2 14 x −2 x 2 + 4 x + 48 (b) (d) (f) (h) (j) 3x 2 18 x 24 5x 2 15x 10 x 4 y + x 3 y − 2x 2 y 3x 2 15x 18 2 x 2 2 x − 112 (a) (d) (g) (j) (m) (p) x2 − 9 49 − x 2 x2 9 y2 2 x 2 18 25 − x16 25x10 − 1 (b) (e) (h) (k) (n) (q) 16 − x 2 9x 2 4 y 2 121 y 2 144 x 2 200 − 2 x 2 x 2 y 2 − 100 1 81x 4 y 6 2 Tip 3 Tip a b a b a b a b a b 62 a b a b Unit 3: Algebra (c) (f) (i) (l) (o) x 2 − 25 81 − 4 x 2 16 x 2 49 y 2 x 4 y2 25x 2 64w 2 − 2 y4 z 10 Straight lines and quadratic equations Exercise 10.2 C Remember to make the righthand side equal to zero and to factorise before you try to solve the equation. 1 (a) (c) (e) (g) (i) (k) (m) (o) (q) Quadratic equations always have two roots (although sometimes the two roots are the same). Mixed exercise x 2 3x 0 (b) 6 x 2 12 x (d) x 2 −11 0 (f) 8 2 2 (h) x 2 + 5x 5x 4 = 0 (j) x 2 x − 20 = 0 (l) x 2 + 15 8 x (n) x 2 + 56 15x (p) 5x 2 20 x 20 0 8 2 32 0 −16 x − 24 x 2 = 0 49 − 4 2 0 x 2 + 6x 6x 8 = 0 x2 4x 5 = 0 x 2 + 8x 8 x 20 −60 − 17 x = − x 2 x 2 − 20 x + 100 = 0 D 1 (a) y 1 x 2 x −1 y 1 x+3 2 (b) y x −1 y (c) y = 2 x −1 y (d) y 2 x − 4 = 0 x −1 y y 2 (a) y (d) y = 2x − 1 1 − 2 (b) y + 6 x (c) x (e) 2 x 3y 3y = 6 (f) y y = −8 x 3 (a) (b) (c) (d) y y y x+8 4 x 5 1 2 y y- − 2 − 3 − − Unit 3: Algebra 63 10 Straight lines and quadratic equations (e) (f) (g) (h) 2 y 4x 4x + 2 0 x+ y=5 x y − y − − 4 9 8 7 6 5 4 3 2 1 y x –9 –8 –7 –6–5 –4–3 –2–10 1 2 3 4 5 6 7 8 9 –2 –3 –4 –5 –6 –7 –8 –9 5 a b y c y d) y y 6 4 2 4 2 –6 –3–3 –6 e 3 6 7 x –4 –2 2 4 f) y –5 Tip 6 x –10 –2 10 20 x –4 –6 3 3 5 x (5,–5) –4 –2 2 4 x –3 6 x D= t (a) (b) D D y = mx + c (i) Unit 3: Algebra t D= t (c) (d) (e) 64 3 –4 y 5 –5 –3 –6 –3 –2 x (ii) (iii) t 10 Straight lines and quadratic equations (f) (i) (ii) 2 1 2 (iii) 3 4 (g) 7 (a) (b) (i) (ii) (iii) (iv) (v) 8 (b) (2 x 2) ( x − 1) (a) ( x − 8)2 (c) (3x (e) (2 (g) (3x (i) x + 1 x − 1 2 2 (k) (x 2y) 2 3x )(3x 2) 2 ) 2 y 1 2) ( x − 7 )2 2 (d) (1 6 y ) 2 (f) (2 5) 2 1 (h) x + y 2 2 1 1 (j) − 2 + 2 x x (l) −2 x ( x − 2) + 8 x 2 2 (m) 2 x ( x 1) − (7 2 x )( 2 x ) 2 9 (a) a 3 4a (c) x 2 x − 2 3 y ) − 4z 2 x2 (g) x 4 − 4 (i) 4 x 2 4 x − 48 (k) 5 20 x16 (e) (2 x 2 (b) x 4 − 1 (d) x 2 22xx 1 (f) x 2 + 16 x + 48 (h) x 2 5x 6 14xx 24 (j) 2 x 2 14 (l) 3x 2 15x 18 x 10 (a) (b) (c) (d) (e) (f) 7x2 2 2 6x 2 3x 2 4x2 5x 2 42 x 35 0 8=0 18 x 12 6 x = −3 6x 16 x 20 0 20 x 20 0 Unit 3: Algebra 65 11 Pythagoras’ theorem and similar shapes 11.1 Pythagoras’ theorem a c2 = a2 + b2 b c c2 = a2 + b2 Exercise 11.1 A Tip hypotenuse 1 (a) (b) (c) 15 cm x 3 cm 13 mm y x 8 cm 5 mm 4 cm (d) (e) x (f) y 0.4 cm 24 cm 0.5 cm 1.2 cm 26 cm (g) (h) x 11 cm 4 cm y 7.3 cm 7 cm 66 Unit 3: Shape, space and measures x 0.6 cm 11 Pythagoras’ theorem and similar shapes 2 Tip (a) (b) 70 mm (c) 6 cm x 80 mm y z 20.2 cm 4 cm 8 cm 120 mm 8.2 cm (d) (e) 5 mm 5 m m (f) y x 16 cm 13 mm 15 cm 9 cm 9 cm z 5 cm 12 cm 16 cm 3 (a) (b) A 10 (c) (d) J H 13 8 7 B D 15 8 9 9 12 L C F E 12 G I 11 K 5 4 (a) (−2, −2) (c) (−4, 5) (e) (−1, −3) Tip (2, 2) (0, 1) (2, −3) (b) (−3, −1) (0, 2) (d) (2, 4) (8, 16) (f) (−2, 0) (4, 3) Exercise 11.1 B 1 2 3 4 Unit 3: Shape, space and measures 67 11 Pythagoras’ theorem and similar shapes 5 11.2 Understanding similar triangles Tip Exercise 11.2 1 (a) (b) 4.47 cm x 2 cm 4 cm β α β 10 cm z 10 cm x (f) z 4 cm 8 cm β α 43° x 8.5 cm α α β α 27 cm y α 21 cm 40 cm 43° y α 5 cm 9.5 cm (h) β x y 8 cm 6 cm 8 cm Unit 3: Shape, space and measures α β y y 4 mm y α 68 8 cm x 3 mm 8 cm 7 cm (d) 2 mm 6 mm 8 mm α 4 mm 5 mm β y (g) β 9 mm 6 mm 1 cm (c) (e) 2 cm 12 mm α α x β 30 cm y 25 cm β 15 cm 11 Pythagoras’ theorem and similar shapes Δ ABC 2 Δ ADE A B C D E 3 building 3m 30 m Nancy 4m 11.3 Understanding similar shapes a b A a2 b2 B 2 volume A a 3 = volume B b surface area A a 2 = surface area B b Exercise 11.3 1 (a) 15 (b) 25 45 x 30 24 y 18 x 30 Unit 3: Shape, space and measures 69 11 Pythagoras’ theorem and similar shapes 2 3 (a) (b) 18 cm 12 cm 69 mm area = 113.10 cm2 23 mm area = 4761 mm2 4 (a) (b) x 1 cm area = 10 cm3 area = 40 cm3 area = 16.2 m3 5 A 6 (a) (b) (c) 11.4 Understanding congruence 70 Unit 3: Shape, space and measures x 3m B A B area = 405 m3 11 Pythagoras’ theorem and similar shapes Remember, the order in which you name a triangle is important if you are stating congruency. If Δ ABC is congruent to Δ XYZ then: AB = XY BC = YZ Exercise 11.4 1 State whether each pair of triangles is congruent or not. If they are congruent, state the condition of congruency. (a) AC = XZ (b) X B x° x° C Y A Z 78° (c) X 35° 35° N M (d) 78° 67° 67° P D x y O Z Y (e) (f) A B (g) Q P N O D A C Z 87° R M (h) Y 87° R F P D C y x E Q E X B Tip 2 MNOP is a trapezium. MP = NO and PN = MO. Show that Δ MPO is congruent to Δ NOP giving reasons. M When you are asked to show triangles are congruent you must always state the condition that you are using. N Q P O PN = MO Unit 3: Shape, space and measures 71 11 Pythagoras’ theorem and similar shapes AB 3 AC C B A D E Mixed exercise 1 (a) (b) Δ ABC AB = 2 BC = AC = 3 (a) (c) (e) (b) (d) 4 5 (a) 4.5 cm α 1.5 cm (b) β x 13.5 cm α 5 cm Unit 3: Shape, space and measures 36 cm 15 cm x 72 30° 10.5 cm α α y β 4.5 cm (c) 30° 87° 63° x 11 Pythagoras’ theorem and similar shapes 6 (a) (b) 1.5 m 0.75 m 3.0 m 150 m 50 m 1.5 m 7 2 18 cm 4.5 cm area = 288 cm2 8 A B 2 B A 50 mm 80 mm area = 60 800 mm2 9 2 (a) (b) 10 (a) H D A F C B (b) D A G E B y C x y F G x E y x I H I (c) (d) A D C 6 7 5 G x 6 F y B y x E I 5 x Unit 3: Shape, space and measures H 7 73 11 Pythagoras’ theorem and similar shapes x 11 8.6 m xm 6.5 m 12 35 mm 17 mm 140 mm 105 mm 35 mm (a) (b) 74 Unit 3: Shape, space and measures 12 Averages and measures of spread 12.1 Different types of average Exercise 12.1 1 (a) (b) (c) (d) (e) (f) Tip 2 (a) (i) (iv) (ii) (v) (iii) (vi) (b) 3 4 Unit 3: Data handling 75 12 Averages and measures of spread 5 Tip 6 (a) (b) 7 Previous salary (a) (b) (c) (d) (e) 12.2 Making comparisons using averages and ranges Tip Exercise 12.2 1 (a) (b) (c) 76 Unit 3: Data handling Salary with increase 12 Averages and measures of spread Tip 2 (a) (b) 3 Runner A Runner B (a) (b) 12.3 Calculating averages and ranges for frequency data fx Mean = total of (score frequency f ) column total of frequency f column Exercise 12.3 1 (a) (b) 3 (c) 3 3 3 5 3 3 (d) 2 2 2 3 2 2 2 2 3 3 3 3 2 2 2 2 (a) (b) Score 2 (c) 3 5 Frequency Score Frequency 25 22 23 Score Frequency Unit 3: Data handling 77 12 Averages and measures of spread 12.4 Calculating averages and ranges for grouped continuous data – – – Estimated mean = sum of (midpoint × frequency) sum off requencies Exercise 12.4 m 1 Tip Marks (m) Midpoint Frequency (f) ≤m m 2 ≤m 5 ≤m ≤m ≤m ≤m (a) (b) (c) (d) Tip 2 Words per minute (w) ≤w ≤w ≤w ≤w ≤w ≤w (a) (b) (c) (d) 78 Unit 3: Data handling Frequency Frequency × midpoint 12 Averages and measures of spread 12.5 Percentiles and quartiles – – – 2 3 3 Exercise 12.5 Tip 1 1 Q 4 Q2 1 2 ( 1) ( 1) 3 3 4 ( (a) 55 (b) 3 (c) 1) (d) 2 3 Mixed exercise 3 2 5 3 5 5 2 1 (a) (b) (c) 2 (a) (b) (c) 3 Supplier Mean (weeks) Range (weeks) A B C Unit 3: Data handling 79 12 Averages and measures of spread 4 Volume (cm3) 2 3 5 Frequency (a) (b) (c) 5 Age in years (a) ≤a ≤a ≤a ≤a ≤a ≤a ≤a (a) (b) (c) (d) 6 (a) (b) (c) (d) 7 (a) (b) 80 Unit 3: Data handling 3 Frequency 13 Understanding measurement 13.1 Understanding units To change to a smaller unit, multiply by conversion factor × 1000 × 100 × 10 km m cm mm kg g cg mg kl l cl ml .. 100 .. 1000 .. 10 To change to a larger unit, divide by conversion factor. Exercise 13.1 Tip 1 (a) = = = = = = = 3 = (b) 2 (a) (c) (e) = = = (b) (d) (f) (a) (c) (e) = = = (b) (d) (f) = = = 3 = = = Unit 4: Number 81 13 Understanding measurement 4 (a) (c) (e) (b) (d) (f) 5 Cubic centimetres or cm3 is sometimes written as cc. For example, a scooter may have a 50 cc engine. That means the total volume of all cylinders in the engine is 50 cm3. (a) (c) (e) 2 (a) (c) (e) 3 = 2 = 2 2 = 2 (b) (d) (f) = (b) (d) (f) 3 2 2 = = 2 2 6 = 3 = 3 = 3 3 3 = 3 3 = 3 3 = 3 7 8 9 2 × 10 3 13.2 Time 15 60 1 4 Tip = 82 Unit 4: Number 12 60 1 5 = =02 × 13 Understanding measurement Exercise 13.2 1 (a) Name Time in 1 4 Time out Lunch Hours worked 3 4 1 2 (b) 2 3 4 (a) (b) (c) (d) 5 Chavez Street Castro Avenue Peron Place Marquez Lane (a) (b) (c) (d) (e) Unit 4: Number 83 13 Understanding measurement 13.3 Upper and lower bounds dp means decimal places sf means significant figures Exercise 13.3 1 (a) (b) (c) (d) (e) (f) (g) (h) 2 (a) (b) 3 (a) (b) 4 (a) (b) 5 (a) (b) 84 Unit 4: Number 13 Understanding measurement 13.4 Conversion graphs x y y Exercise 13.4 Tip 1 Exchange rate Rupiah (in thousands) 600 500 400 300 200 100 0 0 10 20 30 40 50 60 70 80 90 100 Australian dollars (a) (b) (i) (ii) (iii) (c) (i) (ii) 2 Conversion graph, Celsius to Fahrenheit 250 200 Temperature in °F 150 100 Tip 50 –20 0 20 40 60 Temperature in °C 80 100 (a) (b) (i) ° (ii) ° (iii) ° Unit 4: Number 85 13 Understanding measurement (c) ° (d) 3 Conversion graph, pounds to kilograms 60 40 Kilograms 20 0 40 80 Pounds 120 160 (a) (b) (c) (i) (ii) (iii) 13.5 More money Exercise 13.5 Tip Currency exchange rates – October 2011 Currency 1 US $ The inverse rows show the exchange rate of one unit of the currency in the column to the currency above the word inverse. For example, using the inverse row below US $ and the euro column, €1 will buy $1.39. 86 Unit 4: Number 1 Euro 1 UK £ US $ Euro (€) UK £ Indian rupee Aus $ Can $ SA rand NZ $ Yen (¥) 13 Understanding measurement (a) (i) (iii) (v) (ii) (iv) (vi) (b) (i) (ii) (iii) (i) (ii) (iii) (c) Mixed exercise 1 (a) (d) (g) (j) 3 (b) (e) (h) (k) 2 3 (c) (f) (i) (l) 2 3 3 2 3 4 5 (a) (b) 6 (a) (b) (c) Unit 4: Number 87 13 Understanding measurement 7 Conversion graph for imperial gallons to litres (a) (b) 45 40 30 Litres (i) (ii) (i) (ii) (c) 35 (d) (i) (ii) 25 20 15 10 5 0 Currency exchange rates – October 2011 0 4 6 8 2 Gallons (imperial) 10 1 mile = 1.61 km The US gallon is different from the imperial gallon with a conversion factor of 1 US gallon to 3.785 litres. 8 9 (a) (b) (c) 10 88 Unit 4: Number 14 Further solving of equations and inequalities 14.1 Simultaneous linear equations x y x y Exercise 14.1 A 1 x (a) (b) (c) x 3 y x y=0 − <x< x 2 y x 2y = 3 − <x< 2x + 2 2 y 33xx − 1 0 − <x< (d) y (e) x=4 x −7 x 33yy = 3 y 2x + 3 − <x< − <x< Unit 4: Algebra 89 14 Further solving of equations and inequalities 2 (a) (b) A (i) A F (ii) D C (iii) A F E (c) A C 10 y D B 8 F 6 4 2 x –10 –8 –6 –4 –2 0 –2 2 4 6 8 10 E –4 –6 –8 –10 x 3 y (a) y = 2 x+ y=6 (b) y x = 3 y 3x = 5 (c) x + y = 4 2 x + 3 y = 12 (a) x + y = 5 x y=7 (b) 3 2x (c) 2 x 33yy = 12 3x 3y 3 y = 30 (d) 2 x 33yy = 6 4 x 6 y = −4 (e) 2 x 5 y = 11 3x 2y 2y = 7 x 4 Remember that you might need to rearrange one or both equations before solving. y (a) 3x y 7 3x y 5 5x + y = 10 2 x 1 + y=3 4 4 2x 9 y (i) x = 6y 9 3 Unit 4: Algebra y 7 y 8 y 1 y 4 (f) y 2 x = 1 2 y 3x = 5 5 (e) 90 (d) 2 x 3x (b) 2x + y =3 2 x + y=3 2 (c) (f) 3x y + =4 4 2 3y x+ =7 2 7 x 6 y = 17 (g) 2 x 1 + y=6 3 3 3x 2y 2y = 4 2x y 3 x y + =2 (d) 3 2 x + 4 y 11 (h) 3x + y 1 = 2 2 5x 7 y =2 3 14 Further solving of equations and inequalities 6 7 8 Exercise 14.1 B 1 x (a) y (b) 20 (c) – 3x 6y 2 3x y +6 20 2x + 4 2x 9y 2y – 4x –3y – 6 4x – 2y (d) (e) (f) 2x + 7 x+y 1 2 x 5– 3y 2 3x 5x – 2y 9–y 2x + 4 9 + 3y 2 (a) d c (b) 14.2 Linear inequalities – – Unit 4: Algebra 91 14 Further solving of equations and inequalities Exercise 14.2 Tip 1 (a) x (d) (g) a (b) y ≥ (e) n (h) a (c) f (f) –m (i) a s (b) 1 (e) − (c) (f) a 2 (a) (d) x h e 3 a 2 b 3 (a) (b) x x x +8 3 r 1 + (i) 4 8 (e) (l) (o) y+4 1 ≥ y− 12 2 3 8 1 3 2x (f) x −5 4 (j) x (m) 2 9 (3 x (c) 3 33) ≥ 3 5 (g) (h) x (k) x 6 (d) x (n) 4 3 + g n 3 14.3 Regions in a plane Tip Exercise 14.3 y y mx line c y c mx mx 1 (a) y 92 x (c) y > –x 2 (a) (d) y (b) y x mx Unit 4: Algebra c (b) x y x y (e) (c) y y (f) x y x y 4 > 3 2 2 3 e e g n 1 x 4 n 14 Further solving of equations and inequalities 3 5 y 5 4 4 3 3 2 2 1 x –5 –4 –3 –2 –1 0 –1 1 2 3 4 5 1 –2 –3 –3 –4 –4 –5 –5 y 5 4 4 3 3 2 2 1 x –5 –4 –3 –2 –1 0 –1 1 2 3 4 5 x –5 –4 –3 –2 –1 0 –1 –2 5 y 1 2 3 4 5 1 2 3 4 5 y 1 –5 –4 –3 –2 –1 0 –1 –2 –2 –3 –3 –4 –4 –5 –5 x 4 x y x y y 5 (a) x y x y x (b) x y 14.4 Linear programming 5 4 3 2 1 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5 y Exercise 14.4 x 1 2 3 4 5 y 1 x y x 2 (a) y y (b) x x y y x y x x y x x a y Unit 4: Algebra 93 14 Further solving of equations and inequalities 3 Tip 4 14.5 Completing the square 2 a a x+ − 2 2 x2 + x 2 Exercise 14.5 ( x + )2 + b 1 (a) (d) (g) (j) Tip (b) (e) (h) (k) x 2 4x 4x 7 x 2 + 10 x + 17 x 2 − 16 x + 57 x 2 8x 5 (c) (f) (i) (l) x 2 + 14 x + 44 x 2 + 22 x + 141 x 2 + 18 x + 93 x 2 + 20 x + 83 2 x (a) x 2 5x 6 = 0 (b) x 2 x − 6 0 2 (e) x − 16 x + 3 0 (f) x 2 + 7x 7x 1 = 0 (i) x 2 2 x − 100 = 0 x (c) x 2 4x 4x 3 = 0 (g) x 2 + 9x 9x 1 = 0 3 Tip ax 94 x 2 + 6x 6x 4 x 2 − 12 x + 30 x 2 + 24 x + 121 x 2 2x 2 x 10 bx Unit 4: Algebra c (a) 2 x 2 (d) 2 3x − 2 0 3 5= x (b) x ( x − 4 ) = −3 (e) ( x + 1))(( x − 7 ) = 4 (c) 3x 2 2 (3 x 2 ) (f) x + 2 x 2 = 8 (d) x 2 6 x 7 = 0 (h) x 2 + 11x + 27 = 0 14 Further solving of equations and inequalities 14.6 Quadratic formula ax bx c −b ± b2 − 4ac x= 2a Exercise 14.6 Tip 1 (a) (d) (g) (j) Tip x (b) (e) (h) (k) x 2 + 14 x − 120 = 0 x 2 3x 4 = 0 x 2 + 10 x + 25 = 0 x 2 9x 9 x 20 0 (c) (f) (i) (l) x 2 + 5x 5x x 2 4x 4x x 2 + 2x 2x x 2 + 3x 3x (a) 2 + 6x 6x 4 = 0 (d) x 2 6 x 8 = 0 (g) x 2 + 24 x + 121 = 0 (j) x 2 2 x 10 0 (b) (e) (h) (k) x2 + x − 4 0 x 2 + 10 x + 17 = 0 x 2 + 11x + 27 = 0 x 2 8x 5 = 0 (c) (f) (i) (l) x 2 + 14 x + 44 = 0 x 2 + 7x 7x 1 = 0 x 2 + 18 x − 93 = 0 x 2 2 x − 100 = 0 (a) (b) 6 x 2 11x 35 0 (c) 2 x 2 (e) 4 x 2 133 (f) 8 x 2 x 2 − 14 x + 40 = 0 x 2 8x 8 x 15 0 x 2 + 4x 4 x 12 0 x 2 4 x 12 0 6=0 4=0 8=0 40 0 2 Tip 3 Consecutive numbers are one unit apart. 