sample assessment material methods of mathematics unit 1

Sample assessment materials

GCSE

Edexcel GCSE in Methods in Mathematics

Edexcel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn, London WC1V 7BH

Edexcel GCSE in Methods in Mathematics Sample Assessment Materials © Edexcel Limited 2010 1

Contents

General Marking Guidance 2

Unit 1: Methods 1 3

Unit 1: Methods 1 Foundation Paper 5

Unit 1: Methods 1 Foundation Mark scheme 29

Unit 1: Methods 1 Higher Paper 39

Unit 1: Methods 1 Higher Mark scheme 60

Unit 2: Methods 2 69

Unit 2: Methods 2 Foundation Paper 71

Unit 2: Methods 2 Foundation Mark scheme 93

Unit 2: Methods 2 Higher Paper 101

Unit 2: Methods 2 Higher Mark scheme 123

Edexcel GCSE in Methods in Mathematics Sample Assessment Materials © Edexcel Limited 20102

General Marking Guidance

• All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last.

• Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions.

• All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, ie if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate’s response is not worthy of credit according to the mark scheme.

• Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited.

• Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response.

• Mark schemes will indicate within the table where, and which strands of, QWC is being assessed. The strands are as follows:

i) ensure that text is legible and that spelling, punctuation and grammar are accurate so that meaning is clear. Comprehension and meaning is clear by using correct notation and labelling conventions.

ii) select and use a form and style of writing appropriate to purpose and to complex subject matter. Reasoning, explanation or argument is correct and appropriately structured to convey mathematical reasoning.

iii) organise information clearly and coherently, using specialist vocabulary when appropriate. The mathematical methods and processes used are coherently and clearly organised and the appropriate mathematical vocabulary used.

Guidance on the use of codes within this mark scheme M1 – method mark A1 – accuracy mark B1 – working mark C1 – communication mark QWC – quality of written communication oe – or equivalent cao – correct answer only ft – follow through


Unit 1: Methods 1


Centre Number Candidate Number

Write your name hereSurname Other names

Total Marks

Paper Reference

Turn over

*S38369A0126*

Edexcel GCSE

Methods in MathematicsUnit 1: Methods 1For Approved Pilot Centres ONLY

Foundation Tier

Sample Assessment MaterialTime: 1 hour 45 minutes

You must have:

Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used.

5MM1F/01

Instructions• Use black ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions.• Answer the questions in the spaces provided – there may be more space than you need.• Calculators must not be used.

Information• The total mark for this paper is 100. • The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.• Questions labelled with an asterisk (*) are ones where the quality of your written communication will be assessed – you should take particular care on these questions with your spelling, punctuation

and grammar, as well as the clarity of expression.

Advice• Read each question carefully before you start to answer it.• Keep an eye on the time.• Try to answer every question.• Check your answers if you have time at the end.

S38369A©2010 Edexcel Limited.


GCSE Mathematics 2544

Formulae: Foundation Tier

You must not write on this formulae page.Anything you write on this formulae page will gain NO credit.

Area of trapezium = (a + b)h

Volume of prism = area of cross section × length

b

a

h

length

crosssection

12

GCSE Mathematics 2MM01


Answer ALL questions.

Write all your answers in the spaces provided.

You must write down all stages in your working.

You must NOT use a calculator.

1 The shape is made from a right-angled triangle, a parallelogram and a quadrilateral.

(a) Mark with arrows (>>) a pair of parallel lines.(1)

(b) Mark with the letter A an acute angle.(1)

(c) Mark with the letter R a reflex angle.(1)

(Total for Question 1 = 3 marks)


2

The diagram shows a spinner as an 8 sided shape. Part of the spinner is coloured red, part is coloured green, part is coloured blue and the

rest is coloured yellow.

The spinner is spun once and lands on one colour.

(a) Which colour is the most likely? You must explain your answer.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Write down the probability of getting a red or a yellow.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jas spins the spinner once. He gets a yellow.

Yusuf then spins the spinner once.

(c) Has Yusuf got a greater chance, the same chance or a smaller chance of getting yellow than Jas had? You must explain your answer.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


R

G

B

Y


3 Here is a grid of centimetre squares.

(a) Write down the coordinates of the point

(i) B,(1)

( . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . )

(ii) C.(1)

( . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . )ABCD is a quadrilateral.

The area of ABCD is 24 cm2.

(b) Write down the coordinates of D.(2)

( . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . ) E is another point on the grid.

The coordinates of the midpoint of AE are (–1, 2).

(c) Find the coordinates of E.(2)

( . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . )


1

2

3

4

–1

–2

–3

–4

–3 –2 –1 1 2 3 4 5 6–5 –4–6 O x

y

C

A B


4 Here is a list of eight numbers.

(a) Write down two numbers from the list with a sum of 87(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Write down a number from the list which is

(i) a multiple of 9(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) a square number.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) Use a word from the above box to complete this sentence correctly.(1)

11 is a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of 88

Here are the same 8 numbers drawn larger.

l l l 6 l 8 3 66 8 6 9 8 2 8 8

(d) From these numbers, write down a number which has

(i) exactly one line of symmetry,(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) two lines of symmetry and rotational symmetry of order 2,(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(iii) rotational symmetry of order 2 but no lines of symmetry.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


cube multiple factor product

11 16 18 36 68 69 82 88


5 Here are four shapes. The shapes have been drawn accurately.

(a) Write down the mathematical name of shape C.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Neel makes the shapes accurately out of card.

