CambridgeMaths stage 6. Year 11, Mathematics extension 1

955 Pages·32.367 MB·English

by  Brian Dorofaeff, David Sadler, William Pender, Derek Ward, Julia Shea

1111
YEAR
MATHEMATICS
EXTENSION 1
CambridgeMATHS
STAGE 6
BILL PENDER
DAVID SADLER | DEREK WARD | BRIAN DOROFAEFF | JULIA SHEA
Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
Table of contents
Rationale viii
Overview xx
Acknowledgements xxiii
About the authors xxiv
1 Methods in algebra 1
1A Arithmetic with pronumerals 2
1B Expanding brackets 6
1C Factoring 10
1D Algebraic fractions 14
1E Solving linear equations 20
1F Solving quadratic equations 23
1G Solving simultaneous equations 27
1H Completing the square 30
Chapter 1 Review 33
2 Numbers and surds 35
2A Whole numbers, integers and rationals 36
2B Real numbers and approximations 42
2C Surds and their arithmetic 48
2D Further simplification of surds 51
2E Rationalising the denominator 54
Chapter 2 Review 58
3 Functions and graphs 61
3A Functions and function notation 62
3B Functions, relations and graphs 66
3C Review of linear graphs 73
3D Quadratics functions — factoring and the graph 78
3E Completing the square and the graph 85
3F The quadratic formulae and the graph 94
3G Powers, polynomials and circles 99
3H Two graphs that have asymptotes 106
3I Four types of relations 113
Chapter 3 Review 121
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
4 Transformations and symmetry 124
4A Translations of known graphs 125
4B Reflections in the x-axis and y-axis 134
4C Even and odd symmetry 141
4D The absolute value function 146
4E Composite functions 153
Chapter 4 Review 158
5 Further graphs 162 Extension 1
5A Solving inequations 163
5B The sign of a function 170
5C Reciprocals and asymptotes 177
5D Graphing sums and products 184
5E Absolute value and square roots 194
5F Inverse relations and functions 202
5G Inverse function notation 209
5H Defining functions and relations parametrically 214
Chapter 5 Review 220
6 Trigonometry 224
6A Trigonometry with right-angled triangles 225
6B Problems involving right-angled triangles 232
6C Three-dimensional trigonometry 238
6D Trigonometric functions of a general angle 245
6E Quadrant, sign, and related acute angle 250
6F Given one trigonometric function, find another 258
6G Trigonometric identities 262
6H Trigonometric equations 268
6I The sine rule and the area formula 274
6J The cosine rule 283
6K Problems involving general triangles 290
Chapter 6 Review 296
7 The coordinate plane 302
7A Lengths and midpoints of intervals 303
7B Gradients of intervals and lines 308
7C Equations of lines 317
7D Further equations of lines 322
7E Using pronumerals in place of numbers 328
Chapter 7 Review 332
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
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8 Exponential and logarithmic functions 334
8A Indices 335
8B Fractional indices 343
8C Logarithms 349
8D The laws for logarithms 354
8E Equations involving logarithms and indices 359
8F Exponential and logarithmic graphs 365
8G Applications of these functions 371
Chapter 8 Review 376
9 Differentiation 378
9A Tangents and the derivative 379
9B The derivative as a limit 385
9C A rule for differentiating powers of x 391
dy
9D The notation for the derivative 399
dx
9E The chain rule 406
9F Differentiating powers with negative indices 414
9G Differentiating powers with fractional indices 419
9H The product rule 424
9I The quotient rule 428
9J Rates of change 432
9K Continuity 440
9L Differentiability 445
Chapter 9 Review 449
10 Polynomials 452 Extension 1
10A The language of polynomials 453
10B Graphs of polynomial functions 459
10C Division of polynomials 464
10D The remainder and factor theorems 468
10E Consequences of the factor theorem 473
10F Sums and products of zeroes 480
10G Multiple zeroes 490
10H Geometry using polynomial techniques 495
Chapter 10 Review 500
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
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11 Extending calculus 503
11A The exponential function base e 504
11B Transformations of exponential functions 510
11C Differentiation of exponential functions 514
11D Differentiation and the graph 517
11E The logarithmic function base e 522
11F Exponential rates using the base e 528
11G Radian measure of angle size 531
11H Solving trigonometric equations 538
11I Arcs and sectors of circles 543
11J Trigonometric graphs in radians 550
Chapter 11 Review 555
11112222 Probability 558
12A Probability and sample spaces 559
12B Sample space graphs and tree diagrams 569
12C Sets and venn diagrams 575
12D Venn diagrams and the addition theorem 583
12E Multi-stage experiments and the product rule 590
12F Probability tree diagrams 598
12G Conditional probability 604
Chapter 12 Review 615
13 Discrete probability distributions 617
13A The language of probability distributions 618
13B Expected value 628
13C Variance and standard deviation 637
13D Sampling 646
Chapter 13 Review 661
14 Combinatorics 664 Extension 1
14A Factorial notation 665
14B Counting ordered selections 670
14C Ordered selections and grouping 678
14D Ordered selections with identical elements 682
14E Counting unordered selections 687
14F Using counting in probability 696
14G Arrangements in a circle 703
14H The pigeonhole principle 708
Chapter 14 Review 714
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
15 The binomial expansion and Pascal’s triangle 716 Extension 1
15A Binomial expansions and Pascal’s triangle 717
15B Further binomial expansions 724
15C The binomial theorem 728
15D Identities in Pascal’s triangle 734
15E Enrichment — using the general term 740
Chapter 15 Review 743
16 Further rates 745 Extension 1
16A Related rates 746
16B Exponential growth and decay 752
16C Modified natural growth and decay 760
Chapter 16 Review 767
17 Further trigonometry 769 Extension 1
17A Restricting the domain 770
17B Defining the inverse trigonometric functions 777
17C Graphs involving inverse trigonometric functions 785
17D Trigonometric functions of compound angles 790
17E The double-angle formulae 796
17F The t-formulae 800