2 4x 8 = 0 (d) 2 x 2 5x = 25 (g) 9 2 1 6x (h) x ( x − 16) + 57 = 0 (j) 2 5x = 3 5x (k) 6 x 2x 2 9=0 13x 6 = 0 (i) 9 ( (l) 4 2 1 =0 2 4 x = 18 3x + 3x 1) x 2 3=0 4 5 Unit 4: Algebra 95 14 Further solving of equations and inequalities 14.7 Factorising quadratics where the coefficient of x2 is not 1 x Exercise 14.7 1 Tip (a) 2 2 (e) 4 2 (i) −x x− 3 (b) 9 3 (f) 14 x − 51x 51x + 7 (g) 3x (j) (k) 15x 2 16 x − 15 2 (c) 4 x 2 122 6x + 1 6x 2 3x 2 10 x 25 3x 2 77xx − 66 2 9 11xx 20 11 (d) 6 x 2 7x − 5 (h) 6 x 2 11x 7 (l) 8x 2 25x 3 2 (a) 4 x 2 122 (d) 6 x 2 6x (e) 4 7 xy 5 y 2 (g) 2 ( x 1) 4 x (j) (b) 2 x 2 − 9 2 2 4x (h) 2 ( x + 1) 2 ( (c) 50 x 2 + 40x 40 x + 8 2 4 (k) 3 x 1 x − 132 14 14x 1 2 (f) 12 x 2 3 + 3 ( x + 1) + 2 ) 1 + x ( x + 1) 2x 2 (i) 3 ( x 1) 2 (l) 8 x 2 10 ( x + 1) 25 25xy 3 y 2 14.8 Algebraic fractions Exercise 14.8 Tip 1 2x 14 16z (e) 6 3x x 34 x (f) 17 xy (a) (b) (i) 24ab2 36a 2b (a) x2 + x − 6 x 2 2x x 7x 12 x 2 (g) 18 xy (c) 12 x 30 5a (h) 25a 2b (d) 2 (e) (i) 96 Unit 4: Algebra a2 5a 14 a 7a 2 35x 2 + 49 x 15x 2 + 21x (b) a 2 b2 a + 2abb b2 (c) 2 x 2 xy x y2 2 x xy x (d) (f) a 2 b2 a + abb b2 (g) 2 x 2 77xx − 15 2 2 6 (h) 2 2 3x 2 10 x 3 x 2 2 x 15 x4 2x 2 3 x +1 2 14 Further solving of equations and inequalities Tip Tip 3 (a) 8 x 5x × 15 16 (b) 3y 2y × 4 9 (c) 5 7 × a a (d) 7 x 2 14 x ÷ 5 y 25 y 2 (e) a a c ÷ × b b a (f) 4a 2 b 4 b2 2a × ÷ 2a 3 3b2 (g) 4a 2 b2 3ab × ÷ 7b 8a 1 (h) (a) 3 2 + x y (b) 3 4− p 3 − + 2 8p 4p (c) (e) 3 2 + x + 2 x +1 (f) 2) 2m 3( − 3 2 (g) p.a. stands for ‘per annum’, which means ‘per year’. 66xx 8 x + 3 × 3x 9 2−x (i) 6 4 9x 2 2 2 4 (h) Mixed exercise x2 x2 2 3x − 3 (i) x2 + 4x 2 2 1 3 − 3 7 + 2p 5p x2 (d) 5 4 + p +1 2p 2 5 4 + x − 6 x2 + x − 2 2 x2 − 1 x 2 y = 12 1 x y x y 2 x y y 4x 4x = 7 2 y 33xx − 4 0 3 4 (a) −3 < + 2 (b) 5 − 2x 2 ≥7 5 −2 < ≤ 3 6 x y y > 1 5 y ≤ 4 x ≥ −2 y > x + 1 y ≥ x +1 x y x +y 2 7 (a) x 2 2 xy x (d) 2 y 2 133 y 7 (b) a 4 b2 (e) −4 x 2 + 2 x + 6 (c) x 2 + 6x 6 x 55 2 (f) ( x + 1) 5 ( x + 1) − 14 Unit 4: Algebra 97 14 Further solving of equations and inequalities 8 (a) 2a 2 22aa − 6 0 (d) 3x 2 5x 5x + 2 0 (b) 3 2 (e) 5x 2 4=0 3x = −33 2 (c) x 2 + 2x 2 x 15 0 (f) 3x 2 6 x = 6 2 3 5 x 9 (a) 5x 2 8x − 4 0 8x (b) p (b) 16 − 4 x 2 4 8 (c) 1 1 + 2 p2 5 p (e) x 2x z ÷ × 2 y yz 2 xy 3 (f) 3a 5 a a − + 2 3 5a 2 − qx + r = 0 10 (a) 98 Unit 4: Algebra x2 y2 (x + y) 2 (d) 2 7 x 2 15 y × 5 y 14 x (g) 4 x2 + 2 (h) 3x 6 x 2 3x 8 ÷ x2 4 2 − x − 2 x2 + 4x (i) 1 2 x 2 111 5 − 2x 2 1 x 1 15 Scale drawings, bearings and trigonometry 15.1 Scale drawings 1 50000 Exercise 15.1 REWIND 1 (a) Revise your metric conversions from chapter 13. ! (b) 2 (a) (b) × A5 is half A4 and has dimensions 14.8 cm × 21 cm. 3 (a) × (b) (i) (ii) × 5.5 m Centre service line Net Service line Base line Singles side line 1.4 m Doubles side line Unit 4: Shape, space and measures 99 15 Scale drawings, bearings and trigonometry 4 (a) × (b) 15.2 Bearings Exercise 15.2 1 (a) (b) (c) X 2 (a) N (b) Y (c) N N Y 70° 45° Y X 130° X X Y 3 ° (a) (b) (c) (d) 100 Unit 4: Shape, space and measures 15 Scale drawings, bearings and trigonometry 15.3 Understanding the tangent, cosine and sine ratios tanθ = sinθ = cosθ = opp( ) adj d( ) opp( ) hyp y adj d( ) hyp y the opposite side the adjacent side opp( ) adj( adj( ) × tanθ adj d( )= the opposite side the hypotenuse hyp sinθ opp hyp y = the adjacent side the hypotenuse hypotenuse opp( ) sinθ opposite side θ hyp cosθ hyp adj d opp( ) tanθ hyp y = adjacent side adj d( ) cosθ Exercise 15.3 1 (a) (b) (c) B c e g a A A (d) q f p A C b A r hypotenuse opp(A) adj(A) Remember, when working with right-angled triangles you may still need to use Pythagoras. 2 (a) ° = ° = 60° x cm =y 30° y cm (b) p cm 50° =p ° =q ° =q q cm 40° Unit 4: Shape, space and measures 101 15 Scale drawings, bearings and trigonometry 3 The memory aid, SOHCAHTOA, or the triangle diagrams A O S H C (a) (d) O H T A may help you remember the trigonometric relationships. (b) (e) ° ° (c) ° ° ° 4 (a) (b) (c) B A y 55° 1.2 cm 3 cm x d cm 1.8 cm C x= y= (d) 1 cm C A 4 cm A= x x= °= B= (e) Z 5m X A 1.5 cm 12 m y C 2.5 cm B Y y= ∠X = X= AC = B= C= (a) (b) 5 (c) ym B 3 cm C A X 37° c cm 72 m 15° B 65° A x cm Y C (d) B (e) am C A B 55° 13 m 45° A 102 Unit 4: Shape, space and measures 25 cm x cm C 2.5 cm B 15 Scale drawings, bearings and trigonometry 6 (a) (b) 1 4 (e) (c) 13 15 (f) (d) (g) 5 1 2 (h) 61 63 7 (a) (b) (c) 13.5 m A a A 2.4 cm c 15 cm 13 m b B B 17 cm C 0.7 cm C (d) (e) Z 5m A X 55 cm 12 m e d C 70 cm B θ = (iii) θ= Y 8 (i) (a) (ii) = (b) (c) B y θ x (d) θ c a z b a (e) P C b c θ Q θ y Z X q r x p R A z θ Y 9 (i) (a) (ii) (b) (c) 22 13 (iii) 7 (d) B 16 4x 5x D (e) 13 E 63 85 C A 12 10 Unit 4: Shape, space and measures 103 15 Scale drawings, bearings and trigonometry 10 (a) (b) (c) 3 2 1 2 (d) (e) (f) 11 H h 7.5 m H h 61º 27º Tip 12 (a) (b) 12 R 5 cm cm 15 m D E C Unit 4: Shape, space and measures y° 9m B D 75° P 3 cm S E B O 6 cm x° C C x° cm 3x m xm 104 (d) Q 8.6 6 cm 52° y cm B (c) A A D cm x° A 45° 5.6 cm E 15 Scale drawings, bearings and trigonometry 15.4 Solving problems using trigonometry Tip Exercise 15.4 1 horizontal angle of depression 2 lin eo fs igh t (a) line ight of s angle of elevation (b) horizontal 15.5 Angles between 0° and 180° θ= −θ θ θ θ θ Exercise 15.5 1 Tip θ θ (a) (e) − (i) θ – θ (b) (f) (j) (c) − (g) (d) (h) θ 2 (a) (e) (i) θ (b) (f) (j) θ θ θ θ θ θ − (c) (g) (d) (h) θ θ θ θ − 15.6 The sine and cosine rules sin A sin s B sin C = = a b c a2 b2 + c 2 bc cos A a b c = = sin A sin s B sin C b2 a2 + c 2 2ac cos B c 2 = a 2 + b2 ab cosC Unit 4: Shape, space and measures 105 15 Scale drawings, bearings and trigonometry Exercise 15.6 Tip x 1 Tip (a) sin sin 45° = 11 12 (b) x 23 = sin 50° sin 72° (c) sin x sin 38° = 7.4 5.2 (d) x 30 = sin 35° sin 71° (e) x 8 = (f) sin 55° sin 68° sin sin 45° = 4 6 (g) sin sin 35° = 24 36 x 8.5 = sin 59° sin 62° (i) sin sin105° = 4 16 (h) 2 (a) (b) (c) 7.7 cm 7.7 cm 57.2° 42° 5.5 cm x cm 114.5° x cm 54.3° 72° (d) 37° x cm (e) (f) 4.9 cm 25.5° 81.7° 71.5° 45° x cm 23.4 cm x cm 9.7 cm 24° x cm 60° 3 (a) (b) (c) θ 5.4 cm 4.8 cm θ 9.5 c m 129° 42° 9. 9 θ 8 cm cm 104° 5.6 cm (d) (e) 5.5 cm 124° θ 10 cm (f) θ 18.2 cm 7.5 cm 78° 40° 5.2 cm 11.8 cm θ 106 Unit 4: Shape, space and measures 15 Scale drawings, bearings and trigonometry b2 4 a2 + c 2 2ac cos B B θ 5 (a) (b) 3 cm 3.5 cm 2.4 cm 3 cm θ θ 2.4 cm 2.4 cm (c) (d) θ 4.7 cm 2.4 cm 5 cm 2.8 cm θ 4.1 cm Tip 3.2 cm 6 (a) (b) A 5.6 m X 25° 6.8 cm z 6.7 m 31° Y B a Z 7 cm C (c) (d) O P 4.7 cm m r 7 cm 131° 5.4 cm N M Q 43.5° 3.8 cm R Unit 4: Shape, space and measures 107 15 Scale drawings, bearings and trigonometry 15.7 Area of a triangle 1 1 2 ABC area = 2 ab sin C 1 2 area = ac sin B area = bc sin A Exercise 15.7 Tip 1 Draw a rough sketch of each of these figures before calculating their area. (a) (b) (c) (d) (e) (f) BC a XZ q XY p ∆ABC ∆ABC ∆XYZ PQR XYZ ∆PQR ∠C ∠C ∠X ∠R ∠Y ∠Q AC b XY ∠Q ∠X q 2 x 3 (a) (b) (c) 2.5 cm x x x 4.5 5 cm 7 cm cm 4 cm Tip 74° 65° 5 cm 6 cm 15.8 Trigonometry in three dimensions Exercise 15.8 1 Tip A D C B E H G 108 Unit 4: Shape, space and measures 3 cm 4 cm 9 cm F (a) (c) (e) (g) ˘ DEF ! DFE F HF CH (b) (d) (f) (h) FD GH ! GHF H ! CHF H 15 Scale drawings, bearings and trigonometry 2 P 60° Q 15 m (a) QS (c) PQ (d) PO 3 A 8 cm Tip (b) QO E B D 8 F 4 cm (a) (b) Mixed exercise cm C BD BD ! BDF CDEF x 1 (a) (b) sin x sin121° = 5. 2 7. 3 3.5 cm 5 cm x 4 cm (c) 11.5 cm 7.7 cm 39.4° x Unit 4: Shape, space and measures 109 15 Scale drawings, bearings and trigonometry Tip 2 CD 3 AB A 50° B 21° 530 m C 4 BD D AC A 70° B 50° C D 50 cm 5 X 6 Y R S T X Y X RS Y 1500 m 77° 58° R 110 Unit 4: Shape, space and measures T S T 16 Scatter diagrams and correlation 16.1 Introduction to bivariate data Exercise16.1 1 A B D E (a) (d) (b) (e) C (c) Unit 4: Data handling 111 16 Scatter diagrams and correlation 2 Student heights compared to distance jumped Distance jumped (m) 6.0 5.5 5.0 4.5 4.0 150 155 160 165 170 Height (cm) 175 (a) (b) (c) (d) (e) 3 Student Tip Age (years) Distance (m) 8 9 9 Jen (a) (b) (c) (d) (e) (f) (g) 112 Unit 4: Data handling e 180 16 Mixed exercise 1 (a) (b) (c) Accidents at a road junction 40 (i) (ii) 30 (d) (e) 20 10 d 2 0 35 70 105 140 Average speed (km/h) Comparison of car age with re-sale price 15 Price ($ 000) Number of accidents 50 0 Scatter diagrams and correlation 12 9 6 3 0 1 2 3 4 Age (years) 5 (a) (b) (c) (d) (e) Unit 4: Data handling 113 17 Managing money 17.1 Earning money Exercise 17.1 Casual workers are normally paid an hourly rate for the hours they work. Tip 1 2 3 (a) 4 Some workers are paid for each piece of work they complete. This is called piece work. 5 6 (a) (b) (c) 114 Unit 5: Number (b) (c) (d) 42 1 2 17 Managing money 17.2 Borrowing and investing money = × PRT I= 100 = = = 105 100 = 1 05 Exercise 17.2 ‘per annum’ (p.a.) means ‘per year’ 1 (a) (b) (c) (d) (e) Tip I= PRT 100 P= 100I RT R= 100I PT T= 100I PR 2 3 4 5 Unit 5: Number 115 17 Managing money Using the multiplying factor for compound interest gives the new amount. Subtract the principal to find the actual interest. 6 (a) (b) (c) (d) (e) 7 8 17.3 Buying and selling = = Rate of profit (or loss) = = Rate of discount = discount × 100% originall selling ll price Exercise 17.3 1 (a) (b) (c) (d) 2 3 4 116 Unit 5: Number 33 1 3 profit (or loss) ×100% cost price 17 Managing money 5 (a) (b) (c) Mixed exercise 5 1 2 1 (a) (b) (c) 2 (a) (b) 1 (c) 1 2 3 (a) (b) 4 Years 1 2 3 4 5 6 7 8 Simple interest Compound interest (a) (b) (c) 5 6 Unit 5: Number 117 17 Managing money 7 (a) (b) (c) 8 (a) (b) 9 10 11 12 118 Unit 5: Number 18 Curved graphs 18.1 Plotting quadratic graphs (the parabola) y = ax + bx + c y y a a Exercise 18.1 Remember, the constant term (c in the general formula) is the y-intercept. 1 − (a) y x −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y = −x2 + 2 (b) x y = x2 − 3 (c) x y = −x2 − 2 (d) x y = −x − 3 2 (e) x y = x2 + 1 2 Unit 5: Algebra 119 18 Curved graphs 2 13 12 11 10 9 8 7 6 5 4 3 2 1 –4 y 4 x2 –3 –2 (a) (b) (c) x –1 –10 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 x2 + 3 y y 3 4 (d) (e) 3 + x2 y 2 1 y = x2 + 2 y x2 3 (a) x −2 −1 0 1 2 3 4 5 y = x2 − 3x + 2 (b) x −3 −2 −1 0 1 2 3 y = x2 − 2x − 1 (c) x y = −x2 + 4x + 1 120 Unit 5: Algebra −2 −1 0 1 2 3 4 5 6 18 Curved graphs 4 Height in metres 10 8 6 4 2 0 0 1 2 3 4 Time in seconds 5 (a) (b) (c) (d) (e) 18.2 Plotting reciprocal graphs (the hyperbola) y= x y x a x xy = a y x y=x y y = −x Exercise 18.2 Tip 1 y xy = a x a (a) xy = 5 (d) y = − 8 x 16 x 4 y=− x (b) y = (e) (c) xy = 9 2 2 (a) (b) (c) (d) a 2 Unit 5: Algebra 121 18 Curved graphs 18.3 Using graphs to solve quadratic equations x y= x y Exercise 18.3 y 1 8 6 x2 − x − 6 y = x2 − x − 6 4 2 Tip –4 –3 –2 –1 0 –2 (c) 1 2 3 4 5 –4 –6 –8 (a) x 2 x −6 0 2 (a) (b) x 2 x −6 4 2 x −x+2 y (c) x 2 x x = 12 − (b) (i) − x 2 − x + 2 = 0 (ii) − x 2 − x + 2 = 1 (iii) − x 2 − x + 2 = −2 3 (a) x x x2 − x − 6 (b) (i) −6 = 2 − − 6 (ii) x 2 x − 6 0 (iii) x 2 x = 12 18.4 Using graphs to solve simultaneous linear and non-linear equations Exercise 18.4 1 4 y 2x + 2 x 2 (b) y = x + 2 x − 3 y 3 (c) y x2 + 4 y= x (a) y = 122 Unit 5: Algebra x +1 18 Curved graphs 2 (a) y 2 x 2 + 3x − 2 y (b) y = x 2 + 2 x 2x 2 + 2x + 4 (c) y (d) y y = x+2 x+4 y 2x − 4 1 2 0 5 x + x + 1.5 y x 2 18.5 Other non-linear graphs x x x x x3 x3 y ax3 bx2 cx y d xa Exercise 18.5 x 1 Tip (a) y (d) y (g) y y (b) (e) (h) x3 − 4x2 x3 + 4x2 − 5 x 3 − 3x 2 + 6 (c) (f) y = x3 + 5 y = x 3 + 2 x − 10 y 3x 3 + 5x y 2 Tip x –2 –1 0 1 2 x 3 + 5x 2 + 5 2x 3 + 4 x 2 − 7 y y x 3 − 5 x 2 + 10 2 3 4 5 6 y (b) (c) y (i) x 3 5x 2 10 0 (iii) x 3 5x 2 10 x 5 (ii) x 3 − ≤x≤ 5x 2 10 10 x 3 Tip x 3 − 5 x 2 + 10 (a) y (d) y 1 x 3 2x + x x− 1 x 2 3 x − x 4 x 1 x2 − x + x (b) y = x 3 + (c) y = x 2 + 2 − (e) (f) y y 4 x (a) y = 2 x (b) = 10 x y = 2− x y 2x − 1 x x Unit 5: Algebra 123 18 Curved graphs 5 (a) (b) (i) a 12 x + 1 (ii) (c) (i) A (ii) H 18.6 Finding the gradient of a curve = y change vertical change = x change horizontal change Exercise 18.6 y 1 x 2 − 2x − 8 (a) (b) y x 2 (a) y x3 − 1 x (b) A 3 30 y Growth rate of beans Growth in cm 25 20 with light 15 without light 10 5 x 0 124 Unit 5: Algebra 5 Days 10 15 18 Curved graphs Mixed exercise 1 (a) (b) (i) (ii) (c) A B A C C B 2 (a) y x2 y x3 (b) (c) (i) 2 = 4 (ii) x 3 + 8 0 (d) x 3 y 8 7 6 5 4 3 2 1 x 0 –8 –7 –6–5–4–3 –2–1 –1 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 (a) (b) (c) Unit 5: Algebra 125 19 Symmetry and loci 19.1 Symmetry in two dimensions If a shape can only fit back into itself after a full 360° rotation, it has no rotational symmetry. Exercise 19.1 1 (a) (b) A B E D G 2 (a) (b) 3 126 Unit 5: Shape, space and measures C F H 19 Symmetry and loci 19.2 Symmetry in three dimensions Exercise 