Two of the shapes can be fitted together without overlapping to make a trapezium.

(b) Show how this can be done.(1)

Two of the shapes can be fitted together without overlapping to make a hexagon.

(c) Show how this can be done.(1)


A B C D


*6

AB is a straight line.

Pinkal is trying to find the value of the angle marked x

This is what she does.

y = 38° because the angles next to each other on a straight line add up to 160°z = 52° because y + z comes to 90°

x = 318 because angles around a point add up to 360°

Pinkal has made some mistakes.

Find the mistakes and work out the correct value of x.

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .°


Diagram NOTaccurately drawn

x

z

B122°

Ay


7 Dwayne has a bag of counters. The counters are either blue or green. There are 10 blue counters and 2 green counters.

Dwayne takes a counter at random from his bag.

(a) Write down the probability it is green.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mario has a bag of counters. The counters are either blue or green. There are twice as many blue counters as green counters.

Mario takes a counter at random from his bag.

(b) Write down the probability it is blue.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Both boys put back the counter they took out.

They empty the counters from both bags into a vase.

Dwayne says that the probability of taking a blue counter from the vase is 1318

(c) What is the probability of taking a green counter from the vase?(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dwayne knows that there are between 20 and 50 counters in the bag.

(d) Work out the number of counters there are in the bag.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



8

(a) Triangle ABC is equilateral.

Explain why.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PQR is a straight line.SQ = SR.

(b) (i) Work out the size of the angle marked x°.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) Give reasons for your answer.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



A

B60°

60°

C

S

x° 50°P Q R


9 Here are some patterns made from squares.

(a) The diagram below shows part of Pattern number 4 Complete the diagram for Pattern number 4

(1)

(b) Find the number of squares used for Pattern number 10

You must explain your answer.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

One pattern has 31 black squares.

(c) Work out the total number of squares that this pattern has. You must show clearly how you obtained your answer.

(2)

(d) Find a rule to find the number of squares, s, in terms of the Pattern number, n.(2)

s = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Pattern number 4

Pattern number 2 Pattern number 3 Pattern number 1


10 (a) On the grid, plot the graph of y + 2x = 10 for values of x from –1 to 5(3)

(b) Use your graph to find

(i) the value of y when x = – 0.7(1)

y = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) the value of x when y = 5.4(1)

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


1

2

3

4

5

6

7

8

9

10

11

12

13

14

–1

–2

1 2 3 4 5 6–1–2 O x

y


11 Simplify

(a) c + c + c(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) 2a + 3a – a(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) 2xy + 3xy – 6xy(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(d) 3a + 5b – a + 2b + 8 (2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


12 Beth says 20 – 5 × 3 is 45

Pat says 20 – 5 × 3 is 5

(a) Who is right? Give a reason for your answer.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Work out (12 + 9) ÷ 3(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) Put in brackets to make the following calculation correct.(3)

5 + 8 × 3 − 4 + 2 = 13



13 Here are two fractions, 34

and 45

Which is the larger fraction?

You must show clearly how you obtained your answer.



14 There are some beads in a bag. The beads are either green, blue or yellow.

A bead is selected at random from the bag.

The probability that the bead is blue is twice the probability that the bead is green. The probability that the bead is yellow is 0.1 more than the probability that the bead

is green.

Work out the smallest number of beads there could be in the bag.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



15

(a) Enlarge shape S, scale factor 2, centre (0, 2).(2)

(b) Describe fully the single transformation that maps shape S onto shape T.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shape S can be transformed to shape U by a rotation followed by a translation.

(c) Describe the rotation and the translation fully.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


2

4

6

8

10

–2

–4

–6

–8

–6 –4 –2 2 4 6 8 10–10 –8 O x

y

T

U

S


16 Greg thinks that he has a biased dice. He throws the dice 200 times and records the score on the dice each time.

Information about the scores on the dice is shown in the table.

Score on the dice 1 2 3 4 5 6

Frequency 23 37 33 35 20 52

(a) Is the dice biased? You must show clearly how you obtained your answer.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tess has a different dice which is known to be biased. The probability that Tess throws the biased dice once and gets a 6 is 0.7

Tess throws the biased dice 250 times.

(b) Work out an estimate for the number of times she should get a 6.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



17

ABCDEF is a hexagon with one line of symmetry. CD is parallel to AF. The distance between CD and AF is 12 cm. The interior angles at A and F are right angles. EF = 5 cm AF = 12 cm Calculate the area of the hexagon.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2


A

B

C D

E

5 cm

6 cm

12 cm

12 cm

F



18 (a) Solve 4t + 1 = 19(2)

t = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Solve 4w + 8 = 2w + 7(2)

w = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Matt has £x Sunil has twice as much as Matt. Catrin has £5 more than Sunil.

The total amount of money they have altogether is 7 times what Matt has.

