revise pearson edexcel international gcse (9-1 ......international gcse (9-1) mathematics a revision...

Revise Pearson Edexcel International GCSE (9-1)

Mathematics A Revision Guide

Sample

NUMBER1 Factors and primes

2 Indices 1

3 Indices 2

4 Calculator skills 1

5 Fractions

6 Decimals

7 Recurring decimals

8 Surds 1

9 Calculator skills 2

10 Ratio

11 Proportion

12 Percentage change

13 Reverse percentages

14 Repeated percentage change

15 Upper and lower bounds

16 Standard form

17 Problem-solving practice 1


ALGEBRA19 Algebraic expressions

20 Expanding brackets

21 Factorising

22 Linear equations 1

23 Linear equations 2

24 Formulae

25 Arithmetic sequences

26 Straight-line graphs 1

27 Straight-line graphs 2

28 Parallel and perpendicular

29 Quadratic graphs

30 Cubic and reciprocal graphs

31 Rates of change

32 Quadratic equations

33 The quadratic formula

34 Completing the square

35 Simultaneous equations 1

36 Simultaneous equations 2

37 Inequalities

38 Trigonometric graphs

39 Transforming graphs 1

40 Transforming graphs 2

41 Inequalities on graphs

42 Sketching graphs

43 Using quadratic graphs

44 Turning points

45 Quadratic inequalities

46 Gradients of curves

47 Proportion and graphs

48 Proportionality formulae

49 Harder relationships

50 Rearranging formulae

51 Sequences and series

52 Algebraic fractions

53 Quadratics and fractions

54 Surds 2

55 Functions

56 Composite functions

57 Inverse functions

58 Differentiation

59 Gradients and calculus

60 Turning points and calculus

61 Kinematics

62 Algebraic proof



GEOMETRY AND MEASURES65 Speed

66 Density

67 Other compound measures

68 Angle properties

69 Solving angle problems

70 Angles in polygons

71 Perimeter and area

72 Units of area and volume

73 Prisms

74 Circles and cylinders

75 Sectors of circles

76 Pythagoras’ theorem

77 Trigonometry 1

78 Trigonometry 2

79 Volumes of 3D shapes

80 Surface area

81 Translations, refl ections and rotations

82 Enlargements

83 Combining transformations

84 Bearings

85 Scale drawings and maps

86 Constructions

87 Similar shapes 1

88 Similar shapes 2

89 The sine rule

90 The cosine rule

91 Triangles and segments

92 Pythagoras in 3D

93 Trigonometry in 3D

94 Circle facts

95 Intersecting chords

96 Circle theorems

97 Vectors

98 Vector proof



PROBABILITY AND STATISTICS

101 Mean, median and mode

102 Frequency table averages

103 Interquartile range

104 Cumulative frequency

105 Histograms

106 Probability

107 Relative frequency

108 Set notation

109 Venn diagrams

110 Probability and sets

111 Conditional probability

112 Tree diagrams



115 FORMULAE SHEET

116 ANSWERS

A small bit of small printEdexcel publishes Sample Assessment Material and the Specifi cation on its website. This is the offi cial content and this book should be used in conjunction with it. The questions in Now try this have been written to help you practise every topic in the book. Remember: the real exam questions may not look like this.

Contents

F01_EDHM_RG_GCSE_8090_i-ii.indd 2 12/7/18 8:25 PM

SAMPLE

NUMBERHad a look Nearly there Nailed it!

1

Factors and primesThe factors of a number are any numbers that divide into it exactly. A prime number has exactly two factors. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 and so on.

The highest common factor (HCF) of two numbers is the highest number that is a factor of both numbers.

84 � 2 � 2 � 3 � 7 � 22 � 3 � 7

Remember to put in the multiplication signs.This is called a product of prime factors.

Prime factorsIf a number is a factor of another number and it is a prime number then it is called a prime factor. You use a factor tree to fi nd prime factors.

84

422

76

3 2

Remember to circle the prime factors as you go along. The order doesn’t matter.

The lowest common multiple (LCM) of two numbers is the lowest number that is a multiple of both numbers.

To fi nd the HCF, circle all the prime numbers which are common to both products of primefactors. 2 appears twice in both products so you have to circle it twice. Multiply the circlednumbers together to fi nd the HCF.

Draw a factor tree. Continue until every branch ends with a prime number. This question asks you to write your answer inindex form. This means you need to use powers to say how many times each primenumber occurs in the product.