19.2 Tip symmetry Tip symmetry 1 plane of (a) (b) (c) (d) (e) (f) (g) (h) 2 axis of (a) (b) (c) (d) (e) (f) Unit 5: Shape, space and measures 127 19 Symmetry and loci 19.3 Symmetry properties of circles Tip Exercise 19.3 x 1 y (a) (b) A O x 230° X x 80° Z B y Y MN PQ MN 2 M S PQ S MN SO P O R 9 cm N Q (a) MN PQ (b) 3 MN (a) N 128 Unit 5: Shape, space and measures MN S (b) M S PQ SO AB P O T C A O E Q B D T R 19 Symmetry and loci 19.4 Angle relationships in circles Tip Exercise 19.4 The angle relationships for triangles, quadrilaterals and parallel lines (chapter 3), as well as Pythagoras’ theorem (chapter 11), may be needed to solve circle problems. ∠NPO O 1 ∠MOP N M Q O 70° 15° P (a) ∠PNO (b) ∠PON (c) ∠MPN (b) ∠AOD (c) ∠BDC (d) ∠PMN 2 O B C 55° 25° O A (a) ∠ACD D Unit 5: Shape, space and measures 129 19 Symmetry and loci 3 E D 115° A C B 4 AB ABD B C 125° D A 5 SPT O SR ∠PSO ∠TPR S 29° O P Q R T ∠BAC O 6 A 72° O B 130 Unit 5: Shape, space and measures C D ∠BOC 19 Symmetry and loci AB DC ∠ADC 7 ∠DCA D A 64° 22° B C (a) ∠BAC (b) ∠ABC (c) ∠ACB O ∠QMO 8 SNT ∠MPN S M N 18° O Q 56° T P (a) ∠MNS (b) ∠MOQ (c) ∠PNT 19.5 Locus REWIND Make sure you know to bisect an angle as this is often required in loci problems (see chapter 3). ! Exercise 19.5 1 (a) A (b) A B (c) A B C Unit 5: Shape, space and measures 131 19 Symmetry and loci O 2 (a) (b) 3 1m 4 4 km Road HOOL CH SC School 5 132 Unit 5: Shape, space and measures A B 19 Symmetry and loci 6 Cell tower A Radio mast C 30 m 15 m B Flagpole Scale 1 cm = 5 m (a) AB (b) BC D D D X Mixed exercise 1 (a) (b) (c) (d) (e) (i) (ii) 2 (a) (b) (c) (d) Unit 5: Shape, space and measures 133 19 Symmetry and loci O 3 (a) (b) (c) (d) A A x A B 28° O D w yz A O O 160° C C x 29° B y C O (a) (b) 250 mm 18 cm y y x 120 mm O x 15 cm O 5 Route 66 Petrol station AB 6 A 134 Unit 5: Shape, space and measures B A O x y x B x 4 D B 60° C 20 Histograms and frequency distribution diagrams 20.1 Histograms Frequency density = frequency class interval Exercise 20.1 1 Marks class interval Remember there can be no gaps between the bars: use the upper and lower bounds of each class interval to prevent gaps. English frequency Mathematics frequency 3 7 5 6 51 6 3 1 REWIND You met the mode in chapter 12. ! (a) (b) (c) (d) Unit 5: Data handling 135 20 Histograms and frequency distribution diagrams Frequency 2 Ages of women in a clothing factory 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 19 21 23 25 27 Age (years) 29 31 (a) (b) (c) (d) (e) Tip 3 Age (a) a a a a a No. of people 4 House prices in property magazine 30 25 20 Frequency density 15 10 5 0 (a) (b) (c) 136 Unit 5: Data handling 20 40 60 80 100 120 140 Price ($thousands) 160 180 200 20 Histograms and frequency distribution diagrams 20.2 Cumulative frequency 3 1 Exercise 20.2 1 Percentage Number of students m m 3 m 7 m m m 51 m m m 3 m 1 (a) Tip (b) (c) (d) (e) 1 Height of professional basketball players (a) 3n 4 n 4 (b) (c) (d) 1 3 Cumulative frequency 2 Tip n 2 3 200 150 100 50 0 125 140 155 170 185 200 Height (cm) Unit 5: Data handling 137 20 Histograms and frequency distribution diagrams Mixed exercise 1 Heights of trees 10 Frequency 8 6 4 2 0 0 2 4 6 8 10 Height in metres 12 (a) (b) (c) 2 Mass (kg) Frequency ≤m< ≤m< 7 ≤m< 13 ≤m< ≤m< 11 (a) (b) (c) (d) 3 Minutes ≤m< m< No. of teenagers m< m< m< m< 15 4 Heights (h cm) ≤h Frequency (a) (b) (c) 138 Unit 5: Data handling 3 1 3 ≤h< ≤h< 15 ≤h< 21 Ratio, rate and proportion 21.1 Working with ratio a b 3 5 Remember, simplest form is also called ‘lowest terms’. 2 5 Exercise 21.1 1 (a) 2 (c) (e) 3 4 3 2 3 (b) 1 (d) x 2 Tip x 1 2 (a) x (b) (e) x (f) (g) (b) (c) (i) x 1 = 21 3 2 x = 7 4 5 3 (j) = x 8 x (c) x 5 16 = x 6 (d) x x 10 (h) = 4 15 3 4 (a) 5 A B C 6 Unit 6: Number 139 21 Ratio, rate and proportion 21.2 Ratio and scale n n Exercise 21.2 A Tip n 1 n n n (a) n n (b) (c) 1 n 2 (a) (b) (c) Exercise 21.2 B 1 2 3 (a) (b) 4 (a) (b) Map distance (mm) Actual distance (km) 5 (a) (b) (c) 140 Unit 6: Number 50 50 1 2 21 Ratio, rate and proportion 21.3 Rates n Exercise 21.3 Tip 1 2 3 (a) (b) (c) Remember, speed is a very important rate. distan t ce speed = time 4 distan t ce = speed × time time = (a) (c) distan t ce speed (b) (d) 2 5 (a) (c) 1 2 (b) (d) 21.4 Kinematic graphs Unit 6: Number 141 21 Ratio, rate and proportion Exercise 21.4 1 600 Distance (km) 500 400 300 200 100 0 1 2 3 4 5 Time (hours) 6 (a) (i) (ii) (iii) (b) (c) (d) (e) 2 Pam Pam bi lo 100 Da Distance (km) 200 0 60 120 Time (min) 180 (a) (b) (c) (d) (i) (e) (f) (g) (h) (i) 142 Unit 6: Number (ii) Dabilo 240 21 Ratio, rate and proportion 3 120 100 80 Speed (km/h) 60 40 20 0 1 2 3 4 Time (minutes) 5 6 (a) (i) (ii) (b) (c) (d) 4 Tip 30 Speed (m/s) 20 10 ms −2 0 20 40 60 80 Time (seconds) 100 120 (a) (b) (c) (d) Unit 6: Number 143 21 Ratio, rate and proportion 21.5 Proportion Exercise 21.5 Tip x x y x y 1 (a) y (b) A 600 A 5 B (c) A B 2 (a) (b) 3 4 5 (a) (b) (c) 6 (a) (b) 7 (a) (b) (i) (ii) 144 Unit 6: Number 6 B 21 Ratio, rate and proportion 8 (a) (b) 9 10 21.6 Direct and inverse proportion in algebraic terms P kQ PQ k k k Exercise 21.6 Tip a 1 b (a) (b) ∝ a b k a b m 2 T T m m T 3 F m (a) (b) F m xy 4 x 6 y a 5 y∝ (a) (b) 1 x y (a) (b) (c) m k b c x y x 6 y F m F x y x y x y x x y Unit 6: Number 145 21 Ratio, rate and proportion 7 8 x y (a) (b) y x (c) x y x 1 3 a b (a) (b) a b a x y y b a b x 9 (a) (b) y y= y x 10 x y 10 (a) (b) (c) y x x x y y x 21.7 Increasing and decreasing amounts by a given ratio new amount x x × old amount = new amount = old amount y y Exercise 21.7 1 2 3 (a) (b) 4 Mixed exercise 1 (a) (b) 2 146 Unit 6: Number x y x y 21 Ratio, rate and proportion 3 (a) (i) (ii) 4 (i) (ii) 1 2 (iii) (b) (iii) 4 Distance (km) 500 400 300 200 100 0 1 2 3 4 5 Time (hours) 6 7 (a) (b) (c) (d) (e) 5 20 Speed (m/s) 10 0 10 20 30 40 Time (seconds) 50 60 (a) (b) (c) (d) 6 7 Unit 6: Number 147 21 Ratio, rate and proportion P 8 (a) (b) 9 (a) (b) 148 Unit 6: Number V P P V V P V 22 More equations, formulae and functions 22.1 Setting up equations to solve problems x Tip Exercise 22.1 A x 1 (a) (b) (c) x) P A x x 2 (a) (b) x S M x 3 (a) (b) x S x 4 (a) (b) (c) Tip x x Exercise 22.1 B 1 2 3 4 Unit 6: Algebra 149 22 More equations, formulae and functions 5 X Y 6 Z Y Z X X X Y Z 7 8 22.2 Using and transforming formulae – – REWIND You have met this topic in chapter 6. This exercise uses the same principles to transform slightly more complicated formulae. ! Exercise 22.2 1 (a) U T V W (b) V W U T C (c) A = B B (d) A = C (e) P Q (f) P Q (g) P Q R (h) Q P (V) (V) (i) Q P PR (P) (j) Q (k) PQ (P − R ) R (P) (Q) (a) (b) Unit 6: Algebra (B) (Q) (Q) (Q) (P) V 2 150 (B) I I R V RI 22 More equations, formulae and functions A 3 (a) (b) r r 5 9 C 4 (a) (b) (c) F K (F − 32) C K C 22.3 Functions and function notation x x x x (d) m x Exercise 22.3 Tip 1 x x x (a) y (b) (c) x→ x 2 (a) (b) x (i) (ii) (iii) (c) (d) (i) (e) a a (ii) a b) h:x→ 5 x 3 (a) (b) x (i) x 4 (ii) x x (a) 5 (iii) b a x (b) x x x x Unit 6: Algebra 151 22 More equations, formulae and functions 6 (a) x x 7 (a) (d) Mixed exercise (b) x x x x g (x ) = 2 (b) x (e) x x x x (c) (c) (f) x (d) f ( x ) = x x) x) 1 2 3 4 5 6 b 7 (a) a = (b) a f (x ) = 8 9 152 Unit 6: Algebra 5 (b 9 b x x (a) (d) x x ) (b − 3) + 3 b) 2x − 3 5 x g (x ) = x −2 5 (b) (e) (c) a a x −2 23 Transformations and matrices 23.1 Simple plane transformations ′ x y x x The enlargement factor = Tip x y y length of a side on the image length of the corresponding side on the object Exercise 23.1 A ABCD 1 (a) (b) ° A′B′C′D′ D A′B′C′D′ B′D′ Unit 6: Shape, space and measures 153 23 Transformations and matrices 2 y 8 A 6 4 2 B –8 C –6 –4 0 –2 H 2 4 E (a) (b) (c) (i) (ii) F x –4 –6 G 8 I –2 D 6 K J –8 ΔABC y=− DEFG HIJK ° H ′I ′J ′K ′ − 3 10 −9 (a) ΔABC (b) 154 Unit 6: Shape, space and measures 0 −6 PQRS y P′Q′R′S ′ A′B ′C ′ 23 Transformations and matrices 4 10 y 8 D D' A 6 4 A' 2 –10 C' –8 –6 –4 –2 2 0 4 6 8 10 x –2 C –4 B' B –6 –8 –10 x 5 6 10 y 8 6 4 2 –10 –8 –6 –4 –2 0 –2 2 4 6 8 10 x –4 –6 –8 –10 Unit 6: Shape, space and measures 155 23 Transformations and matrices Exercise 23.1 B REWIND 1 You dealt with enlargement and scale factors in chapter 21. ! y 8 You can find the centre of enlargement by drawing lines through the corresponding vertices on the two shapes. The lines will meet at the centre of enlargement. 6 B' 4 When the image is smaller than the object, the scale factor of the ‘enlargement’ will be a fraction. A' –8 –6 –4 A –2 2 0 B –2 C' 4 2 6 D –4 C x 8 D' –6 –8 2 a b c d A C X B X 156 Unit 6: Shape, space and measures X D X 23 Transformations and matrices Exercise 23.1 C ABC 1 ΔA′B′C′ y ΔA″ B″ C″ x y 8 A 6 4 C 2 –1 –1 B 2 4 x 8 6 (a) ΔABC (b) ΔABC ΔA″ B″ C″ 2 −8 1 y 8 6 4 2 –1 –1 A 2 4 6 (a) 8 x x (b) ABCD 3 A x= (a) (b) B C y= D ′ ABCD ′ Unit 6: Shape, space and measures 157 23 Transformations and matrices 23.2 Vectors A a b c AB B ! " AB AB x y 1 2 a x ! " AB –a x y y ! " AB kx ky k x1 x2 x1 + x2 y + y = y + y 1 2 1 2 ! " !" ! " ! " AB − CA = AB + AC y ! " AB x2 + y2 O ! " OA A ! " OA Exercise 23.2 A Remember, a scalar is a quantity without direction, basically just a number or measurement. 10 y 1 8 6 A (a) (c) (e) (g) (i) B 4 2 Remember, equal vectors have the same magnitude and direction; negative vectors have the same magnitude and opposite directions. 158 –6 Unit 6: Shape, space and measures –4 –2 E 2 0 –2 D 4 6 8 10 x (j) (k) (l) ! " ! " !AE" DB ! " CD !" BE ! " CD ! " ! " AB ! " + BC ! " ! " !AE" − AB BC = ED (b) (d) (f) (h) ! " BC ! " !BD" EC !" BE 23 Transformations and matrices 2 Tip ! " 2 (a) AB = 3 ! " −1 (b) CD = 4 ! " 4 (c) EF = 5 ! " 3 (d) GH = −4 3 Remember, A is the object and A’ is the image. 10 y 8 B' 6 C B 4 C' –10 –8 –6 –4 E 2 –2 0 –2 E' 2 D –4 D' –6 A 4 6 8 10 x A' –8 –10 Exercise 23.2 B 1 −2 a= b= 3 4 1 (a) b (f) a + b + c (b) (g) 0 c= −3 a a+ b (c) − c (h) b c (d) a + b (i) − a b (e) b + c (j) a b+ c 2 A B G a D (a) (e) (i) F c O b C ! " ! " = a OC = b ! " DE !" OE !" !" 4 BF 3EF ! " OG = c ! " (b) AD (c) ! " (f) CD (g) ! " !" (j) 1 OC + 3GB ! " AG !" ! " BF + FD a b c !" (d) OB ! " !" (h) DE + EF 2 E Unit 6: Shape, space and measures 159 23 Transformations and matrices ! " OA 3 (a) –a Tip (b) a 1 2 (c) a (d) (e) − 1 a 2 a 4 ! " −2 (b) PQ = 7 ! " −13 (d) ST = −12 ! " 4 (a) MD = 5 ! " 9 (c) XY = −12 O 5 X ! " −6 ! " −2 OA = OB = −4 2 6 (a) x A B !" CA ! " ! " AB BC (b) 7 ! " XYZ y !" YZ x ! " 5 OC = 1 C ! " XY y M ! " ! " XZ ZX Y x y M X 8 a Z ! " (b) OY ! " (d) XY ! " (a) OX ! " (c) OZ Tip Y Z 1 2 b 3 0 c −2 5 (a) x y (i) x (ii) y (iii) z z a b c a b c a b c (b) (i) 9 Tip x X ! " AY ! " XY AX a Y b B a D b Unit 6: Shape, space and measures Y (iii) z AB X A 160 (ii) y C AC D ! " MZ 23 Transformations and matrices (a) ! " (i) XY (b) ! " (ii) AD a b ! " (iii) BC XY BC 10 23.3 Further transformations y x x y Exercise 23.3 1 (a) x (b) y y y 10 9 8 7 6 5 4 3 2 1 –1 0 –1 x 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 –1 0 –1 x 1 2 3 4 5 6 7 8 9 10 Unit 6: Shape, space and measures 161 23 Transformations and matrices (c) Tip y (d) y y y 10 9 8 7 6 5 4 3 2 1 k –1 0 –1 x 1 2 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 2 1 –1 0 –1 2 162 (a) (b) (c) (d) Unit 6: Shape, space and measures x 1 2 3 4 5 6 7 8 9 10 23 Transformations and matrices 23.4 Matrices and matrix transformation A A I –1 c d A ad bc d zero b a 1 = A–1 A a d A–1 I 1 (ad − bc ) A (ad − bc ) A determinant A c B A identity 1 B identity ad bc (ad − bc ) −c B A B a b c d A A b A A 1 0 I 0 1 AI A IA A 0 0 O 0 0 A A B 1 0 1 0 Exercise 23.4 −1 2 3 4 3 −3 A 1 (a) A 2 D (a) (b) B B 1 3 (b) A B 4 2 1 0 (c) B A 1 4 5 8 D D–1 Unit 6: Shape, space and measures 163 23 Transformations and matrices 3 1 A 3 0 2 B (a) (b) AB (i) A (a) A 3 0 2 −1 B B 6 7 (b) 3A C (a) (b) BA (ii) AB 1 A 4 4 5 1 3 −4 2 2 x 1 4 3 −3 2 5 (c) B C x C–1 23.5 Matrices and transformations y BA x y A AB B BA Exercise 23.5 Tip 1 (i) (ii) (a) A 10 9 8 7 6 5 4 3 2 1 –4 –3 –2 –1 0 –1 164 Unit 6: Shape, space and measures (b) y B x 1 2 3 4 5 6 10 9 8 7 6 5 4 3 2 1 y A B –1 0 1 2 3 4 5 6 7 8 9 10 –1 x 23 Transformations and matrices 2 P 1 0 0 4 Q R S (a) (b) 3 0 A 0 3 A 3 (a) (b) B BA B x 4 Mixed exercise −1 0 0 1 1 10 y 8 6 B H A 4 2 E –10 –8 –6 –4 –2 0 –2 2 4 6 C 8 x 10 –4 G –6 F –8 D –10 (a) (i) (ii) (iii) (b) (i) (ii) (iii) DEFG 2 y ° D′E′F′G′ 8 (a) 6 (b) 4 (iv) −4 5 DEFG DEFG D″ E″ F″ G″ D' G' 2 E' F' 2 4 6 8 x Unit 6: Shape, space and measures 165 23 Transformations and matrices ABCD 3 10 y 8 6 A B 4 2 –10 –8 –6 –4 –2 D –2 C 2 4 6 8 10 x –4 –6 –8 –10 B (a) (b) ABCD ABCD (c) x=− ° −3 2 ABCD (d) ABCD −2 c= −4 3 2 a= b= 6 −4 4 (a) (i) a (ii) b + c (iii) a b (iv) a + b (ii) b c b + c (iii) a b a b (iv) a b a + b !" ! " AB y O FA x (b) (i) a a x A F 5 ABCDEF B (a) C E D 166 y (b) Unit 6: Shape, space and measures ! " (i) ED !" (iv) EF x !" (iii) FB ! " (ii) DE ! " (v) FD !" EB E y B 23 Transformations and matrices 6 (a) (b) y B′ B A (c) 7 x C (d) ABCD 1 2 x Unit 6: Shape, space and measures 167 23 Transformations and matrices 8 ABCD y 10 9 8 7 6 5 4 3 2 1 D –1 0 –1 A (a) (b) (c) (d) (e) (f) B 1 2 C 3 4 5 6 7 8 9 10 1 2 P 2 1 9 y x 1 3 Q 2 2 P Q P Q PQ QP P P–I 2 5 a a 45 3 4 6 b = 47 −2 4 −1 3 c 10 a b c 11 y 10 9 8 7 6 A 5 4 3 2 1 B x C –1 0 1 2 3 4 5 6 7 8 9 10 –1 (a) (b) (c) (d) (e) 168 Unit 6: Shape, space and