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©SHINGLEE PUBLISHERS PTE LTD All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the Publishers. First Published 1979 Reprinted 1980, 1982 Second Edition 1983 Reprinted 1984 Third Edition 1985 Reprinted 1986, 1987, 1988 Fourth Edition 1989 Reprinted 1990, 1991, 1992, 1993 Fifth Edition 1995 Reprinted 1995, 1996, 1997 Sixth Edition 1999 Reprinted 1999 Seventh Edition 2001 Reprinted 2001, 2002, 2003, 2004, 2005, 2006 Eighth Edition 2007 Reprinted 2008, 2009, 2010, 2011 Ninth Edition 2013 Reprinted 2013, 2014, 2015, 2016, 2017, 2018, 2019 Tenth Edition 2020 Reprinted 2020, 2021
ISBN 978 981 32 4514 3
ACKNOWLEDGEMENTS We are grateful to Mr Tan Teck Hock for designing the interactive geometry templates. Images and links produced from www.geogebra.org used with permission from GeoGebra The Geometer’s Sketchpad® name and images used with permission of Key Curriculum Press, www.keycurriculum.com/sketchpad All licensed images purchased under standard license agreement with www.shutterstock.com While every effort has been made to trace the copyright holders and obtain permission for material reproduced in this book, if there has been an oversight, the Publisher will be grateful to hear from anyone who has not been appropriately acknowledged and to make the corrections in future reprints or editions of this book.
MINISTRY OF BY E
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TION CA DU
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Printed in Singapore
-2 e fr o m 2020
PREFACE think! Additional Mathematics is an MOE-approved textbook specially designed for students studying O and N(A) Level Additional Mathematics. Chapters and sections which have been excluded from the Normal (Academic) syllabus are clearly indicated with a . The features of this textbook series reflect the important shifts towards the development of 21st century competencies and a greater appreciation of mathematics, as articulated in the Singapore mathematics curriculum and other international curricula. Every chapter begins with a Chapter Opener and an Introductory Problem to motivate the development of the key concepts in the topic. The Chapter Opener gives a coherent overview of the big ideas that will frame the study of the topic, while the Introductory Problem positions problem solving at the heart of learning mathematics. Two key considerations guide the development of every chapter â&#x20AC;&#x201C; seeing mathematics as a tool and as a discipline. Opportunities to engage in Investigation, Class Discussion, Thinking Time and Journal Writing are woven throughout the textbook to enhance studentsâ&#x20AC;&#x2122; learning experiences. Stories, videos and puzzles serve to arouse interest and pique curiosity. Real-life examples, applications and contextual problems also teach students to appreciate the beauty and usefulness of mathematics in their surroundings. Underpinning the writing of this textbook series is the belief that all students can learn and appreciate mathematics. Worked Examples are carefully selected, questions in the Reflection section prompt students to reflect on their learning, and problems are of varying difficulty level to ensure a high baseline of mastery, and to stretch students with special interest in mathematics. The use of ICT helps students to visualise and manipulate mathematical objects with ease, hence promoting interactivity. Coding opportunities are included to cater to students with coding knowledge. We hope you will enjoy the subject as we embark on this exciting journey together to develop important mathematical dispositions that will certainly see you through beyond the examinations, to appreciate mathematics as an important tool in life, and as a discipline of the mind.
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KEY FEATURES Chapter Opener includes rationales for learning the chapter to arouse students’ interest and big ideas that connect the concepts within and across chapters.
Introductory Problem provides students with a more specific motivation to learn the topic using a problem that helps develop a concept, or an application problem that students will revisit after they have gained necessary knowledge from the chapter.
Learning Outcomes help students to be aware of what they are about to study so as to monitor their progress.
Important Results summarise important concepts or formulae obtained from Investigation, Class Discussion or Thinking Time.
Worked Example shows students how to present their working clearly when solving related problems. Guiding questions based on PÓlya’s Problem Solving Model are sometimes included.
Recap revisits relevant prerequisites at the beginning of the chapter or at appropriate junctures so that students are ready to learn new knowledge built on their existing schema.
Practise Now consists of questions that help students achieve mastery of procedural skills. Similar and Further Questions identify appropriate questions for students’ self-practice.
Introductory Problem Revisited revisits an application-based Introductory Problem later in the chapter. This is absent if the Introductory Problem leads directly to the development of a concept. P A G E
Exercise questions are classified into three levels of difficulty – Basic, Intermediate and Advanced. Questions at the basic level are usually short-answer items to test basic concepts and skills. The Intermediate level contains more structured questions, while the Advanced level involves applications and higher order thinking skills.
Contextual Problems are authentic problems that usually include modelling the real-world scenario with the equation of a function and are spread throughout the textbook.
Explanation Questions require students to communicate their explanations in writing and are spread throughout the textbook.
