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Advancing Maths for AQA
CORE MATHS 1
Sam Boardman, Tony Clough and David Evans
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Series editors
Sam Boardman Roger Williamson Ted Graham David Pearson
Core Maths 1
1 Advancing from GCSE maths: algebra review
2 Surds
3 Coordinate geometry of straight lines
4 Quadratics and their graphs
5 Polynomials
6 Factors, remainders and cubic graphs
7 Simultaneous equations and quadratic inequalities
8 Coordinate geometry of circles
9 Introduction to differentiation: gradient of curves
10 Applications of differentiation: tangents, normals and
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13
28
52
70
80
99
114
137
153
rates of change
11
Maximum and minimum points and optimisation
problems
166
12
Integration
185
Exam style practice paper
204
Answers
388
Index
437
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Advancing Maths for AQA
CORE MATHS 2
Sam Boardman,Tony Clough, David Evans and Maureen Nield
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Series editors
Sam Boardman Roger Williamson Ted Graham David Pearson
Core Maths 2
1 Indices
2 Further differentiation
3 Further integration and the trapezium rule
4 Basic trigonometry
5 Simple transformation of graphs
6 Solving trigonometrical equations
7 Factorials and binomial expansions
8 Sequences and series
9 Radian measure
10 Further trigonometry with radians
11 Exponentials and logarithms
12 Geometric series
207
221
239
253
279
288
303
317
337
349
359
372
Exam style practice paper
385
Answers
413
Index
437
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Heinemann Educational Publishers
Halley Court, Jordan Hill, Oxford OX2 8EJ
Part of Harcourt Education
Heinemann is the registered trademark of
Harcourt Education Limited
© Sam Boardman, Tony Clough, David Evans and Maureen Nield 2000, 2004
Complete work © Harcourt Education Limited 2004
First published 2004
08 07 06 05 04
10 9 8 7 6 5 4 3 2 1
British Library Cataloguing in Publication Data is available
from the British Library on request.
ISBN 0 435 51330 3
Copyright notice
All rights reserved. No part of this publication may be reproduced in any form
or by any means (including photocopying or storing it in any medium by
electronic means and whether or not transiently or incidentally to some other
use of this publication) without the written permission of the copyright owner,
except in accordance with the provisions of the Copyright, Designs and Patents
Act 1988 or under the terms of a licence issued by the Copyright Licensing
Agency, 90 Tottenham Court Road, London W1T 4LP. Applications for the
copyright owner’s written permission should be addressed to the publisher.
Edited by Alex Sharpe, Standard Eight Limited
Typeset and illustrated by Tech-Set Limited, Gateshead, Tyne & Wear.
Original illustrations © Harcourt Education Limited, 2004
Cover design by Miller, Craig and Cocking Ltd
Printed in Great Britain by The Bath Press, Bath
Acknowledgements
The publishers’ and authors’ thanks are due to the AQA for permission to
reproduce questions from past examination papers.
The answers have been provided by the authors and are not the responsibility
of the examining board.
Every effort has been made to contact copyright holders of material reproduced
in this book. Any omissions will be rectified in subsequent printings if notice is
given to the publishers.
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About this book
This book is one in a series of textbooks designed to provide you
with exceptional preparation for AQA’s 2004 Mathematics
Specification. The series authors are all senior members of the
examining team and have prepared the textbooks specifically to
support you in studying this course.
Finding your way around
The following are there to help you find your way around when
you are studying and revising:
● edge marks (shown on the front page) – these help you to
get to the right chapter quickly;
● contents list – this identifies the individual sections dealing
with key syllabus concepts so that you can go straight to the
areas that you are looking for;
● index – a number in bold type indicates where to find the
main entry for that topic.
Key points
Key points are not only summarised at the end of each chapter
but are also boxed and highlighted within the text like this:
An equation of the form ax b 0, where a and b are
constants, is said to be a linear equation with variable x.
Exercises and exam questions
Worked examples and carefully graded questions familiarise you
with the syllabus and bring you up to exam standard. Each book
contains:
● Worked examples and Worked exam questions to show you
how to tackle typical questions; Examiner’s tips will also
provide guidance;
● Graded exercises, gradually increasing in difficulty up to
exam-level questions, which are marked by an [A];
● Test-yourself sections for each chapter so that you can check
your understanding of the key aspects of that chapter and
identify any sections that you should review;
● Answers to the questions are included at the end of the book.
