La Mathematics LEARNER’S BOOK ¡ MICHAEL CHIYAKA | _ FREDERICK FINCH | SYLVIA MULENGA *rogress in Viathematics ARNER’S BOOK OXFORD Oxford University Pees ia department ofthe University of Oxford. Itfurters the University’s objective of excellence in research, scholarship, and education by publishing worldwide, Oxford isa registered trade mark of ‘Ostord Univeesity Press in the UK and in certain other countries Published in Zambia by (Oxford University Press ORBIS (Pty) Limited Vasco Boulevaed, Goodwood, N1 City, P © Box 12119, Cape Town South Africa © Onford University Press ORBIS (Pty) Ltd 2014 “The mora rights of the author have been asserted [All rights reserved. No pat of this publication may be reproduced, stored in ‘retrieval system, oF transmitted, in any form or by any means, without the prior permission in writing of Oxdord University Press ORBIS (Pty) td, ‘or as expressly permitted by law, by licence, or under terms agreed swith the appropriate reprographic rights organisation, Enquiries concerning ‘reproduction outside the scope ofthe above shouldbe sent to the Rights Department, ‘Oxford University Press ORBIS (Pty) Ltd,st the above address. ‘You must not circulate this work in any other form, ‘and you must impose this same condition on any acquirer, Progress in Mathematics Grade 11 Learner's Book ISBN 978019 040121 4 Second impression 2016 Publisher/ Commissioning editor: Marie Louise Kriel Project co-ordinators: Anson Banda and Irene Mvula Editor: Lorraine Bowie ‘Designer: Sandi Hall ‘Cover photograph: ‘Typesetter: Laura Brecher Printed and boundby ABC Press “The authors and publisher gratefully acknowledge permission to reproduce copyright material in this book.
Fvery effort has been made to trace copyright holders, but ifany copyright infringements have been made, the publisher would be grateful for information that would enable ‘any omissions or errors tobe corrected in subsequent impressions. to third party websites are provided by Oxford Links in good faith and for information only. Oxford disclaims any responsiblity for the matcrials contained in any third party website referenced in this work. Contents How to use this book s v Topic1 Approximations FT Sub-topic 1 Relative and absolute error 2 Summary, revision and assessment.
8 'T0pic 2 Seqiiences and series: T1 Sub-topic 1 Arithmetic progressions 12 Sub-topic 2. Geometric progressions 17 Summary, revision and assessment 23 Topic 3 Coordinate geometry: 25 Sub-topic 1 The length of a straight line between two points. The midpoint of two points. The gradient of a line segment 30 Sub-topic 4 The equation of a straight line 32 Sub-topic 5 Parallel and perpendicular lines 37 Summary, revision and assessment 3 39 Topic 4 —_ Relations and functions a4 Sub-topic 1 Inverse functions ke 42 Sub-topic 2.
45 Sub-topic 3 Applications 7 47 Summary, revision and assessment Sub-topic 1 Introduction to quadratic functions - : 383 Summary, revision and assessment n 57 Topic 6 Quadratic equatfons 59 Sub-topic 1 Introduction to quadratic equations 60 Sub-topic 2. Solutions of quadratic equations - 61 Summary, revision and assessment 71 Topic7 Variation 73 Sub-topic 1 Introduction to variation z 7 74 Sub-topic2 Direct and inverse variation. Jointand partlal variation et 7 Sub-topic4 Graphs. 2 79 Sub-topic§ Applicaions.
80 Summary, revision and assessment a sia 81 Topic8 Gircle theorems g3 Sub-topic 1 Properties of a circle 84 Sub-topic 2. Angle properties sas 87 Summary, revision and assessmment. 100 Topic 9 Gonstrucfion andToci 105 Sub-topic 1 Constructions. The locus of a point 107 Sub-topic 3 Loci in two dimensions seo 108 Sub-topic 4 Lociin three dimensions 110 Summary, revision and assessment 111 Topïc 10 ˆ Trigonometry Bae Sub-topic 1 Introduction to trigonometry.
114 Sub-topic 2 Trigonometric ratios : 115 Sub-topic 3 Sine and cosine rules 121 Sub-topic 4 Area of triangles 3 125 Sub-topic 5 Trigonometry on the Cartesian plane 126 Sub-topic 6 Applications of trigonometry 130 Summary, revision and assessment 133 Topic 11 Mensuration 135 l Sub-topic 1 Area 136 Sub-topic 2 Volume 141 Summary, revision and assessment 145 Topic 12 Probability. 147 Sub-topic 1 Laws of probability + 148 Sub-topic 2. Tree diagrams and grids 151 Summary, revision and assessment sos 187 Topic 13: StaHsties 159 Sub-topic 1 Cumulative frequency tables 160 Sub-topic 2. Measures of dispersion 165 Summary, revision and assessment Hi 178 Glossary 181 ill ol ‘Welcome to the Progress in Mathematics series for Grades 10-12! ‘This series is based on the Senior Secondary Syllabus for Mathematics issued by the Ministry of Education, Science, Vocational Training and Early Education, All the knowledge, skills and values expressed in the document are addressed in Progress _ in Mathematics Grade 11 Learner's Book, so that you can feel confident about your ae success in this subject.
‘This page will help you understand how the book works. ‘The book is divided into topics so that you can easily see what content will be covered in your Mathematics class. On the first page of every topic, you will find: A table of sub-topics and specific outcomes that will be covered in the topic. —A starter activity helps to introduce the topic with knowledge you already have.