measures ABC x A A'B'C' B A B 24 Probability using tree diagrams 24.1 Using tree diagrams to show outcomes Tip Exercise 24.1 1 2 H A G 1 2 B F 4 3 C E D 3 (a) (b) 24.2 Calculating probability from tree diagrams Exercise 24.2 1 Unit 6: Data handling 169 24 Probability using tree diagrams H Blue T H Yellow T H Black T (a) (b) (c) (d) 2 First toss 1 2 1 2 (a) (b) (c) (d) (e) Mixed exercise 1 (a) (b) (c) (d) Tip 2 (a) (b) (c) (d) (e) 170 Unit 6: Data handling Second toss 1 2 H 1 2 T 1 2 H 1 2 T H T H T H T Answers 1 (a) 2, 3, 5, 7 (b) 53, 59 (c) 97, 101, 103 Chapter 1 Exercise 1.1 1 Number Natural Integer Prime Fraction −0.2 ✓ −57 ✓ 3.142 ✓ 0 0.3̇ ✓ 1 ✓ ✓ 51 ✓ ✓ 10 270 ✓ ✓ 1 − 4 ✓ 2 7 ✓ 9 11 3 512 (s) 4.03 (u) 6.61 (w) −19.10 Exercise 1.3 Exercise 1.7 2 (a) (b) (c) (d) (e) (f) (g) (h) 2×2×3×3 5 × 13 2×2×2×2×2×2 2×2×3×7 2×2×2×2×5 2×2×2×5×5×5 2 × 5 × 127 13 × 151 3 (a) (b) (c) (d) (e) (f) (g) (h) LCM = 378, HCF = 1 LCM = 255, HCF = 5 LCM = 864, HCF = 3 LCM = 848, HCF = 1 LCM = 24 264, HCF = 2 LCM = 2574, HCF = 6 LCM = 35 200, HCF = 2 LCM = 17 325, HCF = 5 ✓ ✓ Exercise 1.4 ✓ ✓ ✓ ✓ ✓ 1 square: 121, 144, 169, 196, 225, 256, 289 cube: 125, 216 2 (a) 7 (d) 10 (b) 5 (e) 3 (c) 14 (f) 25 (g) (h) 5 (i) 2 3 4 (j) 5 (k) (m) –5 (n) 3 (a) 23 cm Exercise 1.2 A 1 (a) 18 (d) 24 (g) 72 (b) 36 (e) 36 (h) 96 (c) 90 (f) 24 2 (a) 6 (d) 3 (g) 12 (b) 18 (e) 10 (h) 50 (c) 9 (f) 1 5 (a) 1024 cm2 (b) 200 tiles (i) 5.65 9.88 12.87 0.01 10.10 45.44 14.00 26.00 (ii) 5.7 9.9 12.9 0.0 10.1 45.4 14.0 26.0 (iii) 6 10 13 0 10 45 14 26 2 (a) 53 200 (c) 17.4 (b) 713 000 (d) 0.00728 3 (a) (c) (e) (g) (b) (d) (f) (h) 36 12 000 430 000 0.0046 5.2 0.0088 120 10 1 natural: 24, 17 3 1 4 2 rational: − , 0.65, 3 , 0 66 integer: 24, –12, 0, 17 prime: 17 2 (a) two are prime: 2 and 3 (b) 2 × 2 × 3 × 3 (c) 2 and 36 (d) 36 3 (a) 2 × 2 × 7 × 7 (b) 3 × 3 × 5 × 41 (c) 2 × 2 × 3 × 3 × 5 × 7 × 7 4 14th and 28th March (b) −9 °C (c) −12 °C 3 (a) 4 (d) −2 (b) 7 (e) −3 (c) −1 1 (a) 26 (c) 23.2 (e) 3.39 (g) 3.83 (i) 1.76 (k) 7.82 (m) 8.04 (o) 8.78 (q) 94.78 4 after 420 seconds; 21 laps (Francesca), 5 laps (Ayuba) and 4 laps (Claire) (o) 6 6 (b) 529 cm2 2 (a) −2 °C Exercise 1.6 3 20 students (l) 12 1 −3 °C 1 18 m 2 120 shoppers 3 1 4 5 Exercise 1.5 Exercise 1.2 B 1 (a) (b) (c) (d) (e) (f) (g) (h) Mixed exercise ✓ 2 (a) 121, 144, 169, 196, . . . 1 1 2 2 (b) , , , , etc. 4 6 7 9 (c) 83, 89, 97, 101, . . . (d) 2, 3, 5, 7 (t) 6.87 (v) 3.90 (x) 20.19 (b) (d) (f) (h) (j) (l) (n) (p) (r) 66 15.66 2.44 2.15 2.79 0.21 1.09 304.82 0.63 5 (a) true (c) false (b) true (d) false 6 (a) 5 (c) 64 (b) 5 (d) 145 7 (a) 16.07 (d) 11.01 (b) 9.79 (e) 0.12 8 (a) 1240 (c) 0.0238 (c) 13.51 (f) −7.74 (b) 0.765 (d) 31.5 9 (a) It is not strictly possible – to have a tile of 790 cm2, the builders must be using rounded values. (b) 28.1 cm (c) 110 tiles Answers 171 (e) –2x + 8x 1 (a) 3(x + 2) (c) 2(11 + x) (e) 3x2 + 4 1 (g) − x (b) (d) (f) (h) 5 (i) 4 + 3x 3 (a) $ 6(x − 1) 18x x2 + 8 1 x+ 3 (j) 12 – 5x (b) p − 4 (c) 4p x 3 x 9 (b) $ , $ 2x 2x , and $ 9 3 (b) 1.875 m2 (d) 8 cm2 2 –104 (b) 91 3x 2 − 2x + 3 4x 2y − 2xy 5ab − 4ac 4x 2 + 5x − y − 5 −30mn (f) 6x 2y 6xy 3 (h) −4x 3y (i) 4b (j) (k) 3b (o) (q) (s) 20 y 3x 2y 2 2 x 2 15a 4 x 6y 1 4y (l) 9m 4 (n) 3x y 172 5x 9 2y 3 x 2 Answers 3 (k) 16 (l) x 8 5 y 2 5 y 16 x 22 3 (a) x (c) x (f) 1 2 2 y 2 1 6 (r) −14 y 5 2 x 9 y 22 2x 1 4 (j) x y −2 2 3 or 1 1 y2 (d) x2 2 (f) x y 29 − 16 xy − 3x −3x + 2 3x + 1 x2 + x + 2 (b) x 2 + xy 1 (a) –1296 (b) –1 1 (g) 32 (h) 4 2 (g) 12 − x x 4 (h) x 3 − x 2 (a) −6 (b) 24 3 (a) −2 (b) or 1 29 x 4 y 16 4 (c) (f) (i) 8 4x y (c) 5 5 (a) (b) (c) (d) 11x − 3 6x 2 + 15x − 8 −2x 2 + 5x + 12 −x 3 + 3x 2 − x + 5 6 (a) 5x 6 5 (b) 15 1 x (d) 16x4y8 4 (e) 64 x (g) 27 x y 9 (f) x 9y 8 15 4y 4 (h) 3 xy 2 6 1 (b) x 2 (c) x −9 y or y x 9 5 3 5 2y3 x Chapter 3 625 Exercise 3.1 A 2 2 3 −14 9 5y 2 3 1 3 (c) 7 (e) −2 9a + b x 2 + 3x − 2 −4a 4b + 6a 2b3 −7x + 4 (d) 2 x y or (e) 3 (f) − 13 (d) 2 (j) (e) 4x 7 (a) 5xy 3 1 Exercise 2.5 B (b) (d) (f) (h) 3 2 (b) x − 4 1 − 14 (f) 2 x 3 (c) 2 4 x y (b) x2 7 3 (e) x y x3 y or −4 (e) 2 (h) (f) 5x − 1 9 1 (c) –4 (d) (e) 8 15 (h) x 3 y 4 (b) 4 (c) 5x (d) 4 (a) (b) (c) (d) 4 (f) 2 x 3 y 3 (g) x 2 y 4 3 4 2 1 (e) 8x (c) y 4 y 1 (d) x 3 4 (a) x x (b) x (k) y −2 or 2 27 x 10 (d) 7 3 3125 x y 16 y (b) 10 8x y 3 16 8 3 25 x y (j) x x y 49 (h) 4 3 1 (a) x + 12 (f) x 7y 3 50 x 27 y 1 Mixed exercise (d) xy 10 (i) x (p) (t) 2x 2 − 4x −2x − 2 −2x 2 + 6x x 3 − 2x 2 − x 2 (a) x 2 + 2 3 (a) (b) 65 (d) –163 (g) 2 (b) 3x 4y 3 Exercise 2.4 1 (a) (c) (e) (g) 2 (e) (e) Exercise 2.3 (m) 6 y 2x 3y (c) 4 17.75 5 (a) 6 x (c) 2 (a) 3 17 1 (a) (b) (c) (d) (e) (g) 1 (a) 2 (a) 17 (c) 15 (d) Exercise 2.5 A (i) x 7y 1 (a) 54 cm2 (c) 110.25 cm2 3y 2 2 (g) –5x2 – 6x (g) Exercise 2.2 + (f) 3x – 6x 2 Exercise 2.1 2 (a) p + 5 x 2 (c) −8x 3 + 4x 2 + 2x (d) Chapter 2 1 (a) (i) 150° (b) 45° 1 3 (ii) 180° (iii) 135° (f) x = 58° (base ∠s isosceles and ∠s in triangle); y = 26° (ext ∠s equals sum int opps) (g) x = 33° (base ∠s isosceles then ext ∠s equals sum int opps) (h) x = 45° (co-int ∠s then ∠s in triangle) (i) x = 45° (base ∠s isosceles); y = 75° (base ∠s isosceles) (c) (i) 810° (ii) 72° (d) quarter to one or 12:45 2 No. If the acute angle is ≤ 45° it will produce an acute or right angle. 3 Yes. The smallest obtuse angle is 91° and the largest is 179°. Half of those will range from 45.5° to 89.5°, all of which are acute. 4 (a) 45° (b) (90 − x)° (c) x° 5 (a) 135° (c) (180 − x)° (e) (90 + x)° (b) 90° (d) x° (f) (90 − x)° Exercise 3.1 B 1 ∠QON = 48°, so a = 48° (vertically opposite) 2 (a) ∠EOD = 41° (∠s on line), so x = 41° (vertically opposite) (b) x = 20° (∠s round point) 2 (a) x = 36°; so A = 36° and B = 72° (b) x = 40°; so A = 80°; B = 40° and ∠ACD = 120° (c) x = 60° (d) x = 72° 3 ∠B = 34°; ∠C = 68° Exercise 3.3 1 (a) (b) (c) (d) (e) (f) Exercise 3.1 C 1 (a) x = 85° (co-int ∠s); y = 72° (alt ∠s) (b) x = 99° (co-int ∠s); y = 123° (∠ABF = 123°, co-int ∠s then vertically opposite) (c) x = 72° (∠BFE = 72°, then alt ∠s); y = 43° (∠s in triangle BCJ) (d) x = 45° (∠s round a point); y = 90° (co-int ∠s) 2 (a) x = 112° (b) x = 45° ( STQ corr ∠s then vertically opposite) (c) x = 90° (∠ECD and ∠ACD co-int ∠s then ∠s round a point) (d) x = 18° (∠DFE co-int with ∠CDF then ∠BFE co-int with ∠ABF) (e) x = 85° Exercise 3.2 1 (a) (b) (c) (d) (e) 103° ( s in triangle) 51° (ext ∠ equals sum int opps) 68° (ext ∠ equals sum int opps) 53° (base ∠s isosceles) 60° (equilateral triangle) (g) (h) (i) 2 square, rhombus rectangle, square square, rectangle square, rectangle, rhombus, parallelogram square, rectangle square, rectangle, parallelogram, rhombus square, rhombus, kite rhombus, square, kite rhombus, square, kite 45° 45° 45° 45° 45° 45° 45° 45° 27° 27° 3 (a) (b) (c) (d) (e) (f) 1 (a) (i) 1080° (b) (i) 1440° (c) (i) 2340° 2 900 7 (ii) 135° (ii) 144° (ii) 156° = 128.57° 3 20 sides 4 (a) 165.6° (b) 360 14.4 = 25 sides 5 (a) x = 156° (b) x = 85° (c) x = 113°; y = 104° Exercise 3.5 1 M (d) tangent D (b) chord ( ) major arc sector 50° (a) O diameter E P N ( ) 1, 2 student’s own diagrams 3 2.1 cm 4 (a) scalene (b) yes 5 6.6 cm 27° 27° x = 69° x = 64° x = 52° x = 115° x = 30°; 2x = 60°; 3x = 90° a = 44°; b = 68°; c = 44°; d = e = 68° 4 (a) ∠Q + ∠R = 210° (b) ∠R = 140° (c) ∠Q = 70° Exercise 3.4 Exercise 3.6 63° 63° 63° 63° 5 (a) ∠MNP = 42° (b) ∠MNO = 104° (c) ∠PON = 56° Mixed exercise 1 (a) (b) (c) (d) (e) (f) (g) (h) x = 113° x = 41° x = 66° x = 74°; y = 106°; z = 37° x = 46°; y = 104° x = 110°; y = 124° x = 40°; y = 70°; z = 70° x = 35°; y = 55° 2 (a) x = 60 + 60 + 120 = 240° (b) x = 90 + 90 + 135 = 315° (c) x = 80° Answers 173 Exercise 4.2 1 Phone Tally calls Frequency 1 / 1 2 // 2 3 4 // //// 2 5 5 //// //// 9 6 //// // 7 7 //// / /// 6 8 9 /// 3 10 // 2 Brown Blue Green Male 4 0 1 Female 2 1 2 (b) Hair colour Brown Black Blonde Male 2 2 1 Female 1 4 0 No. of brothers/ sisters 0 1 2 3 4 Male 0 1 1 2 1 Female 2 1 1 1 0 Exercise 4.3 1 (a) pictogram (b) number of students in each year group in a school (c) 30 students (d) half a stick figure (e) 225 (f) Year 11; 285 (g) rounded; unlikely the year groups will all be multiples of 15 3 No. of mosquitoes 0 1 2 3 4 5 6 Frequency 7 6 9 7 8 7 6 (b) It is impossible to say; frequency is very similar for all numbers of mosquitoes. Answers Eye colour (c) student’s own sentences 2 (a) 174 4 (a) 2 student’s own pictogram 3 (a) number of boys and girls in Class 10A (b) 18 (c) 30 (d) the favourite sports of students in Class 10A, separated by gender (e) athletics (f) athletics (g) 9 (d) C 26° C May–Nov northern hemisphere No 35 mm April none Mixed exercise 1 (a) survey or questionnaire (b) discrete; you cannot have half a child (c) quantitative; it can be counted (d) No. of children in family 0 4 5 6 Frequency 7 10 11 12 5 2 1 1 2 3 (e) student’s own chart (f) student’s own chart 2 student’s own pictogram 3 (a) compound bar chart (b) It shows how many people, out of every 100, have a mobile phone and how many have a land line phone. (c) No. The figures are percentages. (d) Canada, USA and Denmark (e) Germany, UK, Sweden and Italy (f) Denmark (g) own opinion with reason 4 (a) Student’s own chart (b) Student’s own chart (c) Student’s own chart 5 (a) 15 Value of car over time 10 5 0 10 20 11 20 12 20 13 1 (a) gender, eye colour, hair colour (b) height, shoe size, mass, number of brothers/sisters (c) shoe size, number of brothers/ sisters (d) height, mass (e) possible answers include: gender, eye colour, hair colour − collected by observation; height, mass − collected by measuring; shoe size, number of siblings − collected by survey, questionnaire (b) 10 (c) 2 (d) 26 (e) There are very few marks at the low and high end of the scale. 7 (a) (b) (c) (d) (e) (f) (g) 09 Exercise 4.1 6 (a) student’s own chart (b) 6 (c) 50 20 Chapter 4 5 (a) cars (b) 17% (c) 20 (d) handcarts and bicycles 20 5 student’s own diagram 4 (a) student’s own chart (b) student’s own chart Frequency 1 1 7 19 12 6 4 Value ($ 000) 4 student’s own diagram Score 0−29 30−39 40−49 50−59 60−69 70−79 80−100 08 3 (a) 20 3 (a) (i) radius (ii) chord (iii) diameter (b) AO, DO, OC, OB (c) 24.8 cm (d) student’s own diagram Time (years) (b) 49.6% (c) $3900 Chapter 5 3 (a) 83% (d) 37.5% Exercise 5.1 1 (a) x = 65 (b) x = 168 (c) x = 55 (d) x = 117 (e) x = 48 (f) x = 104 Exercise 5.2 1 (a) 25 8 (b) 17 11 (c) 59 5 (d) 15 4 (e) 59 4 (f) 25 9 2 (a) (d) 63 13 108 5 (b) 28 5 (e) 3 (c) 14 (f) 6 19 (g) 120 (h) 3 14 (i) 72 (j) 3 (k) 233 50 (l) 7 4 3 (a) 13 24 (b) 19 60 (c) 19 21 (d) 35 6 (e) 183 56 (f) 161 20 (g) 18 65 −5 6 (h) 41 40 (i) (k) −10 3 (l) 29 21 −26 9 4 (a) 24 (b) (c) 10 27 (e) 96 7 10 9 3 5 4 5 5 12 11 170 (c) (j) (d) (g) 2 5 (a) (d) 38 9 19 4 (g) 0 6 (a) $525 (h) (b) (e) (h) (f) 7 96 9 14 (i) 4 (f) (i) 39 7 215 72 187 9 (b) $375 (b) 62.5% (e) 4% (h) 207% 2 (a) 1 8 (b) 1 2 (d) 3 5 (e) 11 50 60 kg 150 litres $64 18 km $2.08 (b) (d) (f) (h) (j) 5 (a) (c) (e) (g) +20% +53.3% −28.3% +2566.7% (c) 7% (f) 250% $24 55 ml £19.50 0.2 grams 475 cubic metres (c) 29.8% (f) 47% (c) 49 50 (g) 0.000028 (i) 0.00245 (h) 94 000 000 Exercise 5.4 B 1 (a) (b) (c) (d) (e) (f) (g) (h) (i) 6.56 × 10−17 1.28 × 10−14 1.44 × 1013 1.58 × 10−20 5.04 × 1018 1.98 × 1012 1.52 × 1017 2.29 × 108 4.50 × 10−3 1 28 595 tickets 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) 12 × 1030 4.5 × 1011 3.375 × 1036 1.32 × 10−11 2 × 1026 2.67 × 105 1.2 × 102 2 × 10−3 2.09 × 10−8 2 1800 shares 3 (a) the Sun 3 $129 375 4 (a) 500 seconds = 5 × 102 seconds (b) 19 166.67 seconds = 1.92 × 104 seconds (b) −10% (d) +3.3% (f) +33.3% 6 (a) $54.72 (c) $32.28 (e) $98.55 (b) $945 (d) $40 236 (f) $99.68 7 (a) $58.48 (c) $83.16 (e) $76.93 (b) $520 (d) $19 882 (f) $45.24 Exercise 5.3 B 4 21.95% 5 $15 696 7 2.5 grams 8 7 25 = 28% increase, so $7 more is better Exercise 5.3 C (b) 960 (b) 27 750 (c) $114 885 4.5 × 104 8 × 10 4.19 × 106 6.5 × 10−3 4.5 × 10−4 6.75 × 10−3 2 (a) 2500 (c) 426 500 (e) 0.00000915 4 × 5 = 20 70 × 5 = 350 1000 × 7 = 7000 42 ÷ 6 = 7 (b) (d) (f) (h) (j) (l) (b) 3 (d) 243 Mixed exercise 3 $150 1 (a) (c) (e) (g) (i) (k) 1 (a) (b) (c) (d) 2 (a) 20 (c) 12 1 $50 2 (a) 1200 (b) 6.051 × 106 Exercise 5.5 6 $6228 Exercise 5.4 A Exercise 5.3 A 1 (a) 16.7% (d) 30% (g) 112% 4 (a) (c) (e) (g) (i) 4 (a) $12 7 (a) 300 (b) 6 hours 56 min (b) 60% (e) 125% 8 × 105 2.345 × 106 3.2 × 1010 9 × 10−3 8 × 10−7 4.5 × 10−10 (b) 39 000 (d) 0.00001045 (f) 0.000000001 1 (a) 40 (c) 22 (b) 6 (d) 72 2 (a) 4 5 (b) 3 (a) 1 6 (d) 3 44 (g) 361 16 (j) 2 3 (c) 2 3 (b) 63 (c) 5 3 (e) 31 48 (f) 71 6 (h) 334 45 (i) 68 15 14 9 4 (a) $760 (b) $40 000 Answers 175 5 $10 000 6 (a) 720 (b) 11 779 7 67.9% 8 (a) 5.9 × 109 km (b) 5.753 × 109 km 9 (a) 9.4637 × 10 km (b) 1.6 × 10−5 light years (c) 3.975 × 1013 km 11 2 =5 (g) x = 2 (i) x = 4 (k) x = (b) (d) (f) (h) (j) (l) a(12 − a) 2x(11 − 8x) 18xy(1 − 2x) 2xy 2(7x − 3) x(4x − 7y) 7ab(2a − 3b) (h) x = −5 3 1 (j) x = − = −1 4 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (3 + y)(x + 4) (y − 3)(x + 5) (a + 2b)(3 − 2a) (2a − b)(4a − 3) (2 − y)(x + 1) (x − 3)(x + 4) (2 + y)(9 − x) (2b − c)(4a + 1) (x − 6)(3x − 5) (x − y)(x − 2) (2x + 3)(3x + y) (x − y)(4 − 3x) (l) x = 3 Exercise 6.4 A (f) x = 5 2 1 2 2 (a) x = 10 (c) x = 3 5 8 − 3 2 1 m= (b) x = −2 = 2 −2 3 (d) x = 4 3 1 4 1 =1 3 (e) x = 8 (f) x = (g) x = −4 (h) x = −9 (i) x = −10 (j) x = −13 (k) x = −34 (l) x = 3 (a) x = 18 (c) x = 24 176 Answers −11 2 x(x + 8) x(9x + 4) 2b(3ab + 4) 3x(2 − 3x) 3abc2(3c –ab) b2(3a − 4c) (d) x = 4 =3 (n) x = 3 (a) (c) (e) (g) (i) (k) −5a + 5b 8x − 4xy −9x + 9 3 − 4x − y −3x − 7 −3x2 + 6xy (b) x = 4 18 5 (l) x = 10 2(1 + 4y) x(3 − y) 3(x − 5y) 3b(3a − 4c) 2x(7 − 13y) (b) (d) (f) (h) (j) (l) Exercise 6.2 = (k) x = 3 =1 13 (b) (d) (f) (h) (j) 14x − 2y − 9x −5xy + 10x 6x − 6y − 2xy −2x − 6y − xy 12xy − 14 − y + 3x 4x2 − 2x2y − 3y −2x2 + 2x + 5 6x2 + 4y − 8xy –5x – 3 36 10 (j) x = 9 16 13 12(x + 4) 4(a − 4) a(b + 5) 8xz(3y − 1) 2y(3x − 2z) 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (e) x = (i) x = 1 2 2 (a) (c) (e) (g) (i) −2x −2y 6x − 3y −2x2 − 6xy 12 − 6a 3 2x2 − 2xy 1 2 9 2 3 a 4xy xy2z 1 (a) (c) (e) (g) (i) (k) 9 2 (h) x = = 4 1 − 3 5 6 1 (a) (d) (g) (j) Exercise 6.1 (c) x = = 4 (g) x = −1 =3 Exercise 6.3 Chapter 6 1 (a) x = 3 (f) x = (m) x = 42 12 23 6 (e) x = 17 20 13 =1 (b) x = 27 (d) x = −44 7 13 (b) (e) (h) (k) (c) (f) (i) (l) 8 3y pq ab3 5 5ab 7ab 3xy P+c a 4 b= a c x 5 (a) a = c − b (c) a = c d b de − c b (i) (j) a = ef − d bc (k) a = (l) a = d (e c ) b (m) a (n) a d c (b) a = 2c + 3b (d) a = (e) a = bc − d (or a = −d + bc) d b c a= cd − b 2 e d bc c( f d c b de ) b b Exercise 6.4 B P 2 1 (a) b (b) 9 cm 2π 2A h 3 use (b) b = 35.5 cm l C 2 (a) r = (c) T 70P 12 (c) 46 cm ; b = 3.8 cm 4 (a) (i) 70 kg (b) 11 656 kg (ii) 12 kg =B (d) 960 kg 5 (a) t = m 5 (b) 6 seconds Mixed exercise x = −3 x=9 x=2 x = 1.5 2 (a) x = 2 c = y − mx (g) a = (h) a = 1 (a) (c) (e) (g) D k 3 b= (f) a = d + bc m+r np (b) (d) (f) (h) x = −6 x = −6 x = −13 x=5 (b) x = 3 (a) (c) (e) (g) 3x + 2 −8x + 4y − 6 11x + 5 −2x 2 + 11x 4 (a) (c) (e) (g) (h) 4(x − 2) (b) 3(4x − y) −2(x + 2) (d) 3x(y − 8) 7xy(2xy + 1) (f) (x − y)(2 + x) (4 + 3x)(x − 3) 4x(x + y)(x − 2) 5 (a) (b) (c) (d) 4(x − 7) = 4x − 28 2x(x + 9) = 2x 2 + 18x 4x(4x + 3y) = 16x 2 + 12xy 19x(x + 2y) = 19x 2 + 38xy 6 (a) x = 15° (b) x = 26° (c) x = 30° (b) (d) (f) (h) mq − p n −12x 2 + 8x −16y + 2y2 5x 2 + 7x − 4 10x 2 + 24x Chapter 7 Exercise 7.2 Exercise 7.3 A Exercise 7.1 A 1 (a) (b) (c) (d) 1 (a) 2.56 mm2 (c) 13.5 cm2 (b) 523.2 m2 (d) 401.92 mm2 2 (a) 384 cm2 (b) 8 cm 3 (a) 340 cm (c) 4 tins (b) 153 000 cm2 4 (a) (c) (e) (g) (b) (d) (f) (h) 1 (a) 120 mm (c) 128 mm (e) 36.2 cm (b) 45 cm (d) 98 mm (f) 233 mm 2 (a) (c) (e) (g) (b) 44.0 cm (d) 21.6 m (f) 151 mm 15.7 m 54.0 mm 18.8 m 24.4 cm 3 90 m 2 (a) cuboid (b) triangular prism (c) cylinder 3 The following are examples; there are other possible nets. (a) 4 164 × 45.50 = $7462 cube cuboid square-based pyramid octahedron 2 90 000 mm3 20 410 mm3 960 cm3 1800 cm3 5 332.5 cm3 5 9 cm each 6 (a) 224 m3 6 about 88 cm 7 211.95 m3 7 (a) 197.82 cm 332.5 cm2 399 cm2 59.5 cm2 2296 mm2 7261.92 cm2 (b) (d) (f) (h) 1.53 m2 150 cm2 71.5 cm2 243 cm2 2 (a) 7850 mm2 (c) 7693 mm2 (e) 167.47 cm2 (b) 153.86 mm2 (d) 17.44 cm2 3 (a) (c) (e) (g) (i) (b) (d) (f) (h) 288 cm2 373.5 cm2 366 cm2 272.93 cm2 5639.63 cm2 4 (a) 30 cm2 (c) 33.6 cm2 (e) 720 cm2 82 cm2 581.5 cm2 39 cm2 4000 cm2 (b) (c) (b) 90 cm2 (d) 61.2 cm2 (f) 2562.5 cm2 64 000 50 100 64 000 50 Breadth (mm) 40 64 80 80 Height (mm) 20 20 8 16 Mixed exercise 1 (a) 346.19 cm2 (b) 65.94 cm 2 4.55 cm 3 (a) (c) (e) (g) 2000 mm2 40 cm2 106 cm2 175.84 cm2 (b) 33 000 mm2 (d) 80 cm2 (f) 35 cm2 4 15 m Exercise 7.1 C (b) 47.1 cm 2 6668.31 km 4 61.33 cm 64 000 80 2 (a) (i) 1.08 × 1012 km3 (ii) 5.10 × 108 km2 (b) 1.48 × 108 km2 6 70 mm = 7 cm 2 64 000 1 (a) 5.28 cm3 (b) 33 493.33 m3 (c) 25.2 cm3 (d) 169.56 cm3 (e) 65 111.04 mm3 5 11.1 m 3 (a) 75.36 cm2 (c) 92.34 mm2 Volume (mm3) Length (mm) Exercise 7.3 B 2 1 (a) 43.96 mm (c) 8.37 mm (b) 44 people 8 various answers − for example: (b) 219.8 cm Exercise 7.1 B 1 (a) (c) (e) (g) (i) 60 cm3 1120 cm3 5.76 m3 1.95 m3 (b) 732.67 cm2 (d) 5 (a) (b) (c) (d) cuboid is smaller 14 240 mm3 student’s own diagram cylinder 7536 mm2, cuboid 9000 mm2 6 42 7 volume cone = 23.55 cm3 volume pyramid = 30 cm3 difference = 6.45 cm3 Answers 177 8 volume 3 balls = 1144.53 cm3 volume tube = 1860.39 cm3 space = 715.86 cm3 Exercise 8.4 Exercise 8.1 3 9 17 red = , white = , green = 10 25 50 (b) 30% (c) 1 (d) 2 (a) 0.61, 0.22, 0.11, 0.05, 0.01 (b) (i) highly likely (ii) unlikely (iii) highly unlikely 2 (a) (c) = 0.1 (ii) (v) (viii) 1 4 1 2 (b) 1 (iii) 2 5 9 10 (vi) or 0.33 (c) 1 or 0.16 6 13 40 6 7 20 3 10 (b) 1 2 3 5 (b) (e) (c) 2 5 1 3 cola, cake cola, muffin water, biscuit water, cake water, muffin 2 3 3 (a) 0.16 (b) 0.84 (c) 0.6 (d) strawberry 63, lime 66, lemon 54, blackberry 69, apple 48 (b) 0.97 A CA LA CA TA TA C L C T T (c) 11 2 (a) (d) (g) 4 15 1 15 1 30 3 10 (c) (b) (e) U CU LU CU TU TU 1 5 2 15 7 15 28 153 (c) 40 153 40 153 1 (a) 10 000 (b) heads 0.4083; tails 0.5917 1 (c) 2 (d) could be − probability of the tails outcome is higher than the heads outcome for a great many tosses 1 2 (b) (d) 0 (e) 2 (a) (g) 1 (a) (b) Answers (d) 3 Exercise 8.5 1 2 1 5 5 8 178 1 3 1, 3 2, 3 3, 3 fruit juice, muffin (c) (b) Mixed exercise Yellow 2 1, 2 2, 2 3, 2 fruit juice, cake 9 or 0.5 1 0.73 4 (a) 0.6 (d) 114 4 fruit juice, biscuit 1 5 17 (e) The four situations represent all the possible outcomes, so they must add up to one. 1 cola, biscuit (ix) 0 Exercise 8.3 2 (d) Snack (b) 1 3 (d) 1 2 (c) 3 (a) 3 10 1 2 T HT TT 1 1, 1 2, 1 3, 1 1 2 3 Drink 2 5 4 (a) 5 (a) 2 (a) (b) 9 (b) no sugar; probability = 3 (a) (b) 4 Green 1 (a) 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10 (b) 3 (a) 1 3 Exercise 8.2 1 (i) 10 3 (iv) 10 3 (vii) 10 H HH HT H T Chapter 8 1 (a) 3 (a) 1 A CA LA CA TA TA (d) (c) (f) 14 15 1 45 2 5 3 (a) (c) 1 2 1 (b) 5 6 (b) 7, 36 1 (d) 2 4 (a) $1 $5 6 $5 6 $5 6 $5 6 $2 3 50c 1.5 50c 1.5 $1 6 6 6 6 3 1.5 1.5 2 5 9 10 $1 6 6 6 6 3 1.5 1.5 3 14 (f) 1 6 6 1 4 50c 5.5 5.5 5.5 5.5 2.5 1 1 $5 10 10 10 10 7 5.5 5.5 20c 5.2 5.2 5.2 5.2 2.2 0.7 0.7 (d) 1 8 1 3 Chapter 9 Exercise 9.1 1 (a) (b) (c) (d) (e) (f) 1 10 9 10 1 50c 5.5 5.5 5.5 5.5 2.5 1 1 (c) (c) 17, 19, 21 (add 2) 121, 132, 143 (add 11) 8, 4, 2 (divide by 2) 40, 48, 56 (add 8) −10, −12, −14 (subtract 2) 2, 4, 8 (multiply by 2) 20c 5.2 5.2 5.2 5.2 2.2 0.7 0.7 35 56 (g) 11, 16, 22 (add one more each time than added to previous term) (h) 21, 26, 31 (add 5) 2 (a) (b) (c) (d) (e) 7, 9, 11, 13 37, 32, 27, 22 1 1 1 1, , , 2 4 8 5, 11, 23, 47 100, 47, 20.5, 7.25 3 (a) (b) (c) (d) (e) (f) 5, 7, 9 1, 4, 9 5, 11, 17 0, 7, 26 0, 2, 6 1, −1, −3 3 (a) (b) (c) (d) {} {1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18} {1, 3, 5, 7, 9, 11, 13, 15} {1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20} (e) {2, 4, 6, 8, 10, 12} (f) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 18} 4 (a) {–2, –1, 0, 1, 2} (b) {1, 2, 3, 4, 5} T35 = 73 T35 = 1225 T35 = 209 T35 = 42 874 T35 = 1190 T35 = −67 2n + 5 3 − 8n 6n − 4 n2 1.2n + 1.1 1 a 6 31 3 (d) q u (b) 21 (d) (i) 74 99 (c) 79 90 103 900 (e) 943 999 (f) 928 4995 Exercise 9.3 A (c) false (f) true w x 2 (a) The set of even number from two to twelve. (b) 6 (c) {2} (d) {2, 4, 6, 8} (e) {2} (f) {10, 12} 40 59 80 x 36 – x (c) 0.57 Mixed exercise T120 = 596 T120 = −694 T120 = 359 2 (a) –4, –2, 0, 2, 4, 8 (b) 174 (c) T46 3 0.213231234. . ., 2 , 4 π (c) 16 (ii) (v) 41 1 (iii) 80 21 5 80 −1 0 1 2 3 y 4 5 6 7 8 x −1 0 1 2 3 y 1 −1 −3 −5 −7 −1 0 1 2 9 7 5 3 (d) x −1 0 1 2 3 y −1 −2 −3 −4 −5 x 4 4 4 4 4 y −1 0 1 2 3 z (b) (c) x y 3 1 (in fact, any five values of y are correct) (f) x −1 0 1 2 3 y −2 −2 −2 −2 −2 7 1 (a) 5n − 4 (b) 26 − 6n (c) 3n − 1 18 Exercise 10.1 S (b) 21 16 Chapter 10 C (e) 78 – x C 17 1 (a) x v M (b) (b) true (e) false (h) true t e m r k 9 20 {c, h, i, s, y} {c, e, h, i, m, p, r, s, t, y} {a, b, d, f, g, j, k, l, n, o, q, u, v, w, x, z} (f) {c, h, i, s, y} 12 , 0.090090009. . . 4 9 1 (a) false (d) true (g) false o j 2 (a) (b) (c) (d) (e) 45 , 3 90 , π, 8 (b) 2 (a) 16 , n g h i s c y p 286 999 25 (iv) f 3 (a) Exercise 9.2 1 (a) d P l 5 26 b (b) B Exercise 9.3 B T50 = 105 T50 = −397 T50 = 296 T50 = 2500 T50 = 61.1 6 (a) n 1 2 3 4 Tn 6 11 6 21 (b) Tn = 5n + 1 (c) 496 (d) 55th 5 (a) 23 99 21 5 (a) {x: x is even, x ≤ 10} (b) {x: x is square numbers, x ≤ 25} 4 (a) 8n − 6 (b) 1594 (c) 30th (d) T18 = 138 and T19 = 146, so 139 is not a term 5 (a) (b) (c) (d) (e) 4 (a) (g) x y 1 2 3 −1 0 1.5 −0.5 −2.5 −4.5 −6.5 (h) x y 0 1 2 3 −1 −1.2 −0.8 −0.4 0 0.4 (i) x −1 0 y −1 −0.5 (j) 1 2 3 0 0.5 1 x −1 0 1 2 3 y 0.5 −0.5 −1.5 −2.5 −3.5 2 student’s graphs of values above 3 y=x−2 Answers 179 4 (a) (d) (g) (h) no (b) yes (c) yes no (e) no (f) no yes (horizontal lines) yes (vertical lines) 5 (a) m = 1 (b) m = −1 (c) m = −1 (d) m = (f) m = (g) m is undefined, c = 7 (h) m = −3, c = 0 14 3 (j) m = −1, c = −4 (k) m = 1, c = −4 (l) m = −2, c = 5 (m) m = –2, c = −20 2 8 (a) (b) (c) (d) 1 x 2 (b) y (d) y (f) y 2x − 1 2x + 1 1 (h) y x+2 3 (j) y = x + 4 (l) y = x − 3 x = 2, y = –6 x = 6, y = 3 x = –4, y = 6 x = 10, y = 10 (e) x = −5 2 , y = –5 9 (a) 1 (d) 2 (b) 1 (e) 0 (c) –1 (f) 12 10 (a) (0, 0) (b) (–1.5, 0.5) (c) (–2, 3) (d) (5, 10) (e) (0.5, –3) (f) (1, 1.5) Exercise 10.2 A 1 (a) x 2 + 5x 5x 6 (b) x 2 x −6 (c) x 2 + 12 x + 35 (d) x 2 + 2x 2 x 35 (e) x 2 4x 3 4x (f) 2 x 2 (g) y 9 y 14 9y 180 2 Answers 1 (k) 1 − x 2 2 (a) x 2 + 8x 8 x 16 (b) x 2 6x 6x 9 (c) x 2 + 10 x + 25 (d) y 2 4 y 4 (e) x 2 + 2 xy x + y2 2 (f) 4 x 4 xy xy + y 2 (g) 9 2 12 12xx 4 (h) 4 x 2 122 y 9 y 2 (i) 4 x 2 200 x 25 (j) 16 x 2 − 48x 48 x + 36 (k) 9 6x 6 + x2 (l) 16 − 16x 16 + 4 x 2 (m) 36 − 36y 36 + 9 y 2 1 1 2 4 4 , c = −2 5 (g) x = 2 (i) y = −2x (k) y = 3x − 2 (j) x 2 + x − 132 (m) −12 x 2 + 14 x − 4 (e) m = , c = 7 (a) y x (c) y = 2 5 1 (e) y x −1 3 (l) −3x + 11x − 6 m = 3, c = −4 m = −1, c = −1 1 m=− ,c=5 2 m = 1, c = 0 1 3 2 2 16 (i) m = − , c = + (i) 2 7 xy + 2 y 2 4 4 6 7 (e) m = 2 (f) m = 0 (g) undefined 1 (h) m = 6 (a) (b) (c) (d) (h) x 2 x 1 3 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) x 2 − 25 4 x 2 25 49 y 2 − 9 x4 y4 9 2 16 x6 y4 4 4 16 x y − 4 z 4 4x8 4 y2 16 x 2 y 4 − 25 y 2 64 x 6 y 4 − 49z 4 Exercise 10.2 B 1 (a) ( x + 2))(( x + 2) (b) ( x + 4 ))(( x + 3) (c) ( x + 3))(( x + 3) (d) ( x + 1))(( x + 4 ) (e) ( x + 3))(( x + 5) (f) ( x 1))(( x − 8) (g) ( x 5))(( x − 3) (h) ( x 1))(( x − 3) (i) ( x 26) ( x − 1) (j) ( x 8))(( x + 1) (k) ( x + 5))(( x − 2) (l) ( x + 4 ))(( x − 8) (m) ( x 3))(( x − 4 ) (n) ( x + 4 ))(( x − 3) (o) ( x + 9))(( x − 6) 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) 5 ( x 2) ( x + 1) 3 ( x 4 ) ( x − 2) 3 ( 3) ( x − 1) 5 ( x 2) ( x − 1) x ( x + 10) ( x + 2) x 2 y ( x + 2) ( x 1) x ( x + 7 ))(( x − 2) 3 ( x 3) ( x − 2 ) −2 ( x + 4 )( ) ( x − 6) 2 ( x 7 ) ( x − 8) 3 (a) ( x + 3))(( x − 3) (b) ( 4 x ))(( 4 − x ) (c) ( x + 5))(( x − 5) (d) (7 )(7 − x ) (e) (3x 2 y )(3x 2 y ) (f) (9 2 x )(9 2 x ) (g) ( x + 3 y )( x 3 y ) (h) (11 y + 12 x )(11 y 12 x ) (i) ( 4 x 7 y )( 4 x 7 y ) (j) 2 ( x 3) ( x − 3) (k) 2 (10 )(10 − x ) (l) x 2 + y x 2 y (m) 5 8 5 − 8 ( ( )( )( ) ) (n) ( xy + 10)( xy − 10) 5 8w 5 8w − (o) 2 + z y 2 z y ( ( )( ) (p) 5x 5 1 55x 5 1 (q) 1 9 2 3 )(1 9 Exercise 10.2 C 1 (a) (b) (c) (d) x = 0 or x = 3 x = –2 or x = 2 x = 0 or x = 2 2 x = 0 or x = − 3 (e) x = –1 or x = 1 (f) x = − (g) x = − 7 2 1 2 or x = or x = 7 2 1 2 (h) x = –4 or x = –2 (i) x = –4 or x = –1 2 3 ) (j) x = 5 or x = –1 (k) x = 5 or x = –4 (l) x = –10 or x = 2 (m) x = 5 or x = 3 (n) x = 20 or x = –3 (o) x = 7 or x = 8 (p) x = 10 (q) x = 2 (b) 40 x y Distance −1 0 2 3 −0.5 0 1 1.5 x y 0 2 3 2 1.5 0 2 2 2 3 2 −1 2 0 4 2 8 3 10 m = −2, c = −1 m = 1, c = −6 m = 1, c = 8 1 m = 0, c = − 2 3 (i) 2 (e) m = − , c = 2 (f) m = −1, c = 0 3 (a) y (c) y 2 1 x+ 3 2 4 x −3 5 (b) y x −3 (d) y x −2 (e) y 2 x − 3 (g) y = 2 (f) y x+2 (h) x = −4 4 A 0, B 1, C 2, D 1, E 4 5 (a) y 4 3 (b) y = 7 2x − 6 (c) y = x + 4 (d) x = −10 (e) y (f) y = −3 6 (a) t D x 0 0 2 14 4 28 4 6 Time 7 All four plotted on the same graph. 