Edexcel GCSE Maths Foundation Mastering Mathematics Revision Guide PDF

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Edexcel GCSE

MATHEMATICS Foundation

Keith Pledger

The Publishers would like to thank the following for permission to reproduce copyright material. Photo credits Acknowledgements Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked, the Publishers will be pleased to make the necessary arrangements at the first opportunity. 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 Bookpoint Ltd, 130 Park Drive, Milton Park, Abingdon, Oxon OX14 4SE. Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Email [email protected] Lines are open from 9 a.m. to 5 p.m., Monday to Saturday, with a 24-hour message answering service. You can also order through our website: www.hoddereducation.co.uk ISBN: 978 1 4718 8246 3 © Keith Pledger 2016 First published in 2016 by Hodder Education, An Hachette UK Company Carmelite House 50 Victoria Embankment London EC4Y 0DZ www.hoddereducation.co.uk Impression number

10 9 8 7 6 5 4 3 2 1

Year

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 Typeset in Printed in A catalogue record for this title is available from the British Library.

Get the most from this book Everyone has to decide his or her own revision strategy, but it is essential to review your work, learn it and test your understanding. These Revision Notes will help you to do that in a planned way, topic by topic. Use this book as the cornerstone of your revision and don’t hesitate to write in it — personalise your notes and check your progress by ticking off each section as you revise.

You can also keep track of your revision by ticking off each topic heading in the book. You may find it helpful to add your own notes as you work through each topic.

Revision planner page

Tick to track your progress Use the revision planner on pages 4 and 5 to plan your revision, topic by topic. Tick each box when you have: l revised and understood a topic l tested yourself l checked your answers

Example page

Features to help you succeed Rules

The key rules you need to follow when answers questions on this topic. Worked examples

Several worked examples are given for each topic. Key terms

Exam-style questions

Practice exam questions are provided for each topic. Use them to consolidate your revision and practise your exam skills. Answers

Check how you’ve done using the answers at the back of the book.

Key terms are highlighted in each section. Exam tips

Expert tips are given throughout the book to help you polish your exam technique in order to maximise your chances in the exam.

iii

My revision planner Number Number: pre-revision check xx BIDMAS xx Multiplying decimals xx Dividing decimals xx Using the number system effectively xx Understanding standard form xx Calculating with standard form xx Rounding to decimal places, significance and approximating xx Limits of accuracy xx Multiplying and dividing fractions xx Adding and subtracting fractions and working with mixed numbers xx Converting fractions and decimals to and from percentages xx Applying percentage increases and decreases to amounts xx Finding the percentage change from one amount to another xx Reverse percentages xx Repeated percentage increase/decrease Mixed exam-style questions xx Sharing in a given ratio xx Working with proportional quantities xx The constant of proportionality xx Working with inversely proportional quantities xx Index notation and rules of indices xx Prime factorisation Mixed exam-style questions Number: summary of rules

iv

Edexcel GCSE Maths Revision Guide Foundation

Algebra Algebra: pre-revision check xx Working with formulae xx Setting up and solving simple equations xx Using brackets xx Solving equations with the unknown on both sides xx Solving equations with brackets xx Simplifying harder expressions and expanding two brackets xx Using complex formulae and changing the subject of a formula xx Identities xx Linear sequences xx Special sequences xx Quadratic sequences xx Geometric progressions Mixed exam-style questions xx Real-life graphs xx Plotting graphs of linear functions xx The equation of a straight line xx Plotting quadratic and cubic graphs xx Finding equations of straight lines xx Quadratic functions xx Polynomial and reciprocal functions xx Linear inequalities xx Solving simultaneous equations by elimination and substitution xx Using graphs to solve simultaneous equations xx Factorising quadratics of the form x2 + bx + c xx Solve equations by factorising Mixed exam-style questions Algebra: summary of rules

v

Geometry Geometry: pre-revision check xx Bearings and scale drawings xx Compound units xx Working with compound units xx Types of quadrilateral xx Angles and parallel lines xx Angles in a polygon xx Congruent triangles and proof xx Proof using similar and congruent triangles xx Circumference xx Pythagoras theorem xx Arcs and sectors Mixed exam-style questions xx Constructions with a pair of compasses xx Loci xx Enlargement xx Similarity xx Trigonometry xx Trigonometry for special angles xx Finding centres of rotation xx Understanding nets and 2D representation of 3D shapes xx Volume and surface area of cuboids and prisms xx Enlargement in two and three dimensions xx Constructing plans and elevations xx Surface area and 3D shapes xx Vectors Mixed exam-style questions Geometry: summary of rules