At the end of each topic, you will find the following: le topic summary will help you — revise key learning points in the ic quickly. vision exercises help you revise topic’s work and check your jerstanding. How to use this book Assessment exercises help you prepare for tests and exams, —————____ You will see the following throughout the book: New words boxes give you the definitions of key words or explain what a certain new word means. These words and the definitions are also in the glossary at the back of the book.
+ — Did you know? boxes give you more and new knowledge about what you are learning. ‘Worked examples give an example with a + model answer that shows step by step how to doa calculation. Activities are tasks where you apply the —____} i knowledge and skills you learnt in a section. Note: We use the term activity to refer to written exercises and practical activities.
How to use this book T0PIG Approximations Sub-topic 'Speciic Outcomes: Relative and absolute eror | * Work with relative and absolute errors SfErter actiuity, Look at the diagram below, then answer the questions that follow. 1 Say whether or not each of the following can be determined accurately. a) the number of light bulbs b) the length of line AB ©) the mass of the bananas @) the number of locks 2 From your answers to Question 1, write down a statement about: a) numbers of things (found by counting them) b) measurements of things (found by measuring them). ‘Topic 1 Approximations A}, =z SUB-TOPIC 1 Relative and absolute error Introduction When we count, then the number is true and accurate.
However, if we measure a line as 5 cm, it is not necessarily accurate. This is because all measurements are rounded off to a given number of significant digits or decimal places and are therefore subject to error. A dimension of 2 cm, to the nearest centimetre, lies between 1.5 cm (see Figure 1. Similarly, a dimension of 3 cm, to the nearest centimetre, lies ep ÔJy Nl §Pn[te=25 tuy we SỹTi j ppt cac Figure 1.2 Q1 We say that there is an error of 0.5 cm in the measurement of the dimensions of 2a)ˆ the rectangle.
If the breadth of the rectangle is b, then b lies between 1.5 cm and bd)” 2. If the length of the rectangle is J, then Ilies between 2. 3 15; 10: We write this as follows: These can also be written as follows: 1.5 cm b) In each case, the smaller number is called the lower bound and the bigger ° number is called the upper bound. So, dimension:a measurement if 1.5 cmis the Jower bound: the smallest number of the interval within which a dimension can fall lower bound and 2.5 cm is the upper upper bound: the biggest number of the bound.
The lower bound and the upper interval within whicha dimension can fall bound are also called the lower limit lower limit: another name for a lower and the upper limit respectively. bound In general, if a dimension, d, is given upper limit: another name for an upper bound to the nearest 1, then the error is y TT ng 1 Find the error in each of the following dimensions. a) 340 m, to the nearest 10 m b) 3 600 g, to the nearest 100 g ©) 12 ¢, to the nearest litre 4) 10.4 seconds, to the nearest tenth of a second 2 Topic 1 Approximations Wee encom) 2 A dimension is stated as 4. Find: a) the lower bound b) the upper bound of the dimension.
3 The length and breadth of a rectangle, given to the nearest centimetre, are 15 cm and 10 cm respectively. Find: a) the shortest possible length and the shortest possible breadth of the rectangle b) the longest possible length and the longest possible breadth of the rectangle ©) the limits between which the area must lie, Answers 1a) n=10m - ertor= 10mEs m »b) = 100g -. error= © n=1 €2, etor= xế°- OSE d) n=0. error 2 a) The lower bound = 4.
b) The upper bound = 4. 3 15 cm, given to the nearest centimetre, lies between 14. 10 cm lies between 9. a) The shortest possible length = 14.5 and the shortest possible breadth =9.
'b) The longest possible length = 5.5 cm and the longest possible breadth =10. LÍ ©) The smallest possible area = the shortest possible length x the shortest possible breadth = 145 x9, 137.75 cm? = ‘The largest possible area = the longest possible length x the longest possible breadth = 15.75 cm? So, the area lies between 137. 1 Which of the following can be found accurately? a) the number of words on this page b) the dimensions of this book ©) the mass of a cow @) the number of houses in a village ¢) the number of teachers in a school Sub-topic 1 Relative and absoluteenor ở) KT ity1 (continued) 2 Write down the lower and upper bound of each of the following. a) Sm, to the nearest centimetre b) 250 kg, to the nearest 10 kg ©) 33.5 ¢, to the nearest tenth of a litre 4) 5.81, to the nearest hundredth 3 Find the limits between which the following measures must lie.
Write your answers in the form. , where m is the given measure. you knt a) 8 cm, to the nearest es centimetre aa b) 0.5 cm, to the nearest millimetre ©) 3.5 kg, to the nearest tenth of a kilogram measurement, which is based on seven units of 4đ) 14m, to the nearest metre measurement.000 km, to the nearest 10 km the metre, kilogram, Ê) 52 min, to the nearest 10 seconds second, Kelvin, ampere, 8) 20.15 g, to the nearest hundredth of a gram MOl6 afd the candela Al h) 55 m, to the nearest 5 m measurement are derived i) 1.450 km, to the nearest 50 km from these seven basic j) 0.05 kg, to the nearest gram units. For example, the 4 If the length of a rod is given as 12cm+0.5 cm, __unitof force is the newton, find: which is expressed in a) the shortest possible length of the rod ee b) the longest possible length of the rod.
ne eae Absolute, relative and percentage errors The absolute error of a dimension is the.absolute ies difference between the true value and the recorded value. ‘The notation |x| means Absolute error == |recorded ke value ~ true value| “the absolute Fen value of x”. e aioe The relative error of a dimension is the ratio of the vee the number absolute error to the true value. pou =D absolute error Relative error = “2seeror So, +6] oe = 6 and Ƒ-6| = 6.
truevalue hect the ic: relative error as a HH NPPPPPENN - percentage.