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OCR

GCSE

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Gareth Cole, Heather Davis, Sophie Goldie, Linda Liggett, Robin Liggett, Andrew Manning, Richard Perring, Keith Pledger, Rob Summerson

Build your students’ knowledge and understanding so that they can confidently reason, interpret, communicate mathematically and apply their mathematical skills to solve problems within mathematics and wider contexts; with resources developed specifically for the OCR GCSE 2015 specification by mathematics subject specialists experienced in teaching and examining GCSE. We are working with OCR to get these resources endorsed: Mastering Mathematics for OCR GCSE Foundation 1 Mastering Mathematics for OCR GCSE Foundation 2/ Higher 1 Mastering Mathematics for OCR GCSE Higher 2

9781471840012 9781471840029 9781471840036

June 2015 May 2015 July 2015

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Contents NUMBER Strand 1 Calculating Units 1–9 Moving on

Strand 2 Using our number system Units 1–7 Moving on Unit 8 Recurring decimals

Strand 3 Accuracy Units 1–7 Moving on Unit 8 Upper and lower bounds

Strand 4 Fractions Units 1–6 Moving on

Strand 5 Percentages Units 1–7 Moving on Unit 8 Growth and decay

Strand 6 Ratio and proportion Units 1–5 Moving on Unit 6 Formulating equations to solve proportion problems

Strand 7 Number properties Units 1–6 Moving on Unit 7 Fractional indices Unit 8 Surds

ALGEBRA Strand 1 Starting algebra Units 1–11 Moving on Unit 12 Using roots and reciprocals Unit 13 Manipulating more expressions and equations Unit 14 Rearranging more formulae

iii

Contents

Strand 2 Sequences Units 1–6 Moving on Unit 7 Other sequences Unit 8 nth term of quadratic sequences

Strand 3 Functions and graphs Units 1–7 Moving on Unit 8 Perpendicular lines Unit 9 Inverse and composite functions Unit 10 Exponential functions Unit 11 The equation of a circle Unit 12 Trigonometry functions

Strand 4 Algebraic methods Units 1–5 Moving on Unit 6 Solving linear inequalities in two variables Unit 7 Iteration Unit 8 Proof

Strand 5 Properties of non-linear graphs Unit 1 Moving on Unit 2 Using chords and tangents Unit 3 Translations and reflections of functions Unit 4 Area under non-linear graphs

Strand 6 Working with quadratics Units 1–2 Moving on Unit 3 Factorising harder quadratics Unit 4 The quadratic formula Unit 5 Completing the square Unit 6 Simultaneous equations with quadratics Unit 7 Solving quadratic inequalities

GEOMETRY AND MEASURES Strand 1 Units and scales Units 1–11 Moving on

Strand 2 Properties of shapes Units 1–10 Moving on Unit 11 Circle theorems

iv

Contents

Strand 3 Measuring shapes Units 1–6 Moving on Unit 7 The cosine rule Unit 8 The sine rule

Strand 4 Construction Units 1–5 Moving on

Strand 5 Transformations Units 1–10 Moving on Unit 11 Enlargement with negative scale factors Unit 12 Invariance Unit 13 Trigonometry in 2D and 3D

Strand 6 Three-dimensional shapes Units 1–7 Moving on Unit 8 Area and volume in similar shapes

Strand 7 Vectors Units 1–2 Moving on Unit 3 Proofs with vectors

STATISTICS AND PROBABILITY Strand 1 Statistical measures Units 1–4 Moving on Unit 5 Interquartile range