17G Products to sums 805
Chapter 17 Review 808
Answers 812
Online appendices
See the interactive textbook for guides to spreadsheets, the Desmos graphing calculator, and links to scientific
calculator guides.
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
Rationale
CambridgeMATHS Extension 1 Year 11 covers all syllabus dotpoints for Year 11 of the Mathematics
Advanced and Extension 1 courses being implemented in 2019. This rationale serves as a guide to how
the book covers the dotpoints of the syllabus. Further documents are available in the teacher resources.
The Exercises
No-one should try to do all the questions! We have written long exercises with a great variety of questions
so that everyone will find enough questions of a suitable standard — they cater for differentiated teaching
to a wide range of students. The division of all exercises into Foundation, Development and Enrichment
sections helps with this. Each student will need to tackle a selection of questions, and there should be plenty
left for revision.
Compared to our previous Extension 1 textbooks, the Foundation section in each exercise provides a
gentler start. There are more questions, and they are straightforward. Students need encouragement to
assimilate comfortably the new ideas and methods presented in the text so that they are prepared and
confident before tackling later problems.
The Development sections are graded from reasonably straightforward questions to harder problems. They
are intended to bring students up to Extension 1 standard. The examinations have traditionally included
questions that students find difficult, and we assume that this will continue with the HSC examinations
on the new syllabuses — there are questions at the end of the Development sections intended to prepare
students for these demands.
The Enrichment sections contain ‘interesting’ questions that are probably beyond the Extension 1
examination standard (although one can never be sure of this) and often beyond the Extension 1 syllabus.
One of their purposes is to highlight issues significant in the Extension 2 course. Students aiming to do
Extension 2 in Year 12 should ‘have a go’ at some of these questions, but should keep in mind that some of
this material is demanding even at Extension 2 level.
The Extension 1 material
This book presents the Year 11 Advanced course and Year 11 Extension 1 course within the one volume.
Despite the tight integration of the two courses, there are various practical reasons why Advanced
material and Extension 1 material need to be easily identified and separated. In this volume:
The eleven chapters from the Advanced textbook, and their sections, have been retained. Some of the
text and exercises in those chapters has been made slightly more demanding so that it is suitable for the
higher standards customary in Extension 1.
The Extension 1 material has been placed into six separate chapters distributed appropriately through the
volume.
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
Chapter 1: Methods in algebra
1A Arithmetic with pronumerals
1B Expanding brackets
1C Factoring
1D Algebraic fractions
1E Solving linear equations
1F Solving quadratic equations
1G Solving simultaneous equations
1H Completing the square
Chapter Review Exercise
Chapter 1 revises algebraic techniques. The calculus courses rely heavily on fluency in algebra, and the
chapter is intended for students who need the revision. If students are confident with this material already
from Years 9–10, they should not spend time on it, but simply know that it is there if they find later that
some techniques need revision.
MA-F1.1 dotpoints 2–3
F1.1 dotpoint 1 index laws & Sections 7A–7B: The index laws from Years 9–10 are not reviewed here.
They need a clearer and fuller presentation in Chapter 7 in the context of exponential and logarithmic
functions and their graphs.
F1.1 dotpoint 1 surds & Sections 2C–7D: Surds are dealt with thoroughly in Chapter 2.
F1.1 dotpoint 2 & Sections 3D–3F: When quadratics graphs are developed in Chapter 3, the quadratic
techniques of this dotpoint are covered again, but at a higher level.
Chapter 2: Numbers and surds
2A Whole numbers, integers and rationals
2B Real numbers and approximations
2C Surds and their arithmetic
2D Further simplification of surds
2E Rationalising the denominator
Chapter Review Exercise
Chapter 2 reviews the four standard sets of numbers: , , and — students should be clear about
these before tackling functions. In particular, they must know how to manipulate surds. They must also
understand the geometric basis of the real numbers as the points on the number line, and be able to
approximate them in various ways. The syllabus mostly takes all this as assumed knowledge from Years
9–10, but it is mentioned specifically in.
F1.1, dotpoint 1 (surds), F1.2, dotpoint 2 (intervals)
This chapter is mostly intended as a quick review, and time should not be spent on it unnecessarily.
Types of intervals and their line graphs are introduced in Section 2B. ‘Interval notation’ in the form
3 < x < 6 is used throughout the Year 11 book. The alternative ‘interval notation’ (3, 6) is left until
Year 12 to avoid confusion at this early stage between between an interval on a number line and a point
in the coordinate plane.
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Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party
Chapter 3: Functions and graphs
3A Functions and function notation
3B Functions, relations and graphs
3C Review of linear graphs
3D Quadratics functions — factoring and the graph
3E Completing the square and the graph
3F The quadratic formulae and the graph
3G Powers, polynomials and circles
3H Two graphs that have asymptotes
3I Four types of relations
Chapter Review Exercise
Section 3A, 3B and 3I introduce the more sophisticated notions of functions, relations and graphs
required in the course and mentioned in
MA-F1.2, apart from odd and even functions and composite functions
Functions are introduced first in Section 3A using a function machine, f(x) notation, and tables of values.

