New Syllabus Additional Mathematics

556 Pages·2017·72.33 MB·English

ADDITIONAL
New Syllabus
MATHEMATICS
9th
Edition
Consultant • Dr Yeap Ban Har
Authors • Joseph Yeo
Teh Keng Seng
Loh Cheng Yee
Ivy Chow

ADDITIONAL
New Syllabus
MATHEMATICS
9th
Edition
C
M
Y
CM
MY
CY
CMY
K
Consultant • Dr Yeap Ban Har
Authors • Joseph Yeo MEd, PGDE (Distinction), BSc (Hons)
Teh Keng Seng BSc, Dip Ed
Loh Cheng Yee BSc, Dip Ed
Ivy Chow MEd, PGDE, BSc
SHINGLEE PUBLISHERS PTE LTD
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Tel: [phone] Fax: [phone]
email: [email protected]
http://www.shinglee.com.sg
©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
ISBN 978 981 237 499 8
Acknowledgements
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
Printed in Singapore
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3
PREFACE
New Syllabus
Additional
Mathematics
NSAM
New Syllabus Additional Mathematics (NSAM)
is an MOE-approved textbook specially designed to provide
valuable learning experiences to engage the hearts and
minds of students sitting for the GCE O-level examination
in Additional Mathematics. Included in the textbook are
Investigation, Class Discussion, Thinking Time and Alternative
Assessment such as Journal Writing to support the teaching
and learning of Mathematics.
Every chapter begins with a chapter opener which motivates
students in learning the topic. Interesting stories about
mathematicians, real-life examples and applications are
used to arouse students’ interest and curiosity so that
they can appreciate the beauty of Mathematics in their
surroundings and in the sciences.
The use of ICT helps students to visualise and manipulate
mathematical objects more easily, thus making the learning
of Mathematics more interactive. Ready-to-use interactive
ICT templates are available at http://www.shinglee.com.sg/
StudentResources/
The chapters in the textbook have been organised into
three strands — Algebra, Geometry and Trigonometry
and Calculus. The colours purple, green and red at the
bottom of each page indicate these. Chapters and sections
which have been excluded from the Normal (Academic) syllabus
are clearly indicated with a .
iii
KKKEEEYYY FFFEEEAAATTTUUURRREEESSS
CHAPTER OPENER
Each chapter begins with a chapter opener to arouse students’ interest and curiosity in learning the topic.
LEARNING OBJECTIVES
Learning objectives help students to be more aware of what they are about to study so that they can monitor
their own progress.
RECAP
Relevant prerequisites will be revisited at the beginning of the chapter or at appropriate junctures so that students can
build upon their prior knowledge, thus creating meaningful links to their existing schema.
WORKED EXAMPLE
This shows students how to apply what they have learnt to solve related problems and how to present their working
clearly. A suitable heading is included in brackets to distinguish between the different Worked Examples.
PRACTISE NOW
At the end of each Worked Example, a similar question will be provided for immediate practice. Where appropriate,
this includes further questions of progressive difficulty.
SIMILAR QUESTIONS
A list of similar questions in the Exercise is given here to help teachers choose questions that their students can do
on their own.
EXERCISE
The questions are classified into three levels of difficulty – Basic, Intermediate and Advanced.
SUMMARY
At the end of each chapter, a succinct summary of the key concepts is provided to help students consolidate what
they have learnt.
CHALLENGE YOURSELF
Optional problems are included at the end of each chapter to challenge and stretch high-ability students to their
fullest potential.
REVISION EXERCISE
This is included after every few chapters to help students assess their learning.
iv
Learning experiences have been infused into Investigation, Thinking Time, Class Discussion,
and Journal Writing.
Activities are included Key questions are also
to guide students to included at appropriate
investigate and discover junctures to check if
important mathematical students have grasped
concepts so that they various concepts and
can construct their own to create opportunities
knowledge meaningfully. for them to further
develop their thinking.
Questions are provided
Class
for students to discuss
Discussion
Opportunities are provided
in class, with the teacher
for students to reflect
acting as the facilitator.
on their learning and to
The questions will
communicate mathematically.
assist students to
It can also be used as a
learn new knowledge,
formative assessment to
think mathematically,
provide feedback to students
and enhance their
to improve on their learning.
reasoning and oral
communication skills.
MMMAAARRRGGGIIINNNAAALLL NNNOOOTTTEEESSS
This contains puzzles,
This guides students on how
This guides students to fascinating facts and
to approach a problem.
search on the Internet for interesting stories
valuable information or about Mathematics as
interesting online games enrichment for students.
for their independent and
self-directed learning.
This contains important This includes information This contains certain
information that students that may be of interest mathematical concepts
should know. to students. or rules that students
have learnt previously.
v
New Syllabus
Additional
Mathematics
NSAM
CHAPTER 1
Simultaneous Equations, Polynomials 002
CHAPTER 3
and Partial Fractions
1.1 Linear and Non-linear 004 082 Binomial Theorem
Simultaneous Equations 084 3.1 Binomial Expansion of (1 + b)n
1.2 Polynomials 007 086 3.2 Binomial Coefficients
1.3 Remainder Theorem 015 092 3.3 Binomial Theorem
1.4 Factor Theorem 018 097 3.4 Applications of Binomial Theorem
1.5 Cubic Expressions and Equations 021 101 Summary
1.6 Partial Fractions 030 101 Review Exercise 3
Summary 041
Review Exercise 1 042
CHAPTER 4
104 Indices, Surds and Logarithms
CHAPTER 2 106 4.1 Indices
110 4.2 Surds
Quadratic Equations, Inequalities and 044
118 4.3 Introduction to Logarithms
Modulus Functions
124 4.4 Laws of Logarithms and Change of
2.1 Sum and Product of Roots 046
Base Formula
2.2 Nature of Roots of a Quadratic Equation 051
132 4.5 Logarithmic and Exponential Equations
2.3 Maximum and Minimum Values 057
137 4.6 Graphs of Exponential and
of General Quadratic Functions
Logarithmic Functions
2.4 Quadratic Inequalities 063
143 4.7 Applications of Logarithms
2.5 Intersection of a Line and a Curve 068
and Exponents
2.6 Modulus Functions 071
151 Summary
2.7 Graphs of Modulus Functions 073
152 Review Exercise 4
Summary 076