2 (a) (b) (c) (d) 2 (c) 7 (d) 7 (e) (i) 3 hours (ii) 1 h 24 min (iii) 48 min (f) (i) 21 km (ii) 17.5 km (iii) 5.25 km (g) 7 hours (d) y 2 x − 4 = 0 x y 15 5 (c) y = 2 −1 2 20 x −1 0 3.5 3 x y 25 10 1 x+3 2 (b) y a (a + 2))((a − 2) 2 + 1 ( x 1)( x 1) ( x 2))(( x + 1) ( x 1))(( x − 1) 3 y + 2z )(2 x − 3 y 2z ) (2x 3y ( x + 12)( x + 4 ) x 2 x (g) x 2 + x − 2 2 30 1 x 2 6 42 (a) (b) 1 (0.5, 6.5) (ii) 2 (0, 5) (iii) –1 (1, 3) (iv) − (v) 4 3 undefined 8 (a) x 2 − 16 x + 64 (b) 2 x 2 2 (c) 9 2 12 1 xy 4 y 2 12x (d) (e) (f) (g) (k) 10 x − 45 (l) −2 x 3 + 16 x 2 − 8 x (m) 2 x 3 8x 8 x 2 + 16 x 9 (a) (b) (c) (d) (e) (f) 35 Mixed exercise 1 (a) y Caroline’s distance at 7 km/h y 45 1 −12y 12 36 y 2 2 9 4 4 x 2 200 x 25 9 4 y 2 6x 2 y + 1 1 (h) x 2 + xy x + y2 4 1 (i) x 2 − 4 1 − 4 (j) x2 (–0.5, 3) (–1.5, 0.25) (x ) ( x + 1))(( x − 6) 4 ( x 3) ( x − 4 ) 2 ( x 3) ( x − 4 ) (h) (i) (j) (k) (l) 5 1+ 2 8 1− 2 3 ( x 3) ( x + 2 ) 10 (a) (b) (c) (d) (e) (f) x = –5 or x = –1 x = –2 or x = 2 x = 2 or x = 1 x = –1 x = 5 or x = –1 x=2 ( )()( ) 8 Chapter 11 Exercise 11.1 A 1 (a) (c) (e) (g) (b) (d) (f) (h) 5 cm 12 mm 1.09 cm 8.49 cm 2 (a) 55.7 mm (c) 5.29 cm (e) 9.85 cm 3 (a) no (d) yes 17 cm 10 cm 0.45 cm 6.11 cm (b) 14.4 cm (d) 10.9 mm (f) 9.33 cm (b) yes 4 (a) 32 = 5.66 (b) 18 = 4.24 (c) 32 = 5.66 (c) no (d) 180 = 13.4 (e) 3 (f) 45 = 6.71 Exercise 11.1 B 1 20 mm 2 44 cm Answers 181 3 height = 86.6 mm, area = 4330 mm2 4 13 m and 15 m 5 0.7 m Exercise 11.2 1 (a) (c) (e) (f) (g) (h) 2.24 cm (b) 6 mm 7.5 mm (d) 6.4 cm y = 6.67 cm, z = 4.8 cm x = 5.59 cm, y = 13.6 cm x = 9 cm, y = 24 cm x = 50 cm, y = 20 cm 2 ∠ABC = ∠ADE (corr ∠ are equal) ∠ACB = ∠AED (corr ∠ are equal) ∠A = ∠A (common) ∴ ABC is similar to ADE 3 25.5 m Exercise 11.3 1 (a) x = 18 cm (b) x = 27 cm, y = 16 cm 2 9:4 3 (a) 254.48 cm2 (b) 529 mm2 4 (a) x = 2 cm 1 2 ∴ AB = AC Mixed exercise 1 (a) sketch (b) x = 15 cm 3 (a) 18 = 4.24 (b) 20 = 4.47 (c) 8 2.83 (d) 5 (e) 3.5 1 (a) ABC is congruent to ZXY (SAS) (b) although the two triangles look to be the same size, there are no marks to confirm this to be the case so we cannot be certain they are congruent (c) XYZ is congruent to ONM (ASA) (d) DEF is congruent to RQP (ASA) (e) ACB is congruent to ACD (RHS) (f) PMN is congruent to NOP (SSS or SAS) (g) PRQ is congruent to ZYX (SAS) (h) ABC is congruent to EDC (ASA) 2 PO = PO (common) MP = NO (given) MO = NP (diagonals of isosceles trapezium are equal) ∆ MPO is congruent to ∆ NOP (SSS) Answers 1 (a) (b) (c) (d) (e) (f) mean 6.14 27.44 13.08 5 4.89 5.22 median 6 27 13 5 5 5 mode 6 27 and 12 no 4 6 38 mode 2 (a) (iii) and (vi) (b) sensible answer from student, e.g. different sets can still add up to the same total as another set. If divided by the same number they will have the same mean. 5 Need to know how many cows there are to work out mean litres of milk produced per cow. 5 (a) x = 3.5 cm (b) x = 63°, y = 87° (c) x = 12 cm 6 (a) 2.78 (b) 1 7 (a) $20.40 (b) $6 (c) $10 (d) 2 (only the category B workers) (e) The mean is between $20 and $40 so the statement is true. 7 18 cm2 9 (a) 3 cm (b) height = 12 cm, area of base = 256 cm2 10 (a) (b) ABC is congruent to HGI (RHS) ABC is congruent to DEF (ASA) (c) ACB is congruent to EDF (SAS) (d) CAB is congruent to GIH (SAS) 11 5.63 m 140 mm 68 mm 560 mm 420 mm 140 mm (b) 156 mm Exercise 12.1 4 15 4 P = 2250 mm 12 (a) Chapter 12 3 255 8 23 750 mm2 (b) 25 : 1 Exercise 11.4 182 (b) 130 m 2 102 = 62 + 82 ∴ ABC is right-angled (converse Pythagoras) 6 (a) 4 : 1 (b) 1 : 9 5 28 000 cm3 6 (a) 5 : 1 (c) 125 : 1 3 ∠CDB = ∠DBE = ∠BEA (alt ∠ equal) ∠BAE = ∠CBD (corr ∠ equal) AE = BD (given) ∆ BAE is congruent to ∆ CBD (ASA) AB = BC Exercise 12.2 1 (a) mean = 4.3, median = 5, mode = 2 and 5. The data is bimodal and the lower mode (2) is not representative of the data. (b) mean = 3.15, median = 2, mode = 2. The mean is not representative of the data because it is too high. This is because there are some values in the data set that are much higher than the others. (This gives a big range, and when the range is big, the mean is generally not representative.) (c) mean = 17.67, median = 17, no mode. There is no mode, so this cannot be representative of the data. The mean and median are similar, so they are both representative of the data. 2 (a) mean = 12.8, median = 15, mode = 17, range = 19 (b) mode too high, mean not reliable as range is large 3 (a) Runner B has the faster mean time; he or she also achieved the faster time, so would technically be beating Runner A. (b) A is more consistent with a range of only 2 seconds (B has a range of 3.8 seconds). Exercise 12.3 1 Score Frequency Score × frequency (fx) 0 6 0 1 6 6 2 10 20 3 11 33 4 5 20 5 1 5 6 1 6 Total (a) 2.25 (d) 6 2 40 90 (c) 2 mean A 3.5 Frequency f × midpoint 40 1340 70 2695 41 ≤ w < 46 43.5 80 3480 46 ≤ w < 51 48.5 90 4365 51 ≤ w < 55 53.5 60 3210 Chapter 13 55 ≤ w < 60 58.5 20 1170 Exercise 13.1 Total 360 16 260 (b) 230 mm (d) 2450.809 km (f) 0.157 m Exercise 12.5 3 (a) 9080 g (c) 500 g (e) 0.0152 kg (b) 49 340 g (d) 0.068 kg (f) 2.3 tonne 1 (a) Q1 = 47, Q2 = 55.5, Q3 = 63, IQR = 16 (b) Q1 = 8, Q2 = 15, Q3 = 17, IQR = 9 (c) Q1 = 0.7, Q2 = 1.05, Q3 = 1.4, IQR = 0.7 (d) Q1 = 1, Q2 = 2.5, Q3 = 4, IQR = 3 4 (a) (b) (c) (d) (e) (f) 5 (a) 1200 mm2 (c) 16 420 mm2 (e) 0.009441 km2 1 (a) mean 6.4, median 6, mode 6, range 6 (b) mean 2.6, median 2, mode 2, range 5 (c) mean 13.8, median 12.8, no mode, range 11.9 40 4.5 6.5 Exercise 12.4 1 (a) Frequency Frequency × (f ) midpoint 0 ≤ m < 10 5 2 10 10 ≤ m < 20 15 5 75 20 ≤ m < 30 25 13 325 30 ≤ m < 40 35 16 560 40 ≤ m < 50 45 14 630 50 ≤ m < 60 55 13 715 63 2315 1 student’s own diagrams 2 (a) 2600 m (c) 820 cm (e) 20 mm 45.17 46 ≤ w < 51 41 ≤ w < 46 29 Mixed exercise 40 7 (a) 15 scores fell into the bottom 60% of marks (b) Less than half students scored above the 60th percentile, so you can assume the marks overall were pretty low. 36 ≤ w < 41 38.5 C 3 19 5 Q1 = 18, Q3 = 23, IQR = 5 fairly consistent, so data not spread out 31 ≤ w < 36 33.5 4.12 3 and 5 Total Mid point B median Mid point Words per minute (w) 46.14 mode Marks (m) 6 (a) (b) (c) (d) 2 (a) (b) (c) (d) (b) 3 Data set (b) 36.74 ≈ 37 (c) 30 ≤ m < 40 (d) 30 ≤ m < 40 2 (a) 19 (b) 9 and 10 (c) 5.66 3 C − although B’s mean is bigger it has a larger range. C’s smaller range suggests that its mean is probably more representative. 4 (a) 4.82 cm3 (b) 5 cm3 (c) 5 cm3 5 (a) (b) (c) (d) 6 (a) (b) (c) (d) (e) (f) 19 km 9015 cm 435 mm 492 cm 635 m 580 500 cm 100 m 15 cm 2 mm 63 cm 35 m 500 cm (b) 900 mm2 (d) 370 000 m2 (f) 423 000 mm2 69 000 mm3 19 000 mm3 30 040 mm3 4 815 000 cm3 0.103 cm3 0.0000469 m3 7 220 m 8 110 cm 9 42 cm 10 88 (round down as you cannot have part of a box) 36.47 years 40 ≤ a < 50 30 ≤ a < 40 don’t know the actual ages Answers 183 Half past 3 hour 7 1 $55.88 Dawoot 1 past 9 hours 4 2 five 4 8:17 Nadira a.m. 8h 5:30 1 p.m. 2 hour 43 min John 08:23 Robyn 7:22 a.m. 08:08 17:50 45 min 8 h 42 min 4:30 1 hour 8 h p.m. 8 min 18:30 45 min 9 h 37 min Mari $64.94 $64.82 $60.59 $71.64 2 6 h 25 min (b) maximum speed = greatest distance shortest time = 405 33.5 = 12.09 m/s 5 (a) upper bound of area: 15.5563 cm2 lower bound of area: 14.9963 cm2 (b) upper bound of hypotenuse: 8.0910 cm lower bound of hypotenuse: 7.9514 cm Exercise 13.4 3 20 min 4 (a) 5 h 47 min (c) 12 h 12 min (b) 10 h 26 min (d) 14 h 30 min 5 (a) 09:00 (b) 1 hour (c) 09:30 (d) 30 minutes (e) It would arrive late at Peron Place at 10:54 and at Marquez Lane at 11:19. Exercise 13.3 1 The upper bound is ‘inexact’ so 42.5 in table means < 42.5 Upper bound Lower bound (a) 42.5 (b) 13325.5 13324.5 (c) 450 350 (d) 12.245 12.235 (e) 11.495 11.485 (f) 2.55 2.45 (g) 395 (h) 1.1325 41.5 385 1.1315 2 (a) 71.5 ≤ h < 72.5 (b) Yes, it is less than 72.5 (although it would be impossible to measure to that accuracy). 184 4 (a) 195.5 cm ≤ h < 196.5 cm 93.5 kg ≤ m < 94.5 kg Answers 1 (a) 1 unit = 100 000 rupiah (b) (i) 250 000 (ii) 500 000 (iii) 2 500 000 (c) (i) Aus $80 (ii) Aus $640 2 (a) temperature in degrees F against temperature in degrees C (b) (i) 32 °F (ii) 50 °F (iii) 210 °F (c) oven could be marked in Fahrenheit, but of course she could also have experienced a power failure or other practical problem (d) Fahrenheit scale as 50 °C is hot, not cold 3 (a) 9 kg (b) 45 kg (c) (i) 20 kg (ii) 35 kg (iii) 145 lb 2 23 min 45 s 3 2 h 19 min 55 s 4 1.615 m ≤ h < 1.625 m 5 (a) No, that is lower than the lower bound of 45 (b) Yes, that is within the bounds 6 (a) 3.605 cm to 3.615 cm 2.565 cm to 2.575 cm (b) lower bound of area: 9.246825 cm2 upper bound of area: 9.308625 cm2 (c) lower: 9.25 cm2, upper: 9.31 cm2 7 (a) conversion graph showing litres against gallons (conversion factor) (b) (i) 45l (ii) 112.5l (c) (i) 3.33 gallons (ii) 26.67 gallons (d) (i) 48.3 km/gal and 67.62 km/gal (ii) 10.62 km/l and 14.87 km/l 8 €892.06 9 (a) US$1 = IR49.81 (c) US$249.95 10 £4239.13 Chapter 14 Exercise 14.1 A 1 (a) 5 Exercise 13.5 1 (a) (i) (ii) (iii) (iv) (v) (vi) (b) (i) (iii) (c) (i) (iii) US$1 = ¥76.16 £1 = NZ$1.99 €1 = IR69.10 Can$1 = €0.71 ¥1 = £0.01 R1 = US$0.12 2490.50 (ii) 41 460 7540.15 9139.20 (ii) 52 820 145 632 (b) 99 620 y 4 0 (b) Daily earnings 3 –5 –4 –3 2 1 –2 –1 0 –1 –2 (–3,–3) –3 –4 –5 x = −3, y = −3 y= Time Lunch (a) out Hours worked (b) 690 mm (d) 0.0235 kg (f) 29 250 ml (h) 1000 mm2 (j) 7 900 000 cm3 (l) 0.168 cm3 2700 m 6000 kg 263 000 mg 0.24 l 0.006428 km2 29 000 000 m3 =y Time in 1 (a) (c) (e) (g) (i) (k) +3 Name Mixed exercise 2x 1 x– 3 (a) 27.52 m2 (b) upper bound: 28.0575 m2 lower bound: 26.9875 m2 Exercise 13.2 1 2 3 4 5 x (b) =y 4 4x+2 3 2 1 –5 –4 –3 –2 –1–10 1 2 3 4 3 (–1, –2) –2 – 2y = x –3 x 5 –4 –5 x = −1, y = −2 + –2x 5 4 3 2 1 y y= (c) 2 x –5–4 –3–2–1 0 1 2 3 4 5 6 7 8 9 10 –2 –3 (3,–4) –4 –5 –6 –7 –8 –9 –10 2y x– +3 1= 0 x = 3, y = −4 (d) (c) 2 (a) (A) y x+4 3 (B) y = x + 6 y 5 5 4 4 3 = x 2 y– 1 y 2 y 4x − 2 y x y x−4 y = −2 (−2, 6) (ii) (4, 0) 3 (a) (b) (c) (d) x = 4, y = 2 x = −1, y = 2 x = 0, y = 4 x = 3, y = 1 4 (a) (b) (c) (d) (e) (f) x = 6, y = −1 x = 1, y = 2 x = 18, y = −8 1 x = 1, y = 1 3 x = 3, y = −1 x = 3, y = 7 5 (a) (b) (c) (d) (e) (f) (g) (h) (i) x = 2, y = 1 x = 2, y = 2 x = 2, y = −1 x = 3, y = 2 x = 3, y = 2.5 x = 4, y = 2 x = 5, y = 3 x = 0.5, y = −0.5 x = −9, y = −2 –x –7 x = −5.5, y = −1.5 (e) 5 y 3x + 4 3 +3 1 –2 x 2 y= = 3y 3 –5 –4 –3 –2 –1–10 1 2 3 4 5 (2, –1) –2 –3 x = 2, y = −1 x –2 –1 (e) (−2, −2) –7 –6 –5 –4 –3 –2 –1 2 (h) 0.25 0 0.75 0.5 –2 –1 0 2 (a) (b) (c) (d) (e) (f) –1 1.25 1 1 2 3 3 4 5 (b) 1 2 2 3 4 5 6 7 (k) g < 9 (b) x > 11 (d) x ≤ 52 (f) x > 73 1 (h) e ≥ –3 4 1 (j) x ≥ 45 2 1 (l) y ≤ 4 10 11 (n) n > –2 1 28 1 (a) 2 (a) c + d = 15 (b) 3 desks and 12 chairs –5 –4 –3 –2 –1 0 1 Exercise 14.3 7 2 5 3, 4 –2, –1, 0, 1, 2, 3 2 –1, 0, 1, 2, 3 0, 1, 2, 3, 4 4, 5, 6, 7, 8, 9 (o) x ≤ −6 x = 3, y = –4 x = 15, y = 30 x = 4, y = 2 12 x = 15 , y = 1 4 1.75 (m) n > 31 1 (a) x = –3, y = 4 1 (b) x = –1, y = –5 –4 –3 –2 –1 0 3 0 1.5 3 (a) x ≤ –2 (c) x ≥ 4 (e) x ≥ 9 Exercise 14.1 B 1 (a) 2 (i) 8 6 quarters and 6 dimes –5 1 –2 (i) r < 11 Exercise 14.2 1 (g) –4 –3.5 –3 7 number of students is 7 –4 0 (f) –5 –4 –3 –2 –1 0 (g) x > 9 7 1 (d) 6 a chocolate bar costs $1.20 and a box of gums $0.75 (c) (d) (e) (f) 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 y= x –10–9–8 –7–6–5–4–3–2–1 0 1 2 3 4 5 (–5.5, 1.5) –2 –3 –4 –5 –6 –7 –8 –9 –10 (C) (D) (E) (F) (b) (i) (iii) –4 –3.5 –3 3 4 5 5 4 3 2 1 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 y x 1 2 3 4 5 y < 4x + 1 Answers 185 (b) 5 4 3 2 1 (c) y y ≥ –2x + 5 x 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 (c) 5 4 3 2 1 y x 1 2 3 4 5 5 4 3 2 1 y 5 4 3 2 1 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 186 Answers y 5 (a) (e) x –2 < y ≤ 4 1 2 3 4 5 x 1 2 3 4 5 y 5 4 3 2 1 (b) (–2, 2), (–2, 1), (–1, 2), (–1, –1), (0, 1) 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 (b) 1 2 3 4 5 –3 –4 –5 0 –5 –4 –3 –2 –1 –1 –2 2y ≤ –4 x 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 x 3x – y ≥ 3 5 4 3 2 1 2x < 3y – 6 2 4 (d) y > –x + 1 1 (d) y ≥ –x – 3 –3 –4 –5 0 –5 –4 –3 –2 –1 –1 –2 2 (a) (c) y > 0 –5 –4 –3 –2 –1 –1 –2 1 2 3 4 5 3 (a) y ≥ 2x − 1 (b) y < –4x + 3 y 5 4 3 2 1 x 1 2 3 4 5 1 –3 –4 –5 y (f) 1 x 1 2 3 4 5 x–y>7 –5 –4 –3 –2 –1 0 –1 x–y 3 –2 < 2 4 –3 –4 –5 –6 –7 –8 –9 –10 5 4 3 2 1 y x 1 2 Exercise 14.4 3 4 5 6 7 8 9 10 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 y x 1 2 3 4 5 2 (a) Exercise 14.5 2 1 (a) ( x + 3) − 5 2 (b) ( x − 2) + 3 2 (c) ( x + 7 ) − 5 2 (d) ( x − 6) − 6 2 (e) ( x + 5) − 8 2 (f) ( x + 11) + 20 2 (g) ( x + 12) − 23 2 (h) ( x − 8) − 7 2 (i) ( x + 9) + 12 2 (j) ( x − 1) + 9 2 (k) ( x − 4 ) − 21 2 (l) ( x + 10) − 17 y 8 7 6 5 4 3 2 1 x –1 0 –1 1 2 3 4 5 6 7 8 (b) greatest posible value is at the point (2, 5) so 2y + x = 12 3 let x = number of chocolate fudge cakes and y = number of vanilla fudge cakes y 60 x ≤ 30 it 3 of pr 50 40 x+ 30 y ≤ 20 2y x+ 20 y 10 ≤ 40 –1 0 –1 x 10 20 30 40 50 60 to maximise profit x = 30 and y = 10 maximum profit = $110 4 let x = number of litres of orange and y = number of litres of lemon 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 y orange = x pro f it x x ≤ 1800 +1 .9y y ≤ 600 1x y≤ 3 200 400 600 800 1000 1200 x 1500 1800 maximum income ($2940) when x = 1800 and y = 600 x = 6, x = –1 x = 3, x = –2 x = 3, x = 1 x = 7, x = –1 x = 15.81, x = 0.19 x = –6.85, x = –0.15 x = 0.11, x = –9.11 x = –3.70, x = –7.30 x = 11.05, x = –9.05 3 (a) (b) (c) (d) (e) (f) x = 2, x = –0.5 x = 3, x = 1 x = 2.53, x = –0.53 x = 3, x = –0.5 x = 7.47, x = –1.47 x = –2.27, x = 1.77 Exercise 14.6 lemon = y x ≥ 1000 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) 1 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) x = 10 or 4 x = 6 or –20 x = 1 or –6 x = 3 or 5 x = 4 or –1 x=2 x = 2 or –6 x = –5 x = 2 or –4 x = 6 or –2 x = 4 or 5 x = 5 or –8 2 (a) (b) (c) (d) (e) (f) x = –0.76 or – 5.24 x = 1.56 or –2.56 x = –4.76 or –9.24 x = 7.12 or –1.12 x = –2.17 or –7.83 x = –0.15 or –6.85 (g) (h) (i) (j) (k) (l) 3 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) x = –7.20 or –16.80 x = –3.70 or –7.30 x = 4.19 or –22.19 x = 4.32 or –2.32 x = 8.58 or –0.58 x = 11.05 or –9.05 x = 5.46 or –1.46 x = 1.67 or –3.5 x = 1.31 or 0.19 x = 5 or –2.5 x = 2.25 or 1 x = 1.77 or –1.27 x = 0.80 or –0.14 x = 10.65 or 5.35 x = 9.91 or –0.91 x = 0.5 or –3 x = –0.67 or –1.5 x = 1 or –0.75 4 numbers are 57 and 58 or –58 and –57 5 dimensions are 12 cm × 18 cm Exercise 14.7 1 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (2x − 3)(x + 1) (3x +1)2 (2x − 3)2 (2x + 1)(3x − 5) (4x − 3)(x + 1) (7x − 1)(2x − 7) (x + 5)(3x − 4) (2x − 1)(3x + 7) (3x + 5)(x − 5) (3x − 11)(x + 6) (5x + 3)(3x − 5) (8x + 1)(x + 3) 2 (a) (2x + 3)2 (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) 1 2 (2x 1)(2x 1) or 1 x + 2 ( x + ) 2(5x + 2)2 (2x + y)(3x − 5y) (4x2 − 3)(x2 + 1) 2(2x − 