(c) Work out how much Catrin has.(2)

£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



19 Given that 15.8 × 6.8 = 107.44

work out the values of

(a) 0.158 × 6.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) 1074.4 ÷ 0.158

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



20 Jim has two ordinary dice. On both dice the even numbers are coloured blue and the odd numbers are coloured red.

Each dice is thrown once.

(a) Work out the probability that the numbers on the two dice are the same.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Work out the probability of getting a blue number or the number 1(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



21 What is the largest value of x for which 23 + x is less than or equal to

34

?

Give your answer in its simplest form.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



22 The nth term of a sequence is n2 + 4

(a) Show that when n is odd, not all the terms of the sequence are prime numbers.(2)

(b) Show that when n is even, then those terms of the sequence are multiples of 4(2)



23 This is some information about a class.

There are 35 students in a class. 17 of the students study History. 20 of the students study Geography. 8 of the students study both History and Geography.

* (a) Draw a suitable diagram that shows all this information.(4)

A student is chosen at random from the class.

(b) Find the probability that this student studies neither History nor Geography.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


TOTAL FOR PAPER = 100 MARKS


Uni

t 1

Foun

dati

on:

Met

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1

5MM

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Ans

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Tota

l for

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mar

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2.

(a)

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bec

ause

it is

the

big

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reg

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0.5

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sam

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tal f

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(a

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(2

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(b

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or D

plac

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(2

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2

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for

any

clea

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Tota

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4.

(a)

18

& 6

9 1

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or b

oth

(b

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(b

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(d)(

ii)(d

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18

88

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11

69

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B1

for

eit

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or b

oth

B1 c

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Tota

l for

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mar

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5.

(a)

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diag

ram

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(c

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l for

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stio

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mar

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0 –

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no

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3 C1

for

ang

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sum

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reas

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2 is

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318

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g co

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Tota

l for

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7.

(a)

12101

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ao

(b

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321

B1 c

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(c

)

1851

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(d

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36

1 B1

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Tota

l for

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8.

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i)(b

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0 –

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130

13

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a

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lin

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3 M

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50°

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9.

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D

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(b)

41

2

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mor

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ou g

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o pa

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add

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4, o

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plet

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the

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4n+1

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whe

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Tota

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Add

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10.

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in t

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to t

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low

in t

he r

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± 0

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x=

5.4

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w in

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0.0

5)

Tota

l for

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mar

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11.

(a)

3c

1 B1

cao

(b)

4a

1 B2

cao

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for

2a

or 7

b co

rrec

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(c

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–xy

1

(d)

2a

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2To

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or Q

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5 m

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12

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Pat

wit

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1 C1

for

Pat

and

exp

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tha

t yo

u do

5 ×

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– “

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Add

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13.

15 a

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6 pa

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OR

75.043

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201543

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depe

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M1,

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for

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depe

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M1,

M1)

for

sel

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f 0.

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201543

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depe

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for

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f 2016

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Tota

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14.

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t g

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the

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prob

abili

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ello

w

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+ 0.

11

1.02

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41.0

1−

=g

= 0

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403

M1

for

sigh

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g +

2g

+ g

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1 =

1 M

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r so

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or 4

0

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5MM

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Tota

l for

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stio

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16.

(a)

200

52=–= 20

023

== 0

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Yes,

as

the

diff

eren

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n th

e re

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es

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175

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A1

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To

tal f

or Q

uest

ion:

4 m

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17

.

Area

of

rect

angl

e =

60

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of

trap

eziu

m =

63

612

72+

×=

63

123

cm2

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1 ar

ea o

f re

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gle

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× 12

M1

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of

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m =

6

12 2h

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= 7

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ao

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l for

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mar

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18.

(a)

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1

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the

num

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term

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– 2w

= 7

– 8

A1–0

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e

(c)

x +

2x +

2x

+ 5

= 7x

x =

2.50

£1

0 2

M1

for

x +

2x +

2x

+ 5

= 7x

A1 c

ao

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mar

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(i

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58

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Tota

l for

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20.

(a)

Sam

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A13632

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21.

3243

− =

128

129−

121=

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for

3243

−

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ting

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fra

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Add

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22.

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53,

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4

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Tota

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23.

QW

C(i

i, ii

i)

OR X

X /

/ *

X

X /

/ *

X

/ /

/ /

*

X /

/ /

/ *

X

/ /

/ /

*

X /

/ /

/ *

X

/ /

/ /

* *

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one

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per

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1 w

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8 in

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inte

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9 a

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2 in

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cor

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7×

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cle

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for

17 –

8 =

9,

20 –

8 =

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or 9

His

tory

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Geo

grap

hy

A1 f

or 6

nei

ther

B1 f

or 356

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Total Marks

Paper Reference

Turn over

*S38371A0121*

Edexcel GCSE

Methods in MathematicsUnit 1: Methods 1 For Approved Pilot Centres ONLY

Higher Tier


You must have:

Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used.

5MM1H/01


Instructions

• Use black ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions.• Answer the questions in the spaces provided – there may be more space than you need.• Calculators must not be used.