This leads in Section 3B to the graph, and to the use of the vertical line test to distinguish between a
function and the more general concept of a relation.
The final Section 3I returns to the general theory and covers several related matters — reading a graph
backwards using horizontal lines, graphical solutions of f(x) = k (F1.3 dotpoint 3, sub-dotpoint), the

horizontal line test, the classification of relations as one-to-one, many-to-one, one-to-many and many-to-
many (F1.2 dotpoint 4), and mappings between sets (F1.2, dotpoint 1). This prepares for the Extension 1
topic of inverse functions in Chapter 5.
MA-F1.2 dotpoint 7 & section 4E Composite functions: Composite functions are covered in Chapter 4,
after translations, reflections and absolute values have already provided specific examples.
MA-F1.2 dotpoint 6 & section 5D Graphing sums and products: The graphs of sums, differences and
products are treated in some detail in section 5D.
MA-F1.2 dotpoints 6–7 & Chapter 9 Differentiation: The functions of algebra almost all involve sums,
differences, quotients, products and composites of functions. Apart from composites, it would not
be helpful at this stage to formalise rules for domains and ranges because the rules quickly become
very complicated, and things are best done case-by-case. Each of these five ways to build functions is
considered separately as the rules of differentiation are developed in Chapter 9.
Sections 3C–3H present, in the unified manner introduced in Sections 3A–3B, a variety of graphs that
are partly known from Years 9–10, particularly quadratics, and also higher powers of x, polynomials,
circles, and graphs with asymptotes. Section 3G sketches cubics and general polynomials factored into
linear factors, using a table of values to test the sign, but restricted to square linear factors because the
situation at multiple will be addressed later in Chapter 10 using calculus.
MA-F1.3 all dotpoints, and MA-F1.4 dotpoints 1–4
MA-F1.3 dotpoint 1 & Chapter 7 The coordinate plane: Lines are only briefly covered in Section 3C,
and covered in far more detail in Chapter 7. Gradient is the key idea with linear graphs, and gradient in
turn requires tanθ and the ability to solve, for example, tan θ = −1, where 0° ≤ θ < 180° (see also the

mention at MA-C1.1 dotpoint 3).
x
Mathematics E x t e n s io n 1 Y e a r 1 1 ISBN 978-1-108-46907-4 © Bill Pender et al. 2019 Cambridge University Press
Cambridge M a t h s S t a g e 6 Photocopying is restricted under law and this material must not be transferred to another party