Review Exercise 2 077
154 Revision Exercise B

Revision Exercise A 080
excluded from
the N(A) syllabus
vi
CHAPTER 5
Coordinate Geometry 156
5.1 Midpoint of a Line Segment 158
CHAPTER 7
5.2 Parallel and Perpendicular Lines 161
5.3 More Problems on Equations of 166 204 Linear Law
Straight Lines 206 7.1 Why study Linear Law?
5.4 Area of Rectilinear Figures 172 209 7.2 Converting from a Non-linear Equation
Summary 178 to a Linear Form
Review Exercise 5 179 211 7.3 Converting from a Linear Form to
a Non-linear Equation
214 7.4 Applications of Linear Law
CHAPTER 6
221 Summary
Further Coordinate Geometry 182 221 Review Exercise 7
6.1 Equation of a Circle 184
6.2 Graphs of y2 = kx 194 224 Revision Exercise C
6.3 Graphs of Power Functions 196
Summary 200
CHAPTER 8
Review Exercise 6 202
226 Trigonometric Functions and Equations
228 8.1 Trigonometric Ratios of Special Angles
229 8.2 General Angles
232 8.3 Trigonometric Ratios of General Angles
241 8.4 Graphs of Trigonometric Functions
246 8.5 Further Trigonometric Graphs
257 8.6 Graphs of y = |f(x)|, where f(x)
is trigonometric
260 8.7 Cosecant, Secant and
Cotangent Ratios
262 8.8 Trigonometric Equations
267 Summary
268 Review Exercise 8
excluded from
the N(A) syllabus
vii
CHAPTER 11
336 Differentiation and its Applications
338 11.1 Gradient Functions
341 11.2 Five Rules of Differentiation
353 11.3 Equations of Tangent and Normal to
a Curve
CHAPTER 9 357 11.4 Rates of Change
362 Summary
Trigonometric Identities 270
363 Review Exercise 11
and Formulae
9.1 Trigonometric Identities 272
9.2 Proving of Identities 275 CHAPTER 12
9.3 Addition Formulae 279
366 Further Applications of Differentiation
9.4 Double Angle Formulae 286
368 12.1 Higher Derivatives
9.5 Further Proving of Identities 292
371 12.2 Increasing and Decreasing Functions
9.6 R-Formulae 295
375 12.3 Stationary Points
Summary 301
387 12.4 Problems on Maximum and
Review Exercise 9 301
Minimum Values
394 Summary
395 Review Exercise 12
CHAPTER 10
Proofs in Plane Geometry 306
10.1 Basic Proofs in Plane Geometry 308
10.2 Proofs using Congruence and 314
Similarity Tests
10.3 Midpoint Theorem 320
10.4 Tangent-Chord Theorem 324
(Alternate Segment Theorem)
Summary 331
Review Exercise 10 331
Revision Exercise D 334
excluded from
the N(A) syllabus
viii