vi

Edexcel GCSE Maths Revision Guide Foundation

Statistics and probability Statistics and probability: pre-revision check xx Using frequency tables xx Using grouped frequency tables xx Vertical line charts xx Pie charts xx Displaying grouped data xx Scatter diagrams and using lines of best fit Mixed exam-style questions xx Single event probability xx Combined events xx Estimating probability xx The multiplication rule xx The addition rule Statistics and probability: exam-style questions Statistics and probability: summary of rules

Exam preparation xx xx xx xx xx xx

The language used in mathematics examinations Exam technique Key topics for revision One week to go Answers Index

vii

Algebra: pre-revision check 1 This formula gives the value of p in terms of q and r: p = 2q – 3r. Findthe value of p when q = 10 and r = 4. F S1 U4

S = ut + 1 at 2 2

a Find the value of S when u = 5, t = 4 and a = 10. b Make a the subject of the formula. F/H S1 U10

2 Solve the equations a a+4=6 b b =5 3 c 5c + 4 = 6 d 15 – 3e = 24 F S1 U5 2

8 Prove that the sum of the three consecutive numbers (n – 1), n and (n + 1) is a multiple of 3. F/H S1 U11

3 a Expand these brackets i 5(2a + 3) ii h(3h – 6) iii 3x(4x – 2y) b Factorise fully i 6y + 12 ii 6p2 – 9p iii 5e2 + 10ef iv 8x2y – 12xy2 F S1 U6

9 Here are the first 5 terms of a linear sequence. 4

10

16

22

28…

a Find the nth term of the sequence. b Work out the 50th term in the sequence. c Explain if 900 is a member of the sequence. F S2 U3

F/H S1 U8

6 a Simplify i a4 × a6 x8 ii 5 x 12e6f 7 iii 8e9f 5 b Expand and simplify i (t + 2)(t + 5) ii (v – 7)(v + 5) iii (y – 6)(y – 5) F/H S1 U9

10 a Write down the first 5 terms of the quadratic sequence with nth term 2n 2 – 3. b Find the nth term of the quadratic sequence that has the first 5 terms 3, 8, 15, 24, 35. F S2 U4 11 The nth term of quadratic sequence is n 2 + 5. The nth term of a different quadratic sequence is 80 – 2n 2 . Find the number that is in both sequences. F/H S2 U5 12 Here is a geometric sequence. 5

15

45 135 …

a Find the common ratio. b Find the 10th term of the sequence. F/H S2 U6 13 Here is the graph that shows the depth of water in a harbour. A ship needs to enter the harbour between 0800 and 2000. It needs a 4 metre depth of water in the harbour. Between what times can the ship enter the harbour? F S3 U1 10 Depth of water (m)

4 Solve these equations a 5x – 6 = 2x + 3 b 7 – 2p = 6p + 13 y y c 2 – 32 = 5 + 5 F/H S1 U7 4 5 Solve. a 5(3g – 2) = 35 b 4(5h + 7) = 3(2h + 8) c 2(5k + 8) – 6 = 4(2k – 1)

7 This formula is used to find the distance, S, travelled by an object.

5 0 00:00 04:00 08:00 12:00 16:00 20:00 24:00 Time

Algebra

1

Algebra: pre-revision check

14 a On a coordinate grid drawn with values of x from –3 to +3 and values of y from –6 to +8, draw the graph of y = 2x + 1. b Find the value of x when y = 6 F S3 U2

21 Solve this pair of simultaneous equations. 5x + 2y = 8 2x – y = 5 F/H S4 U 3 & 4 22 Here is the graph of the line y + 2x = 3.