1)(3x + 1) 2(1 − 2x)(1 + x) (x + 3)(x + 2) (3x + 8)(x − 4) 2(3x − 11)(x + 6) (4x − 3)(x + 1) (8x + y)(x + 3y) Answers 187 1 (a) (c) x 7 (e) 8z 3 2x (g) 3y 2b (i) 3a x+3 2 (a) x 2x + y (c) x a+2 (e) a x+5 (g) x+2 (i) 3 (a) (c) (e) (g) (i) 4 (a) (c) (e) (g) 4 (a) x > –5 (b) x ≤ –1 5 a b a b 3 1 (d) x −5 a b (f) a b 5 4 3 2 1 (d) y 0 –5 –4 –3 –2 –1 –1 –2 x = –2 –3 –4 –5 (e) (f) x 1 2 3 4 5 x=3 6 7 (h) (i) Chapter 15 3 Exercise 15.1 x2 6 35 a2 a c 1 42 −2 − 3x 2x + 1 y2 (b) 6 5xy (d) 2 1 (a) length = 10 cm, width = 7 cm (b) length = 14.4 cm, width = 7 cm (f) (b )(b − 1) (h) 4−x 3 the integer solutions are: (b) 13 p + 2 8p 29 10p (d) 7 p +1 (f) 18 − 5m 6 5 x + y = 1.5 2 x greatest, x = 2, y = 4, giving + y = 5 2 7 (a) x(x − 2y) 2 2 (b) (a + b)(a − b) (c) (x − 5)(x + 11) (d) (2y – 1)(y + 7) (e) –2(x + 1)(2x − 3) (f) (x − 6)(x + 3) least, x = –1, y = 2, giving 3x 2 y xy 7 ( x + 2)( x (x (g) 5 2p 10 p2 3xy 2 xy 3 z 4 45a 2 + 16a 16a − 50 30a ( x + 2)(( x − 1) x (x + 4) 1 2x 1 −6 )( x + 5))(( − 1) (2x + 1)( (h) x 2 − 3 9 3)( 1) 17 2) ( x − 1) 4) 5 3x )( x + 1)( x 1) (2x 3)( 8 (a) (b) (c) (d) (e) (f) a = 1.30 or –2.30 x = 1.33 or –1 x = 3 or –5 x = 0.67 or 1 x = 1 or –0.625 x=1 9 (a) x = –2 or 0.4 (b) x = 188 (c) 3 $5000 at 5% and $10 000 at 8% (b) 17 − x (h) x ( x − 3)( (i) 2 x = −2, y = 5 2x 5 2 (f) y 1 (h) 5ab (d) 7 10 (a) 1 x = 2, y = −5 (b) 3 1 x y x+y (b) 2 – x Mixed exercise Exercise 14.8 Answers q ± q 2 − 4 pr 2p 2 (a) length = 9.14 cm, width = 5.5 cm (b) Yes. The length of the mini pitch = width of standard pitch and 2 × width of mini pitch < length of standard pitch. It is possible to have two mini pitches on a standard pitch so, with three standard pitches, up to six matches could take place at the same time. 3 (a) 1 200 or 1 150 (b) (i) and (ii) diagram drawn using student’s scale including × for net posts 4 (a) student’s scale drawing − diagonal distance = 11.3 m (b) using Pythagoras’ theorem Exercise 15.2 1 (a) 090° (b) 225° (c) 315° 2 (a) 250° (b) 310° (c) 135° (e) sin E = Exercise 15.3 1 (a) (b) (c) (d) hypotenuse c z f q opp(A) a y g p adj(A) b x e r (b) 1.43 (e) 0 4 (a) tan A = 3 4 (c) 5.14 2 3 1 5 12 12 3 4 (e) AC = 2 cm, tan B = 3 , tan C = 4 x = 1.40 cm y = 19.29 m c = 3.32 cm a = 13 m x = 35.70 cm 6 (a) 26.6° (d) 85.2° (g) 79.7° (b) 40.9° (e) 14.0° (h) 44.1° (c) 51.3° (f) 40.9° 7 (a) 16° (d) 23° (b) 46° (e) 38° (c) 49° 8 (a) hyp = y, adj(θ) = z, cos θ = (b) hyp = c, adj(θ) = b, cos θ = (c) hyp = c, adj(θ) = a, cos θ = (d) hyp = p, adj(θ) = r, cos θ = z y b c c r p z 22 85 13 , tan B = 3 4 3 5 5 4 (c) sin C = , cos C = , tan C = 85 , cos E = , tan E = (b) 64° (e) 30° (c) 57° (f) 27° 11 4.86 m x = 30° and y = 4.69 cm x = 3 m and y = 53.1° x = 48.2° x = 22.9° and y = 8.90 cm (b) 33° 5 –cos 68° cos 105° cos 135° –cos 60° –cos 125° 2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) θ = 15° or 165° θ = 124° θ = 52° or 128° θ = 70° θ = 20° or 160° θ = 48° θ =39° θ =110° θ =7° or 173° θ =84° or 96° (b) (d) (f) (h) (j) sin 24° sin 55° sin 35° sin 82° –cos 50° x = 40.4° x = 61.2° x = 7.1 x = 22.5° x = 14.0° 10 19.6 4 cos B = (b) θ = 43.2° (d) θ = 31.1° 6 (a) a = 2.9 m (b) z = 3.7 cm (c) m = 9.2 m (d) r = 5.0 cm Exercise 15.7 1 (a) 11.2 cm2 (c) 17.0 cm2 (e) 3.8 cm2 (b) 25.4 cm2 (d) 70.4 cm2 (f) 24.6 cm2 2 44.9 cm2 3 (a) x = 115° and area = 31.7 cm2 (b) x = 108° and area = 43.0 cm2 (c) x = 122.2° and area = 16.3 cm2 (b) (d) (f) (h) 1 (a) (c) (e) (g) (b) (d) (f) (h) 90° 36.9° 9.85 cm 10.3 cm 5 cm 4 cm 66.0° 16.9° 2 (a) 21.2 m (c) 21.2 m (b) 10.6 m (d) 18.4 m 3 (a) 9.80 cm (b) 24.1° Mixed exercise 1 (a) x = 37.6° (b) x = 44.0° (c) x = 71.4° 2 35.3° 3 AB = 300 m Exercise 15.6 1 (a) (c) (e) (g) (i) 5 (a) θ = 66.4° (c) θ = 60.7° Exercise 15.8 1 (a) (c) (e) (g) (i) 3 (a) θ = 82.4° (c) θ = 33.3° (e) θ = 38.4° 12 7 sin A = 7 , cos A = tan A = 13 , 12 13 11 84 x = 18.5 x = 18.2 x = 28.1° x = 8.3 4 AC ≈ 41.6 cm 5 ≈ 16 m lookout (L) 50° 25 m 2 (a) x = 6.3 cm (b) x = 6.8 cm (c) x = 4.4 cm (d) x = 13.2 cm (e) x = 8.7 cm (f) x = 10.8 cm a (e) hyp = x, adj(θ) = z, cos θ = x 19.6 13 Exercise 15.5 (b) tan x = 3 , tan y = 2 (b) sin B = 5 , cos B = 84 1 1.68 m 2 (a) 30° (d) tan y = , ∠X = (90 − y)°, tan X = 9 (a) 16 Exercise 15.4 (c) tan 55° = d , tan B = d 5 (a) (b) (c) (d) (e) 63 65 10 (a) 45° (d) 60° 12 (a) (b) (c) (d) 2 (a) opp(30°) = x cm adj(60°) = x cm opp(60°) = adj(30°) = y cm (b) adj(40°) = q cm opp(50°) = q cm opp(40°) = adj(50°) = p cm 3 (a) 0.65 (d) 0.41 16 65 (d) sin D = 63, cos D = , tan D = 3 student’s drawing (a) 223° (b) 065° (c) 11 km (d) 13 km (b) θ = 23.1° (d) θ = 28.9° (f) θ = 82.5° a 2 + c 2 − b2 2ac base of lookout (B) 5m swimmer (W) shark (S) 6 RS = 591 m Chapter 16 Exercise 16.1 1 (a) E (d) D (b) C (e) B (c) A Answers 189 2 (a) student’s own line (line should go close to (160, 4.2) and (175, 5.55)); answers (b) and (c) depend on student’s best fit line (b) ≈ 4.7 m (c) between 175 cm and 185 cm (d) fairly strongly positive (e) taller athletes can jump further 3 (a) distance (m) (b) 600 (d) Distance (m) 500 400 1 (a) $7.50 (d) $448 (b) $160 (c) $210 (e) $343.75 2 5 years 0 6 8 10 12 14 Age (years) (c) weak positive (e) 12 years old (f) Not very reliable because correlation is very weak (g) 600 m Mixed exercise 1 (a) the number of accidents for different speeds (b) average speed (c) answers to (c) depend on student’s best fit line (i) ≈ 35 accidents (ii) < 40 km/h (d) strong positive (e) there are more accidents when vehicles are travelling at a higher average speed 2 (a) there a strong negative correlation at first, but this becomes weaker as the cars get older (b) 0−2 years (c) it stabilises around the $6000 level (d) 2−3 years (e) $5000−$8000 Chapter 17 Exercise 17.1 1 $19.26 2 $25 560 Answers (b) $92.74 (c) student’s own graph; a comment such as, the amount of compound interest increases faster than the simple interest 5 $862.50 6 $3360 7 (a) $1335, $2225 (b) $1950, $3250 (c) $18 000, $30 000 8 (a) $4818 3 2.8% 5 $2281 more 200 100 190 Exercise 17.2 4 $2800 more 300 0 3 (a) $930.75 (b) $1083.75 (c) $765 (d) $1179.38 4 $1203.40 5 $542.75 6 (a) $625 (b) $25 (c) $506.50 6 (a) $7.50 (c) $225.75 (e) $346.08 (b) $187.73 (d) $574.55 (b) 120% 9 $425 10 $211.20 11 $43.36 (each) 12 $204 Chapter 18 7 $562.75 Exercise 18.1 8 $27 085.85 1 Exercise 17.3 1 (a) $100 (d) $900 x (b) $200 (c) $340 (a) (b) 2 $300 (c) 3 $500 2 5 (a) $179.10 (b) $40.04 (c) $963.90 (e) 2 y x −3 y 2 (d) y 4 $64.41 +2 (b) 40 h (c) (b) $1386 (c) $1232 0 1 2 3 1 2 1 −2 −7 1 6 −2 −3 −2 x − 2 −11 −6 −3 −2 −3 −6 −11 −4 −3 −4 −7 −12 6 2 x − 3 −12 −7 2 y=x + 1 2 9.5 4.5 1.5 0.5 1.5 4.5 9.5 12 1 25 h 2 2 (a) $1190 −1 −7 −2 1 Mixed exercise 1 (a) 12 h −3 −2 y 10 (e) 8 3 (a) $62 808 (b) $4149.02 6 4 (a) 4 Years Simple interest Compound interest 1 300 300 2 600 609 3 900 927.27 4 1200 1125.09 5 1500 1592.74 6 1800 1940.52 7 2100 2298.74 –10 8 2400 2667.70 –12 (b) 2 –4 –2 0 –2 x 2 4 –4 –6 –8 (a) (c) (d) 2 (a) y = 3 + x2 (c) y = x2 (e) y = −x2 − 4 (c) 6 m (d) just short of 4 seconds (e) 3 seconds (b) y = x2 + 2 (d) y = −x2 + 3 3 (a) −2 −1 0 1 2 3 4 5 1 (a) 2 x − 3 x + 2 12 6 2 0 0 2 6 12 y y = x2 − 3x + 2 12 10 8 6 4 2 x –3 –2 –1 0 –2 1 2 3 4 5 6 −3 −2 −1 2 x − 2 x − 1 14 0 1 2 3 y 15 y = x2 − 2x + 1 10 1 2 3 4 (c) −2 −1 0 1 2 3 4 5 2 6 x + 4 x + 1 −11 −4 1 4 5 4 1 −4 −11 6 y 4 2 –4 –2 0 –2 x 2 4 6 8 –4 –6 –8 –10 –12 y = x2 + 4x + 1 4 (a) 8 m (b) 2 seconds 4 6 8 10 (e) y 8 7 y = − 4x 6 5 4 3 2 1 0 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8x –2 –3 –4 –5 y = − 4x –6 –7 –8 y 16 y= x x –4 –6 –8 –10 –12 16 –14 y = x –16 x –4 –3 –2 –1 0 –5 y x 2 0 –16–14–12–10 –8–6 –4 –2 –2 2 4 6 8 10 12 14 16 5 x xy = 5 16 14 12 10 8 6 4 2 2 −1 −2 −1 2 7 y (b) (b) x 10 9 8 7 6 5 4 3 2 1 –8 –6 –4 –2 0 –2 –3 –4 –5 –6 xy = 5 –7 –8 –9 (c) 10 9 8 7 6 5 4 3 2 1 –10 –8 –6 –4 –2 0 –2 –3 –4 –5 –6 –7 –8 xy = 9 –9 2 (a) Length 1 Width 24 12 8 6 4 3 (b) y Width (m) 14 y y 8 7 6 5 4 3 2 1 0 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8x –2 –3 –4 –5 y = − 8x –6 –7 –8 y = − 8x Exercise 18.2 x y (d) xy = 9 x 2 4 6 8 10 24 22 20 18 16 14 12 10 8 6 4 2 0 2 3 4 6 8 12 24 2 1 y x 0 2 4 6 8 10 12 14 16 18 20 22 24 Length (m) (c) the curve represents all the possible measurements for the rectangle with an area of 24 m2 (d) ≈ 3.4 m Answers 191 Exercise 18.3 (b) −1 or 2 2 (a) 3 1 (a) y 8 y = 2x + 2 7 6 5 4 3 2 y = 4x 1 x –8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 y 2 1 –4 –3 –2 –1 0 –1 x 1 2 3 –2 –3 –4 (b) (i) −2 or 1 (ii) ≈ −1.6 or 0.6 (iii) ≈ −2.6 or 1.6 3 (a) (b) y 14 y = x2 − x − 6 13 12 11 10 9 8 7 6 5 4 3 2 1 x 0 –5 –4 –3 –2 –1 1 2 3 4 5 –2 –3 –4 –5 –6 –7 (b) (i) 0 or 1 (ii) −2 or 3 (iii) −3 or 4 –4 192 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 –2 –1 –2 –3 –4 –5 –6 –7 –8 Answers 2 (a) Exercise 18.4 1 (a) −2 or 3 (c) −3 or 4 y y = x2 − x − 6 (1, 4) and (−2, −2) (b) y 12 11 10 9 8 7 6 5 y = x2 + 2x − 3 4 3 2 1 x –8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8 –2 –3 –4 –5 y = −x + 1 –6 –7 –8 (c) (−4, 5) and (1, 0) y = −x2 + 4 6 5 4 3 2 1 –6 –5 –4 –3 –2 –10 y y= 3 x (ii) y = 0 x 1 2 3 4 5 (i) y = –6 y 8 7 6 5 y = x2 + 2x 4 3 y = −x + 4 2 1 x –8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 (−4, 8) and (1, 3) y 8 7 6 y = −2x2 + 2x + 4 5 4 3 2 1 x 1 2 3 4 –2 –3 y = −2x + 4 –4 1 2 3 4 5 6 ≈ (−2.3, −1.3), (1, 3) and (1.3, 2.3) (−2, 0) and (1, 3) –3 –2 –1 0 x –2 –3 –4 –5 –6 (iii) y = 6 (c) y 8 y = 2x2 + 3x − 2 7 6 y=x+2 5 4 3 2 1 x –8–7–6–5–4–3–2–10 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 (d) (0, 4) and (2, 0) y 4 3 y =1 x 2 2 y = −0.5x2 + 1x + 1.5 1 x 0 –8–7–6–5–4–3–2–1 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 (−3, −6) and (1, 2) Exercise 18.5 (e) 50 40 30 1 x (a) y 3 x − 4x –5 2 3 (b) y = x + 5 –4 –3 –2 –1 0 1 2 3 4 5 –225 –128 –63 –24 –5 0 –3 –8 –9 0 25 –120 –59 –22 –3 4 5 6 13 32 69 130 (c) y 2 x 3 + 5x 2 + 5 380 213 104 41 12 5 8 9 –4 –43 –120 (d) y x3 + 4x2 − 5 220 123 0 –5 –2 3 4 –5 –30 –145 –82 –43 –22 –13 –10 –7 2 23 62 125 –157 –71 –25 –7 83 185 3 (e) y = x + 2 x − 10 58 (f) y 2x 3 + 4 x 2 − 7 (g) y x 3 − 3x 2 + 6 56 22 6 (h) y 3x 3 + 5x 350 172 66 (a) 50 40 30 20 10 (b) x 2 4 6 8 10 –10 –8 –6 –4 –2 0 –10 –20 –30 y = x3 + 5 –40 –50 2 4 6 x 8 10 –10 –8 –6 –4 –2 0 –10 –20 –30 –40 y = x3 + 2x − 10 –50 (f) –5 –7 –1 25 343 2 4 6 2 –14 –48 –106 –194 20 10 14 –2 0 2 –14 –66 –172 –350 50 40 30 20 10 y –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 2 y = − 2x + 5x + 5 –50 2 4 50 40 30 20 10 –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 2 y = − x + 4x − 5 –50 50 40 30 x 6 8 10 2 4 6 x 8 10 2 4 6 x 8 10 2 4 6 x 8 10 2 4 6 x 8 10 Answers 193 y –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 2 y = 2x + 4x − 7 –50 (g) y 20 10 –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 2 y = −x − 3x + 6 –50 (d) 50 y 40 30 20 10 20 10 50 40 30 (c) y –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 3 y = x − 4x –50 19 y y (h) 2 4 6 x 8 10 50 40 30 y 20 10 –10 –8 –6 –4 –2 0 –10 –20 –30 –40 3 y = −3x + 5x –50 (d) 2 (a) x −2.5 y −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 4 5 50 6 30 10 (b) 50 y 40 20 10 –4 1 y = x3 + x –2 2 4 6 –2 (c) (i) x = –1.3, 1.8 or 4.5 (ii) x = 0 or 5 (iii) x = –1.6, 2.1 or 4.5 x 0 –10 2 –50 4 (e) –20 50 –30 40 –40 30 –50 20 y y = x3 − 2 x 10 3 x –3 1 x 1 3 y = x + (b) x (a) y (f) 4 –40 10 –4 2 –30 20 x 8 10 x 0 –10 –20 30 –10 –8 –6 –4 –2 0 –10 y=x−5 –20 –30 –40 3 2 y = x − 5x + 10 –50 (e) 3 x 20 y 50 40 30 (d) y = 2x + 40 –36.875 –18 –4.625 4 8.625 10 8.875 6 2.125 –2 –5.625 –8 –6 10 46 (b) (c) y x− –2 –1 –0.5 –2.67 –1.5 0 1.5 4.8 9.9 0 0.1 0.2 0.5 1 2 3 –9.9 –4.8 –1.5 0 1.5 2.67 –2 x 0 –10 2 4 –20 –30 –40 4 y = x +2− x 12.33 8 7 10.25 3 y 2x + –7 –5.5 –5 –7 x 2 y x3 − –26.33 –7 1 3.875 x 1 y x2 − x + 11. 67 5.5 1 –1.25 x (a) 25 Answers 22.04 42.01 –50 –37.99 –17.96 –5.75 –1 4 9. 