Check it!22 � 33 � 4 � 27 � 108 ✓

To fi nd the LCM multiply the HCF by each unshared prime from part (b).

(a) Write 108 as the product of itsprime factors. Give your answerin index form. (3 marks)

108

542

96

32 33

108 � 2 � 2 � 3 � 3 � 3 � 22 � 33

(b) Work out the highest common factor(HCF) of 108 and 24. (2 marks)

108 � 2 � 2 � 3 � 3 � 324 � 2 � 2 � 2 � 3HCF is 2 � 2 � 3 � 12

(c) Work out the lowest common multiple(LCM) of 108 and 24. (2 marks)

LCM � 12 � 3 � 3 � 2 � 216

2 X � 2 � 35 � 72 Y � 32 � 5 � 7

(a) Find the highest common factor(HCF) of X and Y. (2 marks)

(b) Find the lowest common multiple(LCM) of X and Y. (2 marks)

1 (a) Express 980 as a product of its prime factors. (3 marks)

(b) Find the highest common factor (HCF) of980 and 56. (2 marks)

You can use numbers given in index form directly:HCF: Choose the lowest power of each primeLCM: Choose the highest power of each prime.

M01_EDHM_RG_GCSE_8090_001-010.indd 1 12/7/18 12:28 PM

SAMPLE

Had a look Nearly there Nailed it! ALGEBRA

19

Algebraic expressionsYou need to be able to work with algebraic expressions confi dently. For a reminder about using the index laws with numbers have a look at pages 2 and 3.

You can use the index laws to simplify algebraic expressions.

a m � a n � a m + n

x 4 � x 3 � x 4 � 3 � x 7

a m ___ a n � a m � n

m 8 ÷ m 2 � m 8 − 2 � m6

(a m)n � a mn

(n 2)4 � n 2 � 4 � n 8

Algebraic expressions may also contain negative and fractional indices.

a�m � 1 ___ am

(c 2)−3 � c 2 � −3 � c −6 � 1 ___ c6

a 1 _ n � n √

__a

(8p3 ) 1 __ 3 � (8 )

1 __ 3 � (p 3 ) 1 __3

� 3 √ __ 8 � p 3 � 1 __

3

� 2p

You can square or cube a whole expression.

(4x 3y)2 � (4)2 � (x 3)2 � (y)2 � 16x 6y 2

Remember that if a letter appears on its own then it has the power 1.

16 � (4)2

(x 3)2 � x 3 � 2 � x 6

You need to square everything inside the brackets.

You can use the index lawsindex lawssimplify algebraic

1 2 3

One at a timeWhen you are multiplying expressions:1. Multiply any number parts fi rst.2. Add the powers of each letter to

work out the new power.6p 2q � 3p 3q 2 � 18p 5q 3

6 � 3 � 18p 2 � p 3 � p 2 � 3 � p 5

q � q 2 � q 1 � 2 � q 3

When you are dividing expressions:1. Divide any number parts fi rst.2. Subtract the powers of each letter to

work out the new power.

12a 5b 3 _______ 3a 2b 2 � 4a 3b

12 � 3 � 4b 3 � b 2 � b 3 � 2 � b

a 5 � a 2 � a 5 � 2 � a 3

(a) This is four lots of m2, so you write itas 4 x m2 or 4m2

(b) Use (a m)n � a mn

(c) Start by simplifying the top part of thefraction. Do the number part fi rst then the powers. Use a m � a n � a m � n

Next divide the expressions. Divide the number part, then divide the

indices using a m ___ a n � a m � n

Simplify fully

(a) m2 + m2 + m2 + m2 (1 mark)4m 2

(b) (x3)3 (1 mark)x 9

(c) 4y2 � 3y7________

6y (2 marks)

4y 2 � 3y 7

___________ 6y � 12y 9

_____ 6y � 2y 8

1 Simplify (h2)6 (1 mark)

2 Simplify fully(a) (2a5 b)4 (2 marks)(b) 5x 4y 2 � 3x 3y7 (2 marks)(c) 18d 8g10 � 6d 2g5 (2 marks)

Apply the power outside thebrackets to everything inside thebrackets.