15 Find the equation of a straight line graph that passes through the point (0, –2) and has a gradient of 3. F S3 U3

y 6

16 a On a coordinate grid drawn with values of x from –2 to +4 and values of y from –6 to +8, draw the graph of y = x 2 – 3x – 2. b Find the values of x when x2 – 3x – 2 = 0. F S3 U4 17 Find the equation of a straight line graph that passes through the point (–2, 3) and is parallel to the line x + 2y = 8. F/H S3 U5

4 2

−3

−2

−1

0

1

2

3

x

−2 −4

18 a Sketch the graph of the quadratic function y = x 2 – 4x + 3 for values of x from 0 to 5. b Write down the roots of the equation x2 – 4x + 3 = 0. c Write down the line of symmetry of the graph. F/H S3 U6

Find graphically the solution to the simultaneous equations: y + 2x = 3 y – 2x = 1 F/H S4 U5

19 a On a coordinate grid drawn with values of x from –3 to +3 and values of y from –10 to +30, draw the graph of y = x 3 + x 2 – 3x. b Find the values of x when x3 + x 2 – 3x = 0. F/H S3 U7 20 a Write down the inequality shown on this number line.

x −5 −4 −3 −2 −1 0

1

2

3

4

5

b Solve these inequalities i 2x + 5 < 9 ii 24 + 2t > 30 – 3t iii 5(y – 3)  3y – 6 F/H S4 U2

2

Edexcel GCSE Maths Revision Guide Foundation

23 a Expand and simplify i (x + 4)(x – 5) ii (y + 8)(y – 8) iii (6 – a)(a + 6) b Factorise i x2 + 7x + 12 ii e2 – 3e – 10 iii b2 – 25 F/H S5 U1 24 Solve the equations a x2 – 5x + 6 = 0 b x2 – 2x = 15 c p2 – 49 = 0 F/H S5 U2

DIFFICULTY LEVEL MID Rules You can replace words or letters in a formula with numbers. Use BIDMAS to find the value of the missing word or letter. 3 Use inverses to write the formula or equation so that the missing letter is on its own on one side of the formula or equation. 1 2

Worked examples

Look out for

Here is a formula to find the perimeter of a rectangle: P = 2l + 2w. Find the value of P when l = 6 and w = 4.

a

P = 2l + 2w

b

1

P=2×6+2×4

2

P = 12 + 8 = 20

2w means 2 × w

Algebra: pre-revision check

Working with formulae

So if w = 5 then 2w is 2 × 5 = 10 and not 25

Cars 2U

Key terms

Tom hires a car from Cars 2U.

Formula

i How much does it cost to hire a car for 7 days? ii Ben has £100. For how many days can he £20 plus £30 a day hire a car?

Substitute Variable Equation

You must explain your answer. i Cost = 20 + 7 × 30 1 Cost = 20 + 210 2 Cost = £230

ii 100 = 20 + N × 30 1 100 – 20 = N × 30 3 80 = N × 30 so N = 80 ÷ 30 = 2.666… 2 Ben can hire the car for 2 days. This costs £80. 3 days cost £110 which is too much.

Exam tips Always show your working when answering algebra questions. Do not use trial and error methods as you may well lose marks.

Exam-style questions 1 Bobbie uses this number machine to work out the number of cartons of orange juice she needs for a party. Number of people

÷5

+2

Number of cartons

a How many cartons will Bobbie need for 40 people? b How many people are in a party that uses 20 cartons? 2 This formula gives the time taken T minutes to cook a chicken of weight w kg. T = 40w + 20 a How long does it take to cook a chicken of weight 2.5 kg? b It takes 3 hours 20 minutes to cook a different chicken. How heavy was the chicken?

[2] [2] [2] [2]

Algebra

3

Algebra: pre-revision check

Setting up and solving simple equations DIFFICULTY LEVEL MID Rules Always use the inverse operations to solve an equation. + and – are the inverse of one another. 3 × and ÷ are the inverse of one another. 4 To set up an equation a variable must be defined. 1 2

Worked examples a

2

3

b

2

2p + 5 = 17

(–5 is the inverse operation of + 5)

2p + 5 – 5 = 17 – 5

(subtract 5 from each side of the equation)

Key terms

2p = 12

(÷ 2 is the inverse of × 2)

Equation

p=6

(divide each side of the equation by 2)

Inverse operation

Ann is two years younger than Ben. Clara is twice as old as Ben. The total of their ages is 58. Work out their ages.

Variable

Let Ben’s age be x,

Firstly, set up the equation:

Ann’s age will be x – 2,

x + (x – 2) + 2 x = 58

Clara’s age will be 2x

Now collect like terms:

Exam tips

4x – 2 = 62

(Add 2 to each side)

4x = 60

(Divide each side by 4)

Always use algebraic methods and show your working to gain full marks.

x = 15

Always check your answer to make sure it is correct.

Ann will be 13, Ben 15 and Clara 30.

