# GCSE Mathematics - ELITE Tuition

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GCSE Mathematics

Higher Tier

Stafford Burndred

Consultant Editor: Brian Seager, Chairman of Examiners

Easingwold School

GCSE Mathematics

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Further copies of this publication, as well as the guides for Foundation and Intermediate tiers may be obtained from: Pearson Publishing

Chesterton Mill, French's Road, Cambridge CB4 3NP Tel 01223 350555 Fax 01223 356484

Email info@pearson.co.uk Web site

? Pearson Publishing

No part of this publication may be copied or reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopy, recording or

otherwise without the prior permission of the publisher.

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Introduction Examiner's tips Number skills Calculator skills Fractions, decimals and percentages Number patterns Equations Variation Algebraic skills

Graphs

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Contents

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vi

........................................................................................... vii

Rational and irrational numbers .........................................

1

Using a calculator: Brackets, memory and fractions ..........

2

Using a calculator: Powers, roots and memory .................

3

Standard form ....................................................................

4

Percentages and fractions..................................................

5

Calculating growth and decay rates ..................................

6

Patterns you must recognise..............................................

7

Product of primes, highest common factor,

lowest common multiple and reciprocals ..........................

8

Trial and improvement .......................................................

9

Equations ........................................................................... 10

Rewriting formulae ............................................................. 11

Iteration .............................................................................. 12

Direct and inverse variation ............................................... 13

Using algebraic formulae ................................................... 14 Rules for indices (powers) .................................................. 15 Expansion of brackets ........................................................ 16 Factorisation ? 1................................................................. 17 Factorisation ? 2................................................................. 18 Factorisation ? 3................................................................. 19 Solving quadratic equations .............................................. 20 Simultaneous equations: Solving using algebra ................ 21 Simplifying algebraic fractions ? 1 ..................................... 22 Simplifying algebraic fractions ? 2 ..................................... 23

Drawing lines...................................................................... 24 Simultaneous equations: Solving by drawing a graph ...... 25 Solving equations using graphical methods...................... 26 The straight line equation y = mx + c ............................... 27 Using tangents to find gradients ....................................... 28 Expressing general rules in symbolic form ? 1 .................. 29 Expressing general rules in symbolic form ? 2 .................. 30 Drawing graphs.................................................................. 31 Sketching graphs ? 1.......................................................... 32 Sketching graphs ? 2.......................................................... 33 Speed, time and distance graphs ...................................... 34 Area under a curve............................................................. 35

Contents

Angles Similarity Congruency Transformations Measurement Circles Perimeter, area and volume

Pythagoras' theorem and trigonometry

Vectors Locus

Intersecting and parallel lines ............................................ 36 Bearings ............................................................................. 37

Similarity............................................................................. 38

Congruent triangles ? 1 ..................................................... 39 Congruent triangles ? 2 ..................................................... 40

Combined and inverse transformations............................. 41 Enlargement by a fractional scale factor............................ 42 Enlargement by a negative scale factor............................. 43

Compound measures......................................................... 44 Time ................................................................................... 45 Upper and lower bounds of numbers ? 1.......................... 46 Upper and lower bounds of numbers ? 2.......................... 47

Length, area and volume of shapes with curves................ 48 Angle and tangent properties of circles ? 1 ...................... 49 Angle and tangent properties of circles ? 2 ...................... 50 Angle and tangent properties of circles ? 3 ...................... 51

Calculating length, area and volume ? 1 ........................... 52 Calculating length, area and volume ? 2 ........................... 53 Calculating length, area and volume ? 3 ........................... 54 Formulae for length, area and volume .............................. 55 Ratio for length, area and volume ..................................... 56

Pythagoras' theorem .......................................................... 57 Trigonometry: Finding an angle......................................... 58 Trigonometry: Finding a side ............................................. 59 Trigonometry: Solving problems........................................ 60 Trigonometry and Pythagoras' theorem for 3-D shapes.... 61 Sine, cosine and tangent of any angle ? 1 ........................ 62 Sine, cosine and tangent of any angle ? 2 ........................ 63 Sine, cosine and tangent of any angle ? 3 ........................ 64 Sine rule, cosine rule, area of a triangle ? 1 ...................... 65 Sine rule, cosine rule, area of a triangle ? 2 ...................... 66

Vectors ? 1.......................................................................... 67 Vectors ? 2.......................................................................... 68 Vectors ? 3.......................................................................... 69 Vectors ? 4.......................................................................... 70

Locus (plural loci) ............................................................... 71

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Contents

Questionnaires Tables and graphs Cumulative frequency

Designing questionnaires................................................... 72 Sampling ............................................................................ 73 Hypotheses ........................................................................ 74

Comparing data ................................................................. 75 Histograms ......................................................................... 76 Grouped data..................................................................... 77

Cumulative frequency ........................................................ 78 Using cumulative frequency diagrams to compare distributions .................................................... 79

Standard deviation

Standard deviation ............................................................. 80 The normal distribution...................................................... 81

Scatter diagrams

Line of best fit .................................................................... 82

Probability

Estimation of probability by experiment ........................... 83 Tree diagrams..................................................................... 84 Conditional and independent probability ......................... 85 Probability (and, or)............................................................ 86 Probability (at least)............................................................ 87

Supplementary material

3-D co-ordinates ................................................................ 88 Inequalities ......................................................................... 89 Critical path analysis .......................................................... 90 Linear programming........................................................... 91 Transformations (matrices) ? 1 ........................................... 92 Transformations (matrices) ? 2 ........................................... 93

Important facts you are expected to know ............................................................................. 94

Diagnostic tests

Diagnostic tests ......................................................................... 98 Answers...................................................................................... 111

Index

........................................................................................... 116

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