Information

• The total mark for this paper is 100.• The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.• Questions labelled with an asterisk (*) are ones where the quality of your written communication will be assessed – you should take particular care on these questions with your spelling, punctuation


Advice

• Read each question carefully before you start to answer it.• Keep an eye on the time.• Try to answer every question.• Check your answers if you have time at the end.


Formulae – Higher Tier


Volume of a prism = area of cross section × length Area of trapezium = (a + b)h

Volume of sphere = r3 Volume of cone = r2h

Surface area of sphere = 4 r2 Curved surface area of cone = rl

In any triangle ABC The Quadratic Equation The solutions of ax2+ bx + c = 0 where a 0, are given by

Sine Rule

Cosine Rule a2= b2+ c2– 2bc cos A

Area of triangle = ab sin C

length

crosssection

rh

r

l

C

ab

c BA

13

a b csin A sin B sin C

xb b ac

a=

− ± −( )2 42

43

12

b

a

h

12




Write your answers in the spaces provided.


You must NOT use a calculator.

1 Work out 1920

340×

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


2 (a) Write 120 as a product of its prime factors.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Find the Highest Common Factor (HCF) of 120 and 96(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



3 Here are the first four terms in an arithmetic sequence.

7 11 15 19

Is the number 203 a term in this arithmetic sequence? You must give a reason for your answer.


4 (a) Expand and simplify 2(x + 2y) + 3(x – y)(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Solve 6 = – 4(x – 1)(3)

x =. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


(c) Solve 3(y + 2) + 2y = y + 9(4)

y =. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


5 Simplify

(i) 26 × 27

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) 48 ÷ 42

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(iii) 3 33

8

6

×(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



6 Jim has two ordinary dice. On both dice the even numbers are coloured blue and the odd numbers are coloured red.

Each dice is thrown once.

(a) Work out the probability that the numbers on the two dice are the same.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Work out the probability of getting a blue number or the number 1(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



7 Here is a sequence of patterns made from centimetre squares

Pattern Number 1 Pattern Number 2 Pattern Number 3

(a) Write down the number of centimetre squares in Pattern Number 4(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Find an expression, in terms of n, for the number of centimetre squares in Pattern Number n.

(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) Can the number of centimetre squares in a pattern ever be a square number? Explain your answer.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



8 (a) On the grid, plot the graph of y + 2x = 10 for values of x from –1 to 5(3)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

–1

–2

1 2 3 4 5 6–1–2 O x

y


(b) Work out the equation of the graph that is perpendicular to y + 2x = 10 and passes through the point (1, 8).

(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



9 This is some information about a class. There are 35 students in the class. 17 of the students study History. 20 of the students study Geography. 8 of the students study both History and Geography.

(a) Draw a suitable diagram that shows all this information.(4)

A student is chosen at random from the class.

(b) Find the probability that this student studies neither History nor Geography.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



10

A

B

C D

E

F12 cm

6 cm

12 cm

5 cm

ABCDEF is a hexagon with one line of symmetry. CD is parallel to AF. The distance between CD and AF is 12 cm. The interior angles at A and F are right angles. EF = 5 cm AF = 12 cm Calculate the area of the hexagon.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .cm2



11

2

4

6

8

10

–2

–4

–6

–8

2 4 6 8 10–2–4–6–8–10 O x

y

ST

U

(a) Enlarge shape S, scale factor 2, centre (0, 2).(2)

(b) Describe fully the single transformation that maps shape S onto shape T.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shape S can be transformed to shape U by a rotation followed by a translation.

(c) Describe the rotation and the translation fully. (2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



12 Hiroshi has two boxes, A and B.

In box A are 6 green and 4 red counters. In box B are 3 green and 7 red counters.

Hiroshi takes a counter at random from box A and a counter at random from box B.

(a) Work out the probability that Hiroshi takes 2 red counters.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Work out the probability that Hiroshi takes 1 red counter and 1 green counter.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



13 Put the following numbers in order. Start with the smallest number.

6.0 × 108 3.2 × 10–2 46 × 107 0.43 × 10–3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


14 Prove, using algebra, that the sum of two consecutive whole numbers is always an odd number.



15 A

B C


DE

F

ABC is a triangle. D is the midpoint of AC. E is the point on AB such that DE is parallel CB. F is the point on BC such that DF is parallel to AB.

* (a) Prove that triangles AED and DFC are congruent.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Show that the area of the parallelogram EDFB is one half of the area of the triangle ABC.

(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



16

T


S

Q

O

R36°P

PT is a tangent to the circle, centre O at T.

R, S and Q are points on the circle. PSO is a straight line.

* (a) Work out the size of angle SRT. Give reasons for your answer.

(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °


x°48°

2x°


AB

C

D

B, C and D are three points on a circle. AB is the tangent to the circle at B. Angle ABC = x° Angle CBD = 48° Angle BCD = 2x°

(b) Find the value of x(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



17 Diagram NOTaccurately drawn

Y

X

O

P3a

6b

OX = 3a and OY = 6b.

P lies on XY such that XP: PY = 2:1

(a) Find OP, the position vector of P. Give your answer in terms of a and b.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G is the point such that YG is parallel to OP and the length of YG = length of OP.