Strand 2 Statistical diagrams Units 1–7 Moving on Unit 8 Histograms

Strand 3 Collecting data Units 1–3 Moving on

Strand 4 Probability Units 1–6 Moving on Unit 7 Conditional probability

v

Photo credits: p. 18 © ayelet_keshet – Fotolia; p. 24 © Sandy Officer Although every effort has been made to ensure that website addresses are correct at time of going to press, Hodder Education cannot be held responsible for the content of any website mentioned in this book. It is sometimes possible to find a relocated web page by typing in the address of the home page for a website in the URL window of your browser. Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB. Telephone: +44 (0)1235 827720. Fax: +44 (0)1235 400454. Lines are open 9.00a.m.–5.00p.m., Monday to Saturday, with a 24-hour message answering service. Visit our website at www.hoddereducation.co.uk © Gareth Cole, Heather Davis, Sophie Goldie, Linda Liggett, Robin Liggett, Andrew Manning, Richard Perring, Keith Pledger, Rob Summerson 2015 First published in 2015 by Hodder Education, An Hachette UK Company 338 Euston Road London NW1 3BH Impression number Year

5

4

3

2

1

2019 2018 2017 2016 2015

All rights reserved. Apart from any use permitted under UK copyright law, no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Cover photo © lightpixel – Fotolia Illustrations by Integra Typeset in ITC Avant Garde Gothic Std Book 10/12 by Integra Software Services Pvt. Ltd., Pondicherry, India A catalogue record for this title is available from the British Library ISBN 9781471840036

vi

Strand 6 Working with quadratics

Unit 1

Band h

Unit 2

Band h

Factorising harder quadratics

Solve equations by factorising

Foundation 2 Higher 1

Foundation 2 Higher 1

Unit 4

Band i

Unit 3

Band i

The quadratic formula

Factorising harder quadratics

Page 12

Page 5

Unit 5

Band j

Unit 6

Band j

Completing the square

Simultaneous equations with quadratics

Page 18

Page 27

Unit 7

Band j

Solving quadratic inequalities Page 33

1

Unit 5 Completing the square

6

Outside the maths classroom

Measuring heights How would you program the path of a computer-generated person to ensure they could reach the treasure at the maximum height of their jump?

Toolbox 3 2 1

–1

0

D

B

A 1

C 2

3

4

The equation of curve A is y = x2. The equation of curve B is y

= x2 + 2.

It is curve A moved up 2 units.

The equation of curve C is y

= x2 – 6x + 9 = (x – 3)2.

It is curve A moved 3 units to the right.

The equation of curve D is y

= x2 – 6x + 11.

D is curve A moved 3 units to the right and 2 units up.

Curve D can be written as:

y = (x – 3)2 + 2

2 up 3 to the right

(x – 3)2 + 2 is x2 – 6x + 11 in completed square form. Any quadratic expression can be written in completed square form.

18

Unit 5 Completing the square

The line of symmetry is x = a

y

y

(a, b)

b b (a, b) x

a

The curve y = (x − a)2 + b has a minimum at the point (a, b).

The line of symmetry is x

x

a

The curve y = − (x − a)2 maximum at the point (a,

+ b has a b).

= a.

The point (a,b) is often referred to as the turning point. All quadratic curves have the same basic shape. To write x2

+ 8x + 13 in completed square form:

1 x2 + 8x + 13

The coefficient of x is 8, 8 ÷ 2 = 4

So the completed square form will be (x 2

+ 4)2 + b.

(x + 4)2 = x2 + 8x + 16

Multiplying out the brackets.

x2 + 8x + 13 = x2 + 8x + 16 + 13 − 16

The first three terms are a perfect square.

= (x + 4)2 − 3 This is completed square form.

y

The curve y = (x + 4)2 − 3 has a minimum at (−4, −3) (see graph on the right).

4 3 2 1 2

In some quadratic expressions the coefficient of x is not 1. For example in 8x2

–6

–5

–4

+ 6x + 9 it is 8.

–2

–1 0 –1

1

2

x

–2

The general completed square form is often written as:

p(x + q)2 + r

–3

(−4, −3)

–3

Once you know the turning point and y-intercept of a quadratic expression, you can sketch its graph.

19

Strand 6 Working with quadratics

Example – When the coefficient of x² is 1 a Write x2 −

3x + 5 in the form (x + a)2 + b. b Sketch the curve y = x2 − 3x + 5.