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Contents C1
1 Advancing from GCSE maths: algebra
review
Learning objectives
1.1 Advancing from GCSE
1.2 Solving linear equations
An historical problem
1.3 Solving simultaneous linear equations
1.4 Linear inequalities
1.5 Function notation
1.6 Use of calculators
Key point summary
Test yourself
1
1
2
4
5
7
9
10
11
12
2 Surds
Learning objectives
2.1 Special sets of numbers
2.2 Surds
Order of a surd
2.3 Simplest form of surds
2.4 Manipulating square roots
2.5 Use in geometry
2.6 Like and unlike surds
2.7 Adding and subtracting surds
2.8 Multiplying surds
2.9 Rationalising the denominator
2.10 Equations and inequalities involving surds
Key point summary
Test yourself
13
13
14
14
15
16
17
17
18
19
21
23
26
26
3 Coordinate geometry of straight lines
Learning objectives
3.1 Cartesian coordinates
3.2 The distance between two points
3.3 The coordinates of the mid-point of a line
segment joining two known points
3.4 The gradient of a straight line joining two
known points
3.5 The gradients of perpendicular lines
3.6 The y mx c form of the equation of a
straight line
28
28
30
33
35
37
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Contents
3.7 The y y1 m(x x1) form of the equation of
a straight line
3.8 The equation of a straight line passing
through two given points
3.9 The coordinates of the point of intersection of
two lines
Key point summary
Test yourself
40
42
43
50
51
4 Quadratic functions and their graphs
Learning objectives
4.1 Parabolas and quadratic functions
4.2 Factorising quadratics and sketching graphs
4.3 Completing the square
4.4 Translations parallel to the x-axis
4.5 General translations of quadratic graphs
4.6 General quadratic equation formula
4.7 The discriminant
Key point summary
Test yourself
52
52
53
55
61
62
63
64
68
69
5 Polynomials
Learning objectives
5.1 Introduction and definitions
5.2 Adding and subtracting polynomials
Notation
5.3 Multiplying polynomials
5.4 Finding coefficients
Key point summary
Test yourself
70
70
71
72
74
75
78
79
6 Factors, remainders and cubic graphs
Learning objectives
6.1 Factorisation
6.2 The factor theorem
6.3 Further factorisation
6.4 Graphs of cubic functions
6.5 Dividing a polynomial by a linear expression
6.6 The remainder theorem
Key point summary
Test yourself
80
80
80
84
88
91
94
97
98
7 Simultaneous equations and quadratic
inequalities
Learning objectives
7.1 Simultaneous equations
7.2 The intersection of a line and a curve
99
99
101
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Contents
7.3 The condition for a line to be a tangent to the
graph of a quadratic function
7.4 Quadratic inequalities
7.5 Quadratics with irrational roots
7.6 Completing the square
7.7 The discriminant revisited
Key point summary
Test yourself
103
105
107
108
109
112
113
8 Coordinate geometry of circles
Learning objectives
8.1 The Cartesian equation of a circle
8.2 Sketching and applying translations on circles
8.3 Finding the equation of a circle using circle
properties
8.4 Conditions for a line to meet a circle
8.5 The length of the tangents from a point to a
circle
8.6 The equation of the tangent and the equation
of the normal at a point on a circle
Key point summary
Test yourself
114
114
118
121
125
128
129
134
136
9 Introduction to differentiation: gradient
of curves
Learning objectives
9.1 Introduction
9.2 Gradients of chords
9.3 The tangent to a curve at a point
9.4 The gradient of a curve as the limit of the
gradient of the chord
9.5 The derivative or derived function
9.6 Differentiation notation
9.7 General rules for differentiation
9.8 Finding gradients at specific points on a
curve
9.9 Finding points on a curve with a given
gradient
9.10 Simplifying expressions before
differentiating
Key point summary
Test yourself
137
137
138
139
141
143
144
145
147
149
150
151
152
10 Applications of differentiation: tangents,
normals and rates of change
Learning objectives
10.1 The equation of a tangent to a curve
10.2 The equation of a normal to a curve
153
153
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10.3 Rates of change
10.4 Increasing and decreasing functions
Key point summary
Test yourself
157
161
164
165
11 Maximum and minimum points and
optimisation problems
Learning objectives
11.1 The least value of a quadratic function
11.2 Maximum and minimum points
Maximum points
Minimum points
11.3 Second derivatives
11.4 The second derivative test
11.5 Stationary points of inflection
11.6 Optimisation
Key point summary
Test yourself
166
166
168
168
168
171
171
176
177
182
184
12 Integration
Learning objectives
12.1 Finding a function from its derivative
12.2 Use of an arbitrary constant
12.3 Finding the equation of a curve
12.4 Simplifying expressions before integrating
12.5 Indefinite integrals
12.6 Integration as an area
12.7 Definite integrals