(b) Find OG, the position vector of the point G. Give your answer in terms of a and b in its simplest form.

(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



18 (a) Rationalise the denominator of 37 (2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Expand ( )( )2 6 3 12+ +

Give your answer in the form p q2 3+ where p and q are integers.(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



19 Prove that the answer to every line of the pattern below is 8.

3 × 5 – 1 × 7 4 × 6 – 2 × 8 5 × 7 – 3 × 9

(4)


*


20 (a) Simplify 2 5 24 16

2

2

x xx− +

− (3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Complete the square for the expression

12x2 – 12x(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) Find the nth term of the quadratic sequence

1, 7, 17, 31, ...(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




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A ro

tati

on o

f 18

0o abo

ut(a

, b)

follo

wed

by

a

tran

slat

ion

of

−−

−b

a2

24

Corr

ect

rota

tion

and

tr

ansl

atio

n

M1

for

a ro

tati

on o

f 18

0o abo

ut a

sta

ted

poin

t an

d a

tran

slat

ion

cons

iste

nt w

ith

the

cent

re o

f ro

tati

on.

The

tran

slat

ion

may

be

poor

ly

desc

ribe

d

2

A1 f

or a

ful

ly c

orre

ct d

escr

ipti

on,

such

as

rota

tion

of

180o a

bout

O

follo

wed

by

a tr

ansl

atio

n of

− 04

Tota

l for

Que

stio

n: 6

mar

ks


=5M

M1H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

12.

(a)

107104

×10

028

3B1

sig

ht o

f tw

o of

107

,106

,103

,104

M1

107104

×

A1 f

or 10

028

oe

(b)

103104

107106

×+

×10

054

3M

1 107

106×

or

103104

×

M1

adds

tog

ethe

r pr

obab

iliti

es f

or G

R an

d RG

A1

oe

Tota

l for

Que

stio

n: 6

mar

ks

NPK=

0.43

×10-3

,3.

2×10

-2,

46×1

07 ,6×

108

2M

1 (n

umbe

rs p

ut in

com

mon

for

m a

t le

ast

in p

airs

acc

ordi

ng t

o si

gn o

f in

dex)

A1 c

orre

ct o

rder

Tota

l for

Que

stio

n: 2

mar

ks

14.

Proo

f 3

M1

atte

mpt

to

arri

ve a

t n

and

n +

1 (o

e)

M1

sigh

t of

2n

+ 1

(oe)

A1

exp

lana

tion

of

2n +

1To

tal f

or Q

uest

ion:

3 m

arks


=5M

M1H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

15.

(a)

QW

Cii

AD

= D

C(g

iven

)A

DE

= FC

D(c

orre

spon

ding

an

gles

for

ED

and

FC)

EAD

= F

DC

(co

rres

pond

ing

angl

es f

or B

A a

nd F

D)

Hen

ceA

ED�

DFC

(2 a

ngle

s an

d th

e co

rres

pond

ing

side

)

Proo

f 3

M1

any

3 co

rrec

t st

atem

ents

whi

ch w

ould

lead

to

a co

ngru

ence

pro

of

M1

all t

hree

sta

tem

ents

jus

tifi

ed

M1

corr

ect

reas

on f

or c

ongr

uenc

e Q

WC:

Sta

tem

ent

supp

orte

d by

co

rrec

t pr

evio

us w

orki

ng a

nd j

usti

fica

tion

(b)

Area

of

DFC

= a

rea

AED

(con

grue

nt)

FC =

ED

= B

F (c

ongr

uenc

y an

d op

posi

te s

ides

of

a pa

ralle

logr

am a

re e

qual

)

So a

rea

of p

aral

lelo

gram

ED

FB =

2 ×

are

a D

FC=

2 ×

area

AED

But

area

A

ED =

41×

area

ABC

Proo

f 2

Hen

ce s

how

n

M1

area

of

DFC

= a

rea

AED

(co

ngru

ent)

FC

= E

D =

BF

(c

ongr

uenc

y an

d op

posi

te s

ides

of

a pa

ralle

logr

am a

re

equa

l)so

are

a of

par

alle

logr

am E

DFB

= 2

× a

rea

DFC

= 2

× ar

ea

AED

A1 b

ut a

rea

AED

=41

× ar

ea

ABC

and

conc

lusi

on

Tota

l for

Que

stio

n: 5

mar

ks


5MM

1HQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e

16.

(a)

QW

C

SOT

= 90

o–

36 o

SQT

= 27

o15

3oM

1 SO

T =

90 o

– 36

ow

ith

angl

e be

twee

n ta

ngen

t an

d ra

dius

= 9

0 o4

C1 S

QT

= SO

T ÷

2 w

ith

angl

e at

cen

tre

2 a

ngle

at

circ

umfe

renc

e.