Solution a

x2 − 3x + 5 = (x + a)2 + b

a needs to be half of the coefficient of x which is −3, so a = −1.5.

x2 − 3x + 5 = (x − 1.5)2 + b x2 − 3x + 5 = (x − 1.5)2 + 5 – 2.25

(1.5)2 = 2.25

= (x − 1.5)2 + 2.75 b

(a = –1.5)

7

y

6 The y -intercept is 5.

5 4 3 2.75

The turning point is (1.5, 2.75).

2 1 x –1

1 1.5 2

3

4

5

Example – Finding turning points when the coefficient of x² is not 1 The graph shows the function y = −2x2 + 12x – 8. Write the function y = −2x2 + 12x − 8 in completed square form. Hence identify the turning point.

Solution −2x2 + 12x − 8 = p(x + q)2 + r y = −2x2 + 12x − 8 = −2(x2 − 6x) − 8 = −2[(x − 3)2 − 9] − 8 = −2(x − 3)2 + 18 − 8 = −2(x − 3)2 + 10 The turning point is (3, 10).

20

y

Use this form when the coefficient of x² is not 1.

p is the coefficient of x². Halve the coefficient of x to get 3.

(x − 3)² = x² − 6x + 9 so (x − 3)² − 9 = x² − 6x. Multiply out the square bracket.

x

Unit 5 Completing the square

Practising skills 1 Copy and complete this multiplication table to show that (x + a)2 = x2 + 2ax + a2.

2 Expand and simplify. a (x + 3)2 3 Starting from (x + a x2 + 12x + 37

b

x2 + 10x − 4 = (x + e x2 + 7x + 6 = (x +

a

x a

(x − 5)2

c

4(x + 1)2 − 4

)2 + )2 +

x2 + 6x + 1 = (x + 3)2 + d x2 + x + 16 = (x − 5)2 +

b

5 Write the following quadratic functions in the form a x2 − 4x + 3 b x2 + 2x + 7 d

x

6)2 = x2 + 12x + 36, write the following in the form (x + 6)2 + c. b x2 + 12x + 30 c x2 + 12x − 4

4 Fill in the missing term or terms. a x2 + 8x − 3 = (x + 4)2 + c

×

x2 + 9x + 20

e

(x + a)2 + b. c

x2 − 14x + 30

c

−x2 + 3x − 5

x2 − x + 1

6 Write the following quadratic functions in the form p(x a 2x2 + 12x + 10 b 2x2 + 4x + 5

+ q)2 + r.

Developing fluency 1 The graph shows the curve

y = (x − r)2 + s.

Exam-style

Its turning point is (2, 1). a Write down the values of r and s.

p

b The curve crosses the y-axis at (0, p). Find the value of p, the y-intercept. c Write down the equation of the curve in the form y = x2 + cx + d. 2

Exam-style

y

(2, 1)

x

0

y = x2 − 8x + 13 is a quadratic function. a What is the y-intercept? b Write the function in completed square form. c Write down the turning point of the function. d What is the equation of the line of symmetry of the function? e Sketch the graph of the function, labelling the points identified in parts a and c.

21

Strand 6 Working with quadratics

3 For each of the following quadratic expressions: a Write it in completed square form. b Write down its turning point. c Sketch the curve.

y = x2 + 8x − 7 iv y = 2x2 + 7x − 2

y = x2 − 5x + 4 v y = −2x2 − 5x + 1

Exam-style

Exam-style

i

4

ii

iii

y = 2x2 − 5x + 3 is a quadratic function. a Sketch the graph of the function. b Label the y-intercept and turning points with their co-ordinates. c Solve the equation 2x2

− 5x + 3 = 0.

5 The graph of a quadratic function crosses the y-axis at (0, 1). It has a turning point at (2, −3). What is the equation of the function? Express it both in completed square form and in the form y

= ax2 + bx + c.