12.8 Regions bounded by lines and curves
12.9 Regions below the x-axis
Key point summary
Test yourself
185
185
186
187
188
189
191
192
194
195
202
203
Exam style practice paper
204
Answers
388
Index
437
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Contents C2
1 Indices
Learning objectives
1.1 Introduction
1.2 Index notation
1.3 The laws of indices
1.4 Negative indices
1.5 The zero index
1.6 Fractional indices
1.7 Solving equations with indices
Key point summary
Test yourself
207
207
208
208
211
212
214
217
220
220
2 Further differentiation
Learning objectives
2.1 Review of differentiation techniques
2.2 Negative and fractional powers
2.3 Simplifying expressions before
differentiating
2.4 Finding gradients of curves and equations
of tangents and normals
2.5 Stationary points
2.6 Optimisation
Key point summary
Test yourself
221
221
221
225
227
229
233
237
237
3 Further integration and the trapezium rule
Learning objectives
3.1 Basic rule for the integration of xn
3.2 Simplifying expressions before integrating
3.3 Indefinite integrals
3.4 Definite integrals
3.5 Interpretation of an area
3.6 Estimation of area using the trapezium rule
Key point summary
Test yourself
239
239
241
243
244
245
247
251
252
4 Basic trigonometry
Learning objectives
4.1 Introduction
4.2 The sine function – an informal approach
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253
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Contents
4.3
4.4
The cosine function – an informal approach
Definitions of the sine and cosine functions
Sine
Cosine
4.5 The graph of the sine function
Key features of the graph of y sin 4.6 The graph of the cosine function
Key features of the graph of y cos 4.7 Finding angles with the same sine or
cosine
4.8 The tangent function
4.9 The graph of the tangent function
Key features of the graph of y tan 4.10 Solving trigonometrical equations
4.11 Sine rule
Ambiguous case
4.12 Cosine rule
4.13 The area of a triangle
Key point summary
Test yourself
255
256
256
256
257
257
257
258
259
261
262
262
263
267
270
271
272
275
278
5 Simple transformations of graphs
Learning objectives
5.1 Translations parallel to the y-axis
5.2 Translations parallel to the x-axis
5.3 General translations of graphs
5.4 Reflections
5.5 Stretches in the y-direction
5.6 Stretches in the x-direction
5.7 Scale factors
Key point summary
Test yourself
279
279
280
281
282
283
284
284
286
287
6 Solving trigonometrical equations
Learning objectives
6.1 Solving equations of the form sin bx k,
cos bx k and tan bx c
6.2 Solving equations of the form sin(x b) k,
cos(x b) k and tan(x b) c
6.3 Solving quadratic equations in sin x,
cos x and tan x
6.4 The relationship cos2 sin2 1
6.5 The relationship between sin , cos and
tan Key point summary
Test yourself
288
288
291
292
295
295
300
301
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Contents
7 Factorials and binomial expansions
Learning objectives
7.1 Factorial notation
7.2 Inductive definition of factorial and 0!
n
7.3 The notation
r
7.4 Pascal’s triangle
7.5 Binomial coefficients
Key point summary
Test yourself
303
303
304
305
306
310
315
316
8 Sequences and series
Learning objectives
8.1 Sequences
8.2 Suffix notation
8.3 Inductive definition
8.4 Limit of a sequence
8.5 Arithmetic sequences
8.6 Arithmetic series
8.7 Sum of the first n natural numbers
8.8 Sum of the first n terms of an arithmetic
series
Alternative formula for Sn
8.9 Sigma notation
8.10 Use of the sigma notation for arithmetic
series
8.11 Review of arithmetic series
Key point summary
Test yourself
317
317
318
318
319
322
323
324
326
327
329
329
332
334
335
9 Radian measure
Learning objectives
9.1 Radians as a unit of measure of angles
9.2 Changing between degrees and radians
9.3 Arc length of a circle
9.4 Area of a sector of a circle
Key point summary
Test yourself
337
337
337
340
342
348
348
10 Further trigonometry with radians
Learning objectives
10.1 Introduction
10.2 Solving simple trigonometrical equations
in terms of radians
10.3 Using trigonometrical identities to solve
equations where answers are given in
radians
Key point summary
Test yourself
349
349
350
354
358
358
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Contents
11 Exponentials and logarithms
Learning objectives
11.1 Exponential functions
11.2 Introduction to logarithms and the
notation used
11.3 Laws of logarithms
11.4 Solving equations of the form ax b
Key point summary
Test yourself
359
359
361
363
365
370
370
12 Geometric series
Learning objectives
12.1 Would you like to be a millionaire?
12.2 Geometric series
12.3 Sum of the first n terms of a geometric
series
12.4 Sum to infinity of a geometric series
12.5 Use of logarithms to find number of terms
Key point summary
Test yourself
372
372
373
375
376
379
384
384
Exam style practice paper
385
Answers
413
Index
417
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