QW

C: C

orre

ct a

ttri

buta

ble

wor

king

sho

wn

wit

h re

ason

sta

ted

×SR

T =

73o

C1 S

RT =

180

– S

QT

wit

h op

posi

te a

ngle

s of

a c

yclic

qua

drila

tera

l are

su

pple

men

tary

QW

C: C

orre

ct a

ttri

buta

ble

wor

king

sho

wn

wit

h re

ason

sta

ted

A1 f

or 1

53o ca

o

(b)

BDC

= x

44o

3B1

BD

C =

xx

x+

+M

1 0

248

182

4818

0x

x+

+=

=A1

cao

To

tal f

or Q

uest

ion:

7 m

arks

17

.(a

)→

→=

OX

OP

+

→ XP =

3a+

32(6

b–

3a)

= 4b

+a

4b+

a3

M1

→ XY =

6b

– 3a

M1

→ XP =

32(6

b–

3a)

A1 c

ao

(b)

→ OG

=

=

→→

+YG

OY

6b+

a+

4b =

a +

10b

a +

10b

3M

1

=(p

aral

lelo

gram

) → OP

→ YG

M1

= 6b

+ ‘

a +

4b’

→ OG

A1 f

t on

(a)

Tota

l for

Que

stio

n: 6

mar

ks


=5M

M1H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

18.

(a)

7773

×77

32

M1

7773

×

A1 c

ao

(b)

23

212

1872

++

+

26

23

34

32

++

+

29

36

+4

M1

corr

ect

exp

or 3

out

of

4 te

rms

corr

ect

M

1 72

362

=×

B12

672

= o

r 3

212

=A1

cao

Tota

l for

Que

stio

n:6

mar

ks

19.

QW

C(i

ii)

The

nth

line

of t

he

patt

ern

is:

(n+

2)(n

+ 4)

–n(

n+

6)

Proo

f su

ppor

ted

by

attr

ibut

able

wor

king

. 4

M1

for

(n +

2)(

n +

4)M

1 fo

r –n

(n +

6)

A1 f

or c

orre

ct s

impl

ific

atio

n B1

con

clus

ion

QW

C: A

ppro

pria

te c

oncl

usio

n su

ppor

ted

by

attr

ibut

able

wor

king

.To

tal f

or Q

uest

ion:

4 m

arks


5MM

1H/X

Q

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 20

.(a

)B1

)2)(1

2(−

−x

x)2

)(12(

25

22

−−

=+

−x

xx

x

)2)(2

(4)4

(416

42

2+

−=

−=

−x

xx

x)2

(41

2+−

xx3

B1 o

r )2

)(2(4

+−

xx

)42

)(42(

+−

xx

B1

cao

M1

for

12x2 –

12x

= 2

(x2 –

6x)

2(x

– 3)

2 – 9

(b)

12x2 –

12x

= 2

(x2 –

6x)

3

=

2[(

x –

3)2 –

(3)2 ]

M1

for

Z=2

[(x

– 3)

2 – (3

)2 ]

= 2

(x –

3)2 –

9

A1 f

or ===================Z

=2(x

– 3

)2 – 9

2n2 –

1

(c)

1

7

1

7

31

3

M1

for

usin

g m

etho

d of

dif

fere

nces

(oe

)

6

10

14

4

4

A1 f

or s

ight

of

2n2

2nd

diff

eren

ce is

4,

so u

se

2n2

= =

n =

1

2

=

n =

2

8

n =

3

18

n =

4

32

nth

term

is 2

n2 – 1

A1 c

orre

ct g

ener

alis

atio

n To

tal f

or Q

uest

ion:

9 m

arks


Unit 2: Methods 2




Total Marks

Paper Reference

Turn over

*S38370A0122*

Edexcel GCSE

Methods in MathematicsUnit 2: Method 2For Approved Pilot Centres ONLY

Foundation Tier


You must have:

Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.

5MM2F/01

Instructions• Use black ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions.• Answer the questions in the spaces provided – there may be more space than you need.• Calculators may be used.

• If your calculator does not have a π button, take the value of π to be 3.142 unless the question instructs otherwise.







Formulae: Foundation Tier


Area of trapezium = (a + b)h

Volume of prism = area of cross section × length

b

a

h

length

crosssection

12





1

The arrow points to a number.

(a) What has to be added to the number to make 6?(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Write down the number that is 1000 times the number the arrow points to.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


2 Find the number that should be written in the empty boxes.

(i)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


0 1 2 3 4 5 6

+ – =38 64 77

× ÷ =2.3 0.765.75


3

This shape has been drawn on a grid of centimetre squares.

(a) Find the area of this shape.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) On the grid, draw a rectangle that has the same area as this shape.(2)


(c) Work out the area of this triangle.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



13 cm5 cm

12 cm


4 Anisha has four cards, with a number written on each card.

Anisha puts all four cards on the table to make a number.

(a) (i) Write the numbers on the cards to show the smallest number Anisha can make with the four cards.

(ii) Write the numbers on the cards to show the largest number Anisha can make with the four cards.

(2)

Anisha uses the cards to make a true statement.

(b) Write the number on the cards to make this true. Use each of Anisha’s cards once.

(1)

A fifth card is needed to show the result of the multiplication 4915 × 10 She needs a fifth card.