Exam-style questions 1 Solve the equations b a a – 3 = 7 [1]; c b = 3 [1]; 5 5d + 4 = 29 [2]; d e 6 – 2e = 3 [2] 2 2 Here is a rectangle. 2x + 5 The length is 2 x + 5. The width is x – 3. The perimeter is 46 cm. Work out the area of the rectangle in cm2. [4]

4

Solve

Edexcel GCSE Maths Revision Guide Foundation

3c + 9 = 7 [2];

x–3

DIFFICULTY LEVEL MID Rules When you expand a bracket you multiply what is inside the bracket by the number or variable outside the bracket. 2 When you factorise an algebraic expression you take out the common factor from each term of the expression and put it outside the bracket. 1

Worked examples a Expand 4(3x + 5);

i

Look out for t(3t – 4)

ii

1

x × x = x2 using index laws

1

4 × (3x + 5) 4 × 3x + 4 × 5 12x + 20

Key terms

t × (3t − 4) t × 3t − t × 4 3t2 − 4t

Brackets Variable Expression

b Factorise i

2

Algebra: pre-revision check

Using brackets

4p + 6 2×2×p+2×3 2 is in 4 and in 6 2(2p + 3)

ii

2

Expand

6p2q – 9pq2 3×p×q×2×p–3×p×q×3×q 3 and p and q are in both terms 3pq(2p – 3q)

Factorise

Exam-style questions 1 Beth is 3 years older than Amy. Cath is twice as old as Beth. The total of their ages is 41. How old are the three girls? [3] 2 PQRS is a rectangle. The length of the rectangle is (2x – 5) cm (2x – 5) cm. P Q The width of the rectangle 6 cm is 6 cm. The area of the rectangle R S is 72 cm2. Work out the perimeter of the rectangle. [4]

Exam tips Always check your answer by multiplying out the brackets. Always define your variable e.g. let Amy’s age be x. Then set up your equation.

Algebra

5

Mixed exam-style questions (Strands 1, 2 and 3) 1 The diagram show the position of two villages. N Redford is on a bearing of 050° from Brownhills. Redfor d Karen walks from Brownhills to Redford. She walks at an average speed of 6 km/h. Brownhills She takes 1 h 30 mins to cover the distance. a Work out the distance between Brownhills and Redford. [2] b Using a scale of 1 cm to 4 km, make an accurate scale drawing showing the position ofthe two villages. [3] 2 The diagram shows a square attached to four similar regular polygons. Calculate the number of sides on the polygons.

[3]

3 ABC is an isosceles triangle. D and E are points on BC. AB = AC, BD = EC Prove that triangle ADE is isosceles.

[3]

A

B

D

E

C

4 The wheel on a bicycle has a diameter of 70 cm. John cycles 15 km on the bicycle. a How many revolutions will the wheel make during the journey? b The journey takes John 1 h 20 mins. Calculate John’s average speed. 5 In the diagram PQ is parallel to ST. QX = XS Prove that triangles PQX and STX are congruent.

[4] [2]

P S X

6 The diagram shows a plan of a garden design. Q A circular pond with a diameter of 1.5 m is dug in the lawn. The centre of the pond is in the centre of the lawn. a Make an accurate scale drawing of the plan using a scale of 2 cm: 1 m. b Calculate the area of the lawn. 2m

4m

patio

pond lawn 8m

6

Edexcel GCSE Maths Revision Guide Foundation

T

[4] [4]

[3] [4]

A 3 cm B

C

8 A paper cone is made from a folding a piece of paper in the shape of sector of a circle. The angle at the centre of the sector is 100°. The radius of the sector is 6 cm. a Calculate the length of the arc of the sector. [2] b Calculate the diameter of the base of the finished cone [2] 6 cm

100°

9 The diagram shows a trapezium PQRS. PQ is parallel to SR. PS = QR. Show that triangles PXS and QXR are similar. P

Mixed exam-style questions

7 The diagram shows a right-angled triangle drawn inside a quarter circle. The chord AC is 3 cm. a Calculate the radius of the circle. b Calculate the area of the segment ABC.

[3]

Q X

S

R

10 ABCD is a rhombus. AC = 10.8 cm, BD = 15.6 cm. Calculate the length of the sides of the rhombus. A

[4]

B 10.8 cm

15.6 cm D

C

Geometry

7

Exam technique ●

Be prepared and know what to expect.

●

Read each question carefully.

●

Don’t just learn key points.

●

Show stages in you...