67 –15.4 –30.2 30.2 9.992 19.99 15.4 7 5 5.5 7 –19.99 –9.992 –3.875 –1 7 26.33 –4.76 –9.89 9.91 (c) y 50 4.84 1.75 1 2.5 6.33 (f) y y = x2 + 2 − 50 4 x 40 15 30 30 20 20 y=x− 1 x 0 –5 40 10 5 –2 10.001 5.008 2.125 2 8.5 27.33 20 10 194 –0.1 –27.33 –8.5 –2 –2.125 –5.008 –10.001 2 –4 –0.2 –4 x 2 4 –4 –2 0 –10 4 y = x2 − x + 1 x 10 x 2 y –4 –2 0 –10 –10 –20 –20 –15 –30 –30 –20 –40 –40 –25 –50 –50 x 2 4 2 (a) 4 (a) x –4 –1 0 1 2 3 4 y = 2x 0.0625 0.125 0.25 0.5 1 2 4 8 16 y = 2–x –3 16 8 18 y = 2– –2 y 4 2 1 0.5 0.25 0.125 0.0625 5 (a) & (b) (i) y=2 16 300 Number of organisms 14 12 10 8 6 y y = 3x 250 200 150 100 50 y = 12x + 1 4 –1 0 2 –6 –4 –2 0 2 4 6 x 0 0.2 0.4 0.6 0.8 y = 10 x 1 1.58 2.51 3.98 6.31 10 1 2 x − 1 –1 –0.6 –0.2 0.2 0.6 12 10 8 1 y y = 10 x 6 4 2 0 –2 y = 2x − 1 0.2 0.4 0.6 0.8 1 1 2 3 4 Time (hours) (b) (ii) 12 per hour (c) (i) ≈ 3.4 hours (b) y 60 y 3 55 y = x − 1 50 45 40 35 30 25 20 15 10 5 x 1.2 5 x 6 (ii) ≈ 42 Exercise 18.6 1 x 1 2 3 4 5 6 (b) gradient = 12 10 y y = x2 − 2x − 8 9 8 7 6 5 4 3 2 1 0 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 0 –6 –5 –4 –3 –2 –1 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50 –55 –60 1 2 3 4 5 6 7 3 answers should be close to: with light ≈ 1.4 cm per day, without light ≈ 1 cm per day Mixed exercise 1 (a) A: y B: y x 3x − 2 x2 + 3 x 1 4 4 C: y = − − 5 (b) (i) (–1, –5) (ii) answer should be the point of intersection of graphs A and C. (c) maximum value y = 3 (a) –2 (b) 6 and –6 Answers 195 2 (a) 10 C y Exercise 19.4 D 1 (a) 15° (isosceles ) (b) 150° (angles in a ) (c) 35° (∠MON = 80°, and MNO in isosceles, so ∠NMO = ∠NOM = 50°, so ∠MPN = 35°) (d) 105° (∠PON = 210° so ∠PMN = 105° – half the angle at the centre) 8 y = x2 6 E 4 F 2 x –3 –2 y = x3 –1 0 –2 1 2 2 (a) 55° (angles in same segment) (b) 110° (angle at centre twice angle at circumference) (c) 25° (∠ABD = ∠ACD, opposite angles of intersecting lines AC and BD, so third angle same) 3 G –4 –6 H has no line symmetry –8 –10 (b) (0, 0) and (1, 1) (c) (i) x = 2 or x = –2 (ii) x = –2 (d) gradient for y x 2 when x = 2 is 4 and gradient for y x 3 when x = 2 is 12 3 (a) y = (c) 4 − x (b) y Exercise 19.1 1 (a) B A 2 (a) 2, student’s diagram (b) 2 3 student’s own diagrams but as an example: 6 144° 7 (a) 22° (b) 116° (c) 42° 8 (a) 56° (b) 68° (c) 52° DIAGRAMS ARE NOT TO SCALE BUT STUDENTS’ SHOULD BE WHERE REQUESTED 1 (a) Exercise 19.2 1 (a) 3 (b) 4 (c) infinite number corresponding to the number of diameters of the circle face (d) as per part (c) (e) 2 (f) 3 (5 if face is a square) (g) 1 (h) infinite number corresponding to the number of diameters of the sphere 2 (a) 4 (b) 3 (d) infinite (e) 4 (c) A A B C 2 O 4 cm (c) 1 (f) 8 3 2 6.5 cm 3 (a) 49.47 cm (b) A 3 cm 1 (a) x = 25° (b) x = 160°, y = 20° Answers 4 35° Exercise 19.5 Exercise 19.3 196 3 ∠DAB = 65°, ∠ADC = 115°, ∠DCB = 115°, ∠CBA = 65° 5 59.5° x y 8 y = −x 7 6 5 4 3 y = − 4x 2 1 0 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8x –2 –3 –4 –5 –6 –7 –8 Chapter 19 (b) A = 0, B = 3, C = 4, D = 4, E = 5, F = 2, G = 2, H = 2 (b) 177.72 cm 1m B 4 (ii) eight (d) (i) Chapter 20 Exercise 20.1 4 km 3 km (ii) none School 5 A B 6 NOT TO SCALE – note that on student drawing point X should be 2 cm (10 m) from B. D X C Radio mast B Flagpole 2 (a) (b) (c) (d) a hexagonal prism the axis of rotational symmetry 6 7 3 (a) w = 72° (base angle isosceles ∆OCB), x = 90° (angle in semicircle), y = 62° (angles in a ∆) z = 18° (base angle isosceles ∆ODC) (b) x = 100° (reflex ∠ADB = 200°, angle at circumference = half angle at centre) (c) x = 29° (∠ADB is angle in a semicircle so ∠BDC = 90°, then angles in a ∆) (d) x = 120° (angle at centre), y = 30° (base angle isosceles ∆) 4 (a) x = 7.5 cm, y = 19.5 cm (b) x = 277.3 mm, y = 250 mm 5 NOT TO SCALE Mixed exercise 1 (a) (i) (ii) none Petrol station 6 NOT TO SCALE X (b) (i) (ii) none A (c) (i) (ii) four 5 cm B Route 66 Frequency 1.5 km A Cell tower 1 (a) Frequency Road (e) (i) SCHOOL SC SCHOOL 60 50 40 30 20 10 0 60 50 40 30 20 10 0 English exam marks 0 10 20 30 40 50 60 70 80 90 100 Mock exam score Maths exam marks 0 10 20 30 40 50 60 70 80 90 100 Mock exam score (b) 51−60 (c) 51−60 (d) students to compare based on histograms, but possible comments are: The modal value for both subjects is the same but the number in the English mode is higher than the Maths. Maths has more students scoring between 40 and 70 marks. Maths has more students scoring more than 90 marks and fewer scoring less than 20. 2 (a) bars are touching, scale on horizontal axis is continuous, vertical axis shows frequency (b) 55 (c) 315 (d) 29−31 (e) scale does not start from 0 X Answers 197 3 Histogram of ages Mixed exercise Chapter 21 1 (a) Exercise 21.1 Frequency density Frequency one person 12 10 8 6 4 2 0 Heights of trees 0 2 1 (a) 3 : 4 (c) 7 : 8 2 (a) (c) (e) (g) (i) 4 6 8 10 12 Height in metres (b) 19 (c) 6−8 metres 15 20 25 30 35 40 45 50 55 Age (years) 4 (a) 300 (b) 480 (c) 100 Frequency 2 (a) Exercise 20.2 P60 x 0 10 20 30 40 50 60 70 80 90 100 Percentage (b) Median = 57%, Q1 = 49% and Q3 = 65% (c) IQR = 16 (d) 91% (e) 60% of students scored at least 59%; 80% of the students scored at least 67% 2 (a) (b) (c) (d) 166 cm Q1 = 158, Q3 = 176 18 12.5% 198 Answers 3 60 cm and 100 cm 4 (a) 20 ml oil and 30 ml vinegar (b) 240 ml oil and 360 ml vinegar (c) 300 ml oil and 450 ml vinegar Exercise 21.2 A 56 58 60 62 Mass (kg) 64 1 (a) 1 : 2.25 (b) 1 : 3.25 2 (a) 1.5 : 1 (c) 5 : 1 (b) 5 : 1 (c) 1 : 1.8 Exercise 21.2 B 1 240 km 3 2 30 m Histogram of airtime minutes Q2 Q1 (e) 1 : 4 x=4 x=3 x = 1.14 x = 2.67 x = 13.33 6 810 mg (b) 60−62 kg (c) 7.4% (d) 10 kg P80 Q3 (b) (d) (f) (h) (j) x=9 x = 16 x=4 x = 1.875 x=7 5 60°, 30° and 90° Frequency density 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 5 teenagers 20 30 40 50 60 70 80 90 100 110 120130140150 Time (minutes) 4 (a) 6.5 cm (b) Cumulative frequency Number of students 1 (a) Masses of students 20 18 16 14 12 10 8 6 4 2 0 54 (b) 6 : 1 (d) 1 : 5 Cumulative frequency of plant heights y 30 20 10 0 Q1 0 1 2 3 4 Q2 Q3 5 6 7 8 Height (cm) median height = 6.8 cm (c) IQR = 8.3 – 4.7 = 3.6 x 9 10 11 12 3 (a) 5 cm (b) 3.5 cm 4 (a) it means one unit on the map is equivalent to 700 000 of the same units in reality (b) Map distance 10 71 50 80 1714 2143 (mm) Actual distance 7 (km) 50 35 56 1200 1500 5 (a) 4 : 1 (b) 14.8 cm (c) 120 mm or 12 cm Exercise 21.3 1 25.6 l 2 11.5 km/l 3 (a) 78.4 km/h (b) 520 km/h (c) 240 km/h 4 (a) 5 h (c) 40 h (b) 9 h 28 min (d) 4.29 min 5 (a) 150 km (c) 3.75 km (b) 300 km (d) 18 km 1 (a) (i) 100 km (ii) 200 km (iii) 300 km (b) 100 km/h (c) vehicle stopped (d) 250 km (e) 125 km/h (e) (f) (g) (h) (i) 2 hours 190 min = 3 h 10 min 120 km/h (i) 120 km (ii) 80 km 48 km/h 40 min 50 min 53.3 − 48 = 5.3 km/h Pam 12 noon, Dabilo 11:30 a.m. 3 (a) (i) 40 km/h (ii) 120 km/h (b) 3.5 min (c) 1200 km/h2 (d) 6 km 5 4 (a) 0–30 s, m/s 2 6 (b) after 70 s, 0.5 m/s2 (c) 90 km/h (d) 2 km Exercise 21.5 1 (a) Yes, (ii) 1 2 (b) (i) 0.35 h (ii) 4.7 h (iii) 1.18 h day 8 (a) 12 days (b) 5 days 9 5 h 30 min Exercise 21.4 2 (a) (b) (c) (d) 1 (b) (i) 2 2 days A 1 = B 150 8 1 15 is not = 2 10 1200 km/h Exercise 21.6 1 (a) k = 7 (b) a = 84 m = 0.4587 , T so m varies directly with T 3 (a) F = 40 (b) m = 4.5 2 ratio of m to T is constant, 4 a = 2, b = 8, c = 1 1 3 5 (a) y = 2 (b) x = 0.5 6 (a) y = 2x2 (c) x = 9 (b) y = 1250 4 (a) (b) (c) (d) (e) 150 km after 2 hours; stopped for 1 hour 100 km/h 100 km/h 500 km 5 (a) (b) (c) (d) 20 seconds 2 m/s2 200 m 100 m 6 4.5 min 7 187.5 g k or PV = k V (b) P = 80 8 (a) P = 9 (a) 7 : 4 (b) 54.86 mm Chapter 22 7 (a) y x = 80 (b) y = 8 (c) x = 15.49 Exercise 22.1 A 8 (a) b = 40 (b) a = 17 7 9 (a) y = 2.5 (b) x = 2 9 10 (a) xy = 18 for all cases, so relationship is inversely proportional 18 (b) xy = 18 or y = x (c) y = 36 Exercise 21.7 1 (a) x − 4 (b) P = 4x − 8 (c) A = x2 − 4x 2 (a) S = 5x + 2 5x + 2 (b) M = 3 3 (a) x + 1, x + 2 (b) S = 3x + 3 4 (a) x + 2 (b) x − 3 (c) S = 3x − 1 1 $300 Exercise 22.1 B 2 $72 000 1 14 3 (a) $51 000 (b) $34 000 2 9 cm 4 14 350 3 80 silver cars, 8 red cars 3 $12.50 Mixed exercise 4 60 m 1 (a) 90 mm, 150 mm and 120 mm (b) Yes, (150)2 = (90)2 + (120)2 4 father = 35, mother = 33 and Nadira = 10 (b) No, (c) Yes, 2 (a) $175 A B = 10 1 (b) $250 5 (a) 75 km (b) 375 km (c) 3 h 20 min 6 (a) 15 litres (b) 540 km 7 (a) inversely proportional 2 1 : 50 3 (a) (i) 85 km (ii) 382.5 km (iii) 21.25 km 5 breadth = 13 cm, length = 39 cm 6 X cost 90c, Y cost $1.80 and Z cost 30c 7 9 years 8 97 tickets Answers 199 Exercise 22.2 1 (a) V = U + T − W U T2 (b) V = − W or 3 V (U − T 1 3 W ) P P 2 P Q=± R Q2 P= 2 Q2 P= R P = Q2 + R R2 Q= P V I= R 20 amps A r= (note, radius cannot be π negative) r = 5.64 mm (f) Q = ± (g) (h) (i) (j) (k) 2 (a) (b) 3 (a) (b) 4 (a) F 9 5 C + 32 (b) 80.6 °F (c) 323 K Exercise 22.3 1 (a) 11 (c) 5 (b) –1 (d) 2m + 5 2 (a) f(x) = 3x2 + 5 (b) (i) 17 (ii) 53 (iii) 113 (c) f(2) + f(4) = 17 + 53 = 70 ≠ f(6) which is = 113 (d) (i) 3a2 + 5 (ii) 3b2 + 5 (iii) 3(a + b)2 + 5 (e) a = ±3 200 Answers 4 (a) 4(x – 5) (b) 4x – 5 Chapter 23 Exercise 23.1 A 1 5 18 C A (d) B = AC (c) B = (e) Q 3 (a) h ( x ) 5 x (b) (i) h(1) = ±2 (ii) h(–4) = ±3 6 (a) f–1(x) = x – 4 (b) f–1(x) = x + 9 x (c) f −1 ( x ) = 5 (d) f–1(x) = –2x x 7 (a) −3 2 x −3 (b) 2 (c) 2(x + 3) (d) 2x + 3 (e) 2x + 3 (f) 2(x + 3) 2 Mixed exercise 1 10 2 41, 42, 43 3 4 years 4 Nathi has $67 and Cedric has $83 5 Sindi puts in $40, Jonas $20 and Mo $70 6 44 children 9a − 26 7 (a) b = 8 a2 − 4 (b) b = 17 8 5 +3 2 x−4 −1 f (x ) = 3 3 a=6 9x + 16 37 3 A P QQ C S f −1 ( x ) = 9 (a) (b) (c) (d) (e) P B S P' R R A' Q' S' B' R' C' 4 A: y = 5 B: x = 0 C: y = −1.5 D: x = −6 5 10 (c) y 8 D D' 3 (a) A x B'' 4 2 2 4 6 8 10 X 2 y –10 –8 –6 –4 –2 0 –2 A'' D' D 1 (a) u r 5 1 (a) AB = 0 y A' A 7 6 5 4 3 B' 2 1 2 C' y B x A 2 4 2 (a) u r −5 (g) CD = −6 u r −5 (h) BE = −6 −5 (k) −6 (a) B 10 8 4 A 6 C B 4 2 –4 u r 9 (f) EC = 6 2 y X u r 1 (e) DB = 6 y X –4 –2 0 B –2 u r −1 (d) BD = −6 (l) Yes (b) rotation 180° about (0, 0) –4 u r 0 (c) AE = −6 9 (j) 0 –4 –5 –6 A'' –7 2 u r 4 (b) BC = 0 (i) they are equal 1 2 3 4 5 –2 B'' –3 C'' 4 C x –5 –4 –3 –2 –1 0 B' 10 Exercise 23.2 A Exercise 23.1 C (b) 8 (b) rotation 180° about (4, 5) 1 A: centre (0, 2), scale factor 2 B: centre (1, 0), scale factor 2 C: centre (−4, −7), scale factor 2 1 D: centre (9, −5), scale factor 4 –2 x 6 –10 Exercise 23.1 B 0 C 4 –8 x for MNOP: rotate 90° clockwise about (2, 1) then reflect in the line x = 5.5 –2 2 –6 for ABC: rotate 90° clockwise about B then reflect in the line x = −6 –4 D –4 6 possible answers are: A' B A 4 x –10 (a) F' 6 (d) B y 8 C A' C' –10 –8 –6 –4 –2 0 –2 C –4 B' –6 D'' –8 10 C' 6 C'' (b) enlargement scale factor 2, using (8, −1) as centre y 2 4 A 2 x –4 –2 0 x 2 4 6 8 –2 –4 Answers 201 (g) b (b) (j) D (c) F G H 4 C −3 2 B 3 9 E 3 Exercise 23.2 B 2 (b) 6 0 (c) 12 −1 (d) 7 −2 (e) 1 −1 (f) 4 −4 (g) 18 (h) −8 22 0 (i) −20 10 (j) −16 2 (a) –a (d) 2c 202 Answers 4 (a) (b) (c) (d) 6.40 cm 7.28 cm 15 cm 17.69 cm 5 (a) (b) (c) (d) 5.10 5 8.06 9.22 u r 7 BC = 5 u r −11 CA = 1 u r 7 (a) XZ = x + y u r (b) ZX = –x – y u r 1 (c) MZ = x + y 2 2 8 (a) (i) x = 7 (d) −8 1 (a) 16 (i) −7a + 7c 6 (a) A(–6, 2), B (–2, –4), C (5, 1) u r 4 (b) AB = −6 E −3 D −3 (h) –c + 3c 3 (a) – (e) student’s own diagrams C 8 3 A 1 b 2 −3 (ii) y = −3 10 (iii) z = −4 (b) (i) 7.28 (ii) 4.24 (iii) 21.5 u r 9 (a) (i) XY = b – a u r 1 (ii) AD = (a + b) u r 2 (iii) BC = 2(b – a) u r u r (b) XY = b – a and, BC = 2(b – a) so they are both multiples of (b u r– a), and hence u r parallel, and is double XY 10 28.2(3sf) (b) 2b (e) 2b (c) −a + c (f) 2c Exercise 23.3 1 (a) y 10 9 8 7 6 5 4 3 2 1 –1 0 –1 (b) x 1 2 3 4 5 6 7 8 9 10 y 10 9 8 7 6 5 4 3 2 1 –1 0 –1 (c) x 1 2 3 4 5 6 7 8 9 10 y 10 9 8 7 6 5 4 3 2 1 –1 0 –1 x 1 2 3 4 5 6 7 8 9 10 (d) 3 (b) 12 y 10 9 8 7 6 5 4 3 2 1 –1 0 –1 9 6 2 (a) & (b) 0 3 y 7 6 5 4 3 2 1 −12 −6 6 (c) −14 4 −10 5 (a) x = 3 (b) Exercise 23.5 x 1 2 3 4 5 6 7 8 9 10 2 (a) shear, x-axis invariant, shear factor 1.5 (b) 2-way stretch: stretch of scale factor 4, invariant line x = 1 and stretch of scale factor 2, invariant line y = 2 (c) shear, x = 0 (y-axis) invariant, scale factor 0.75 (d) shear, line y = 4 invariant, shear factor 1 Exercise 23.4 0 6 5 1 (a) 7 2 −3 2 (c) −11 3 (a) AB = (b) 3 9 3 (a) enlargement, scale factor 3, centre of enlargement the origin −3 0 (b) 0 3 3 10 (ii) (d) 6 (c) 4 (a) 2 x –10 –8 –6 –4 –2 –2 2 (b) 4 6 8 10 –4 Mixed exercise 8 −5 1 y 8 –10 12 −4 6 6 −10 −4 ; so AB ≠ BA (b) (i) –6 (ii) –84 –2 –3 –4 –5 –6 –7 –6 (b) (i) −1 1 2 3 4 5 0 3 4 3 0 x –5 –4 –3 –2 –1 2 (a) R' (2, 8) (b) a stretch parallel to the y-axis (x-axis invariant), scale factor 4 –8 reflect in the line x = −1 (iii) reflect in the line y = −1 9 and BA = 1 7 0 −3 4 (a) 11 0 4 4 −1 2 (ii) rotate 90° clockwise about the origin 2 −1 4 3 2 (a) –12 5 −3 1 (a) (i) reflection in the y-axis (ii) shear with x-axis invariant and shear factor 2 −1 0 (b) (i) 0 1 1 2 (ii) 0 1 1 (a) (i) −2 −2 1 (b) 4 −3 1 1 rotate 90° anticlockwise about (0, 0) then translate 2 −1 reflect in the line y = −1 −8 then translate 0 (a) B′ (−6, −6) (c) B′ (−1, 8) (b) B′ (6, −2) (d) B′ (3, 9) 12 0 (ii) −8 1 (iii) 10 12 (iv) 0 4 (a) (i) (b) (i) a 2a (iii) rotate 180° about origin 6 then translate 0 (iv) reflect in the line x = 0 (y-axis) then translate 0 −2 Answers 203 (ii) c b 11 (a) & (c) y 7 10 8 8 6 4 6 b+c 4 A a b a b 8 2 4 u r 5 (a) (i) =y u r (ii) DE = –y ur (iii) FB = x + y ur (iv) EF = x – y ur (v) FD = 2y – x (b) 4. 47 6 (a) one-way stretch with y-axis invariant and scale factor of 2 (b) one-way stretch with x-axis invariant and scale factor of 2 (c) a shear with x-axis invariant and shear factor = 3 (d) a shear with y-axis invariant, shear factor = 2 Answers B 4 6 C 2 C 8 10 9 8 7 6 5 4 3 2 1 –8 1 (b) 0 A 0 1 2 0 (d) 0 2 2 (e) 0 0 2 x 1 2 3 4 5 6 7 8 9 10 5 (c) 4 7 (d) 6 Chapter 24 Exercise 24.1 1 Card Coin H T H T H T H T R 7 8 Y G 5 6 (e) |P| = –3 1 2 − (f) P–1 = 3 3 2 1 − 3 3 10 a = 7, b = 2 and c = –9 B x 8 10 A –6 –10 2 5 9 (a) 4 3 1 0 (b) 1 0 2a + 3b 6 x y –1 0 –1 A –4 D –2 –2 a–b (iv) 204 2 C –10 –8 –6 –4 –2 0 –2 B 2 (iii) y B 2 1 2 3 4 3 (a) & (b) 2 3 1 3 G H A B C D E F 7 11 Green 4 11 Yellow 8 11 Green 3 11 Yellow Green Yellow Exercise 24.2 Mixed exercise 1 (a) 1 2 Blue 1 3 Yellow 1 6 (b) Black 1 4 1 2 H 1 2 1 2 T H 1 2 1 2 T H 1 2 (c) 2 (a) 1 (a) & (b) 1 2 1 2 1 2 1 2 1 2 (d) H H 1 2 1 2 1 2 T 1 2 (b) (d) 1 8 1 2 (c) 1 6 T 1 12 1 2 H T H T T H T 5 12 1 6 2 1 6 3 1 6 1 2 H 1 6 1 2 1 2 T H 1 6 1 2 1 2 T H 1 2 1 2 T H 1 2 T 4 5 6 (c) 1 8 (d) 2 (a) & (b) 2 7 1 2 (e) 0, not possible on three coin tosses 1 5 7 (c) 5 7 1 2 H 1 2 1 2 T H 1 2 1 2 T H 1 2 1 2 T H 1 2 1 2 T H 1 2 1 2 T H 1 2 1 2 1 2 1 2 H T H T 1 2 1 2 1 2 1 2 H T H T 1 2 1 2 1 2 1 2 H T H T 1 2 T 1 12 1 6 5c 5 6 10c 5c 1 10c 5 5 10c (d) 1 21 (e) 1 (there are no 5c coins left) Answers 205