(c) Write the number that should be on the fifth card.(1)


4 9 1 5

+ =


5 Crystal wrote down the temperature at different times on 1st January 2009.

Time Temperature

4 am – 10°C

8 am – 4°C

noon 7°C

3 pm 6°C

7 pm – 2°C

midnight – 6°C

(a) Write down(2)

(i) the highest temperature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °C

(ii) the lowest temperature.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °C

(b) Work out the difference in the temperature between(2)

(i) 4 am and 8 am

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °C

(ii) 3 pm and 7 pm.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °C



6 Imran thinks of a number. He multiplies the number by 3 He then adds 19 His answer is 61

What number did Imran first think of?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


7 Find the reciprocal of:

(i) 0.2(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) 0.2 × 1

5 (2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



8

Arthur, Beth and Charlie each have some marbles. They keep their marbles in boxes each with their own name on.

Arthur has a marbles, Beth has b marbles and Charlie has c marbles.

(a) Write down an expression, in terms of a, b and c, for the total number of marbles that they have altogether.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Beth puts 4 of her marbles and Charlie puts 2 of her marbles into Arthur’s box. Charlie also puts 1 of her marbles into Beth’s box.

*(b) Complete the diagram to show expressions for the number of marbles in each box. Give your answers in their simplest forms.

(4)

After this has been done the number of marbles in Arthur’s box is the same as the number of marbles in Beth’s box and is the same as the number of marbles in Charlie’s box.

(c) Find an expression, in terms of a, for the total number of marbles that they have.(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Arthur

a

Beth

b

Charlie

c

Arthur Beth Charlie


9 (a) Find two square numbers which have a sum which is also a square number.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Find two cube numbers which have a difference which is a prime number.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


10 (a) Shade1

4of this shape.

(1)

(b) Work out the number that is half way between(3)

(i) 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

(ii) 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6

(iii) 1

4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

(c) Find the point on the line AB that is 1

3of the way along the line from A.

Mark this point with a cross (×).(1)


A B


11

Work out the size of the angle marked x. You must show clearly how you obtained your answer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °


Diagram NOT260°accurately drawn

130°

70° x


12 The diagram shows a mathematical rule.

It multiplies the input by 3 and then subtracts 3 to get the output.

(a) Complete the diagram.(1)

(b) Complete the diagram.(1)

In this diagram the input and the output have the same value, t

(c) Work out the value of t.(2)

t = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

input × 3 – 3 output

10 × 3 – 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117× 3 – 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t× 3 – 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .t


In this diagram the input is x and the output is y

(d) Write x in terms of y(2)


y× 2 + 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x


13 (a) Write1

20as a decimal.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Find7

25 as a percentage.

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Danny shares a bag of sweets with his friends.

He gives Mary3

5of the sweets.

He gives Ann1

10of the sweets.

He has 12 sweets left.

(c) How many sweets does Danny give to Mary?(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



14

(a) What special name is given to this polygon?(1)

………………………….. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .………..

One of the polygons below is congruent to the polygon shown in part (a).

(b) Draw a circle around this polygon.(1)

(c) On the grid, show how the shape below will tessellate. You should draw at least 8 shapes.

(2)



15

XYZ is an isosceles triangle. PQRS is a square.

PQ = XY

XZ = YZ

XZ is 14 cm longer than PQ.

The perimeter of both shapes is the same.

Find the length of XZ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm



X Y

Z

P Q

RS


16 Here is a triangular prism.

(a) Calculate the volume of the triangular prism.(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm3

A cube has the same volume as this triangular prism.

(b) Calculate the surface area of the cube. Give your answer correct to the nearest whole number.

(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2



8 cm

9 cm

14 cm


17

In the diagram, AD = BD = BE.

ADEC is a straight line. Angle DBA = 28° Angle CBF = 32°

BF is parallel to AC.

Find the size of the angle marked y You must show clearly how you obtained your answer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °


D E C

BF

32°y


28°

A


18

The diagram shows a circle enclosed by a square. Work out the area of the region shaded in the diagram. Give your answer correct to one decimal place.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2



12 cm

12 cm


19 (a) Solve 3y + 4 = 2 – 2y(2)

y = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Solve 3(2x – 4) + x = 2x – 25(4)

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



20 There were 800 students at Prestwick School at the start of the year.

48% of these 800 students were girls.

(a) What is the number of girls attending the school?(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

There are 176 students in Year 10.

(b) What percentage of the students at the school are Year 10 students?(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %

By the end of the year1

6of the girls had left and 12

1

2% of the boys had left.

(c) Work out the number of students at Prestwick School at the end of the year.(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



21

ABCDP is a regular pentagon.QPEX is part of a regular octagon.PARQ is a parallelogram.

Calculate the size of angle PAR.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °



R

A

B

D E

P

Q

X



Uni

t 2

Foun

dati

on:

Met

hods

2

5MM

2FQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 1.

(a)

2.8

2M

1 6

– 3.

2 or

6 –

3.4

A1

cao

(b

)32

00

1B1

cao

Tota

l for

Que

stio

n: 3

mar

ks

2.(i

)

(ii)

38 +

64

= 10

2 10

2 –

77 =

25

2.3

÷ 5.

75 =

0.4

0.

76 ÷

0.4

= 1

.9

25 1.9

2 2

M1

38 +

64

and

‘102

’ –

77

A1 c

ao

M1

2.3

÷ 5.