Open-ended Problems are mathematics problems with more than one correct answer and expose students to real-world problems.
Looking Back complements the Chapter Opener and helps students internalise the big ideas that they have learnt in the chapter.
Summary compounds the key concepts taught in the chapter in a succinct manner.
Challenge Yourself extends students’ learning at the end of the chapter. In most chapters, the first problem includes guiding questions based on PÓlya’s Problem Solving Model.
Hints for Challenge Yourself are provided at the end of the textbook to guide students where necessary.
Revision Exercise helps students revise and assess their learning after every few chapters. Review Exercise at the end of each chapter helps students consolidate their learning.
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Guided investigation provides students the relevant learning experiences to explore and discover important mathematical concepts. It usually takes the Concrete-Pictorial-Abstract (C-P-A) approach to help students construct their knowledge meaningfully. The connections between concrete experiences (manipulative or examples), different pictorial representations and symbolic representations are explicitly made. Some investigations may also involve the use of Information and Communication Technology (ICT).
Questions are provided to engage students in discussion, with the teacher acting as the facilitator. Class discussions provide students the relevant learning experiences to think and reason mathematically, enhance their oral communication skills, and learn new concepts and skills.
Key questions are included at appropriate junctures to provide students the relevant learning experiences to think critically on their own before sharing their thoughts with their classmates. Mathematical fallacies are sometimes included to check and test studentsâ&#x20AC;&#x2122; understanding.
Journal writing provides opportunities for students to reflect on their learning and to communicate mathematically in writing. It can also be used as a formative assessment for the teacher to provide feedback for their students. In some chapters, journal writing includes incomplete proofs to help students learn how to write mathematical proofs.
Students are usually required to reflect on what they have learnt at the end of each section so as to monitor and regulate their own learning. The reflection questions provided can be generic prompts or specific to the topics in the section or chapter, to check if students have understood the key ideas.
MARGINAL NOTES Big Idea
This provides additional details of the big idea mentioned in the main text.
Unlike the key feature â&#x20AC;&#x2DC;Recapâ&#x20AC;&#x2122; in the main text, this contains justin-time recall of prerequisite knowledge that students have already learnt.
This includes information that may be of interest to students.
This guides students to think about different methods used to solve a problem.
Internet Resources This guides students to search the Internet for valuable information or interesting online games for their independent and self-directed learning.
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Just For Fun This contains puzzles, fascinating facts and interesting stories about Mathematics as enrichment for students.
Attention This contains important information that students should know.
Problem-solving Tip This guides students on how to approach a problem in Worked Examples or Practise Now.
Coding Coding opportunities are provided for students who know how to code or are interested to learn.
CONTENTS CHAPTER 1
Quadratic functions of the form y = a(x – p)(x – q)
Quadratic functions of the form y = a(x – h)2 + k
Conditions for quadratic curve to lie completely above or below x-axis
Quadratic functions in real-world contexts
Review Exercise 1
Revision Exercise A1
Revision Exercise A2
Equations and Inequalities
Solving quadratic equations by completing the square
Solving quadratic equations using Quadratic Formula
Solving linear and non-linear 24 simultaneous equations
Solving quadratic inequalities 28
Polynomials, Cubic Equations and Partial Fractions
Simplifying expressions involving surds
Remainder and Factor Theorems
Solving equations involving surds
Cubic expressions, equations and identities
Review Exercise 2
Review Exercise 3
Review Exercise 4
Binomial Theorem and its Applications
Binomial expansion of (1 + b)
Binomial expansion of (a + b)n
Applications of Binomial Theorem in real-world contexts
Review Exercise 5
Revision Exercise B1
Revision Exercise B2
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Exponential and Logarithmic Functions
Exponential expressions and equations
Introduction to logarithms
Laws of Logarithms and Change of Base Formula
Logarithmic and exponential equations
Exponential and logarithmic functions and graphs
Applications of logarithmic and exponential functions
Review Exercise 6
Revision Exercise C1
Revision Exercise C2
Trigonometric Functions and Graphs
Midpoint of line segment
Why study Linear Law?
Parallel and perpendicular lines Equation of straight line
Converting non-linear equation into linear form
Trigonometric ratios of acute 200 angles and special angles
Area of rectilinear figures
Converting linear form into non-linear equation
Trigonometric ratios of general angles
Equation of circle
Applications of Linear Law
Trigonometric functions and graphs
Review Exercise 7
Review Exercise 8
Review Exercise 9
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Trigonometric Equations and Identities
10.1 Trigonometric equations
10.2 Trigonometric identities
10.3 Addition Formulae
10.4 Double Angle Formulae
10.5 Proving of identities
Review Exercise 10
Revision Exercise D1
Hints for Challenge Yourself
Revision Exercise D2