3 (a) Simplify (16p10 ) 1 _ 2 (2 marks)

(b) Simplify (64x9y2 )− 1 _ 3 (2 marks)

(2 marks)

Worked

solution

video

Aiminghigher


SAMPLE

Had a look Nearly there Nailed it!GEOMETRY & MEASURES

65

Speed

UnitsThe most common units of speed are:• metres per second: m / s• kilometres per hour: km / h

• miles per hour: mph.To convert between measures of speed you need to convert one unit fi rst then the other. Write the new units at each step of your working. To convert 72 km/h into m/s:72 km/h → 72 � 1000 � 72 000 m/h72 000 m/h → 72 000 � 3600 � 20 m/s1 hour � 60 � 60 � 3600 seconds

Minutes and hoursFor questions on speed, you need to be able to convert between minutes and hours.Remember there are 60 minutes in 1 hour.To convert from minutes to hours you divide by 60.24 minutes � 0.4 hours 24 ___ 60 � 2 __ 5 � 0.4

To convert from hours to minutes you multiply by 60.0.2 hours � 12 minutes 3.2 � 60 � 1923.2 hours � 3 hours 12 minutes

This is the formula triangle for speed.

S T

D

Average speed � total distance travelled ______________________ total time taken

Time � distance ______________ average speed

Distance � average speed � time

DistanceAverage speed Time

Using a formula triangleCover up the quantity you want to fi nd with your fi nger.

S T

D

T � D __ S S � D __ T D � S � T

The position of the other two quantities tells you the formula.

Speed checklistDraw formula triangle. Make sure units match. Give units with answer.

The speed of light in a vacuum is approximately 1.08 × 109 km/h.

Light from the Sun takes approximately 8 minutes and 15 seconds to travel to Earth.

Estimate the distance from the Earth to the Sun. (3 marks)

DS T

8 mins 15 secs = 8.25 mins = 8.25 _____ 60 = 0.1375 hours

D = S × T= 1.08 × 109 × 0.1375= 1.485 × 108 km

Rosa lives in Durham and works in Newcastle. She takes the train to work every day.

Last Tuesday her train journey to work took 12 minutes, at an average speed of 108 km/h.

Her journey home from work took 15 minutes.

Calculate Rosa’s average speed on her journey home. (3 marks)

Be careful with the units. You need to convert 12 minutes and 15 minutes into hours before doing your calculations.

If you’re answering questions involving speed, distance and time you must always make sure that the units match. Speed is given in km/h here, so convert the time into hours before calculating.

LEARN IT!


SAMPLE

Had a look Nearly there Nailed it!PROBABILITY & STATISTICS

101

Mean, median and modeYou can analyse data by calculating statistics like the mean, median and mode.

Mean Median ModeAdd up all the

values

Divide by thetotal number of

values

Do not roundyour answer

More thanone value

with highestfrequency

All valueshave samefrequency

Lookfor the most

commonvalueOne value


This value isthe mode

These values areall modes

There is nomode

Evennumber

of values

Write thevalues in order ofsize, smallest first

Count thenumber of

valuesOddnumber

of values

The median isthe middle value

The median ishalf-way between

the two middle values

Add up all thevalues


values


More thanone value



Lookfor the most





There is nomode

Evennumber

of values


Count thenumber of

valuesOddnumber

of values




Add up all thevalues


values


More thanone value



Lookfor the most





There is nomode

Evennumber

of values


Count thenumber of

valuesOddnumber

of values




Which average works best?

Mean Uses all the data Affected by

extreme valuesMedian Not affected by

extreme valuesMay not be one of the values

Mode Suitable for data that can be described in words

Not always near the middle of the data

You can work out the sum of the 6 remaining

cards using this formula:

Sum of values � mean � number of values

Subtract this sum from the sum of all 8

cards. This tells you the sum of the 2 cards

Kayla removed. The removed cards were

either 5 and 7 or 8 and 4.

Check it!Work out the mean of the remaining 6 cards.

Kayla has eight numbered cards.

1 2 3 4 5 6 7 8She removes two cards. The mean value of the remaining cards is 4.Which two cards could Kayla have removed? Give one possible answer. (4 marks)

6 � 4 � 241 � 2 � 3 � 4 � 5 � 6 � 7 � 8 � 3636 � 24 � 12The removed cards add up to 12 so Kayla could have removed 7 and 5Check:

1 � 2 � 3 � 4 � 6 � 8 _______________________ 6 � 4 ✓

101

Takeshi scored these marks out of 20 in six maths tests.

11 9 5 13 15 12

How many marks must he score in the next test so that his new mean mark and his new median mark are the same as each other? (3 marks)

Make sure you check your answer by

calculating the new mean and median.

Remember that the median is not

affected by extreme values.


SAMPLE

revise pearson edexcel international gcse (9-1 ......international gcse (9-1) mathematics a revision...

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