6 This diagram has five regions. Show how they can be rearranged to show that x2

+ 2ax + d = (x + a)2 + d − a2

x

x

Exam-style

x + 2a 7 The trajectory of a golf ball is described by the function y = The units are metres. What is the greatest height of the golf ball? 22

 ac bb  bb22 −− 44ac 8 Show that: ax ax ++ bx bx ++ cc == aaxx ++  −− 44aa  22aa  22

22

1 x2 + x. − 40

y = −x2 + 3x + 2

Unit 5 Completing the square

Exam-style

Problem solving

5

1 The blue square has an area of x2 + 8x + n. The orange rectangle has a width of 5 units. Find the area of the orange rectangle, when a n = 16 and x = 3

x2 + 8x + n

n = 52 and x = 4 c n = 16 and x is unknown. b

2 Here is a quadratic function. a Write the equation of the function in the form

6

Exam-style

y = (x + a)2 + b.

4

b Find the value of a and the value of b.

2

c Write down the co-ordinates of the maximum point of the curve y = − (x + a)2 + b. –2

–1

0

1

2

3

4

–2

Exam-style

–4 3 The quadratic expressions (9x2 + square for all integer values of x. a Find the value of b.

bx + 25) is a perfect

b A square has an area (9x2 + bx + 25) cm2 and perimeter 68 cm. Find the value of x.

9x2 + b + 25

Exam-style

Exam-style

4 The area of this isosceles right-angled triangle is 8x2 + 24x + 18. a Find an expression in terms of x for the length of the base of the triangle. b Find the value of x if the area of the triangle is 98 cm2.

8x2 + 24x + 18

5 Robin fires an arrow into the air. The equation for the height h metres, of the arrow, at time t seconds is h a Write 20t − 5t2 in completed square form.

= 20t − 5t2.

b Use your answer to find the greatest height the arrow reaches and the time it takes to do so.

23

Strand 6 Working with quadratics

Exam-style

6 Andy says:

I think of a number. I multiply my number by 4. I subtract 2 and square my answer. My final answer is 36. What numbers could I be thinking about?

Exam-style

3x + 5 7 A square of side 6 cm is cut from a square of side 3x The blue area remaining is 108 cm2. a Find the length of the original square.

+ 5. 3x + 5

b Explain why there is only one possible answer.

6 6

Exam-style

8 The radius of the large circle is (5x

+ 9) cm. The radius of the small circle is (3x − 5) cm. a Prove that the yellow region has an area of

((4x + 15)2 − 169)π cm2. b Find the radius of the blue circle when the yellow area is 1056π cm2.

Reviewing skills 1 Expand and simplify. a (x − 1)2

b

3(x − 2)2

c (x

2 Write the following quadratic functions in the form p(x + q)2 + r. a 5x2 − 10x + 6 b 2x2 − 6x + 3 c

3x2 − 4x − 1

3 Sketch the graph of the function y = x2 + 3x − 7. Label the y-intercept and turning points with their co-ordinates. 4 The graph of a quadratic function crosses the y-axis at (0, −1). It has a turning point at (−1, −3). What is the equation of the function? Express it both in completed square form and y

24

− 12 )2 + 34

= ax2 + bx + c form.

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This sample chapter is taken from Mastering Mathematics for OCR GCSE Higher 2

MaSTERING Build your students’ knowledge and understanding so that they can confidently reason, interpret, communicate mathematically and apply their mathematical skills to solve problems within mathematics and wider contexts. Supports you and your students through the new specifications, with topic explanations and new exam-style questions Measure progress and assess learning throughout the course with graduated exercises and worked examples Enables students to identify the appropriate remediation or extension steps they need in order to make the best progress, through easy to follow progression strands Series Editor: Roger Porkess is a highly experienced author who held the position of MEI Chief Executive for 20 years.

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M aT H E M aT I C S

First teaching from September 2015