75 a

nd 0

.76

÷ ‘0

.4’

A1 c

ao

Tota

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Que

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3.(a

)18

cm

22

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show

s cl

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at is

cou

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g sq

uare

s or

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es a

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atte

mpt

s b

×h

÷ 2

A1 c

ao

(b)

rect

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e 2

by 9

or

3 by

6

2M

1 re

ctan

gle

draw

n A1

9 b

y 2

or 6

by

3

(c)

25

12×

30 c

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2M

1 2

512

×

A1 c

ao

Tota

l for

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stio

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ks

4.(a

) (i

)

(ii)

14

59

9541

2

B1 c

ao

B1 c

ao t

rans

pose

d an

swer

s aw

ard

one

mar

k (b

)9

+ 5

= 14

1

B1 f

or 9

+ 5

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4 or

5 +

9 =

14

(c)

01

B1 c

ao

Tota

l for

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5MM

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(a)(

i)

(ii)

7 −10

2B1

cao

B1

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)(i)

(i

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6 82

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t −

6 B1

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− 8

Tota

l for

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stio

n: 4

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6.61

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= 4

2 42

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4 14

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1 fo

r –1

9 an

d ÷

3 or

42

seen

A1

cao

To

tal f

or Q

uest

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2 m

arks

7.

(i)

(ii)

5 253

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ao

M1

0.04

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l for

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stio

n: 3

mar

ks

8.(a

)a

+ b

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1

B1 c

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QW

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i)(b

)a

+ 6

b –

3 c

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4B4

all

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ect

QW

C: c

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and

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ons

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all c

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(B

2 an

y 2

corr

ect)

(B

1 an

y 1

corr

ect)

(c

)b

= a

+ 9

c =

a +

93a

+ 1

8 3

M1

for

b =

a +

9 o

r c

= a

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for

a s

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men

t th

at A

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less

tha

n Be

th (

or C

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A1 f

or a

cor

rect

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r b

or c

in t

erm

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aA1

any

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ress

ion

that

wou

ld s

impl

ify

to 3

a +

18To

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8 m

arks

9.

(a)

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s sq

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bers

1,

4, 9

, 16

, 25

9

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ts s

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at

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cube

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bers

1 8

1

and

8 2

M1

for

1 an

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A1 1

and

8 a

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ent

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7 is

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me

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=5M

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Que

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Add

itio

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10.

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1B1

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(b)(

i) (

ii)

(iii)

40 6.55

83

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B1

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B1 0

.375

, 83 o

e

(c)

Mar

k 2.

1-2.

5 cm

fro

m A

1

B1 f

or m

ark

2.1-

2.5

cm f

rom

A

Tota

l for

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ks

1136

0 –

260

= 10

0 10

0 +

70 +

130

= 3

00

360

– 30

0 =

60

180

– 60

= 1

20

120o

4B1

for

100

cle

arly

sho

wn

in t

he c

orre

ct p

lace

M

1 fo

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0 –

(100

+ 7

0 +

130)

M

1 fo

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0 –

‘60’

A1

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n: 4

mar

ks

12.

(a)

271

B1 c

ao

(b)

401

B1 c

ao

(c)

3t–

3 =

t1.

52

M1

3t–

3 =

tA1

cao

(d

)y

x=

+4

24

2−

=y

x2

4−

=y

x2

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for

a cl

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to

wor

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thr

ough

the

num

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by

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on t

he d

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am o

r fr

om

24

÷−

=y

xA1

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tal f

or Q

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6 m

arks


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.(a

)0.

05

1B1

cao

(b

)

10028

44257

=×

28%

1B1

cao

(c)

107101

53=

+

103107

1=

− 103 o

f th

e to

tal =

12

103

12×

÷ =

40

244

B1107

M1

1071

− o

f th

e to

tal i

s 12

sw

eets

M1

'10

3'12

×÷

A1 2

4

Tota

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ks

14.

(a)

Hex

agon

1

B1 c

ao

(b)

The

one

on

the

far

righ

t 1

B1 c

ao

(c)

Tess

ella

tion

2

B2 f

or a

t le

ast

7 ad

diti

onal

sha

pes

draw

n (B

1 fo

r at

leas

t 5

addi

tion

al s

hape

s dr

awn,

or

shap

es o

nly

on o

ne li

ne,

or in

com

plet

e te

ssel

lati

on)

Tota

l for

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ks


5MM

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Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 15

.Le

tX

Y =

xx

xx

x4

1414

=+

++

+28

=x

423

B1 o

r 14

14+

++

+x

xx

x4M

1 x

xx

x4

1414

=+

++

+A1

cao

Tota

l for

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stio

n: 3

mar

ks

16.

(a)

89

362×

=

3614

504

×=

504

2M

1 8

936

2×=

, o

r '3

6'14×

214

98

÷×

×

A1 c

ao

(b)

350

4x

=x

= 7.

958(

1..)

Su

rfac

e ar

ea =

6 ×

7.9

58(1

..)2

380

3M

1 x

=3

504

M1

6 ×

‘7.9

58(1

..)2 ’

A1 c

ao

Tota

l for

Que

stio

n: 5

mar

ks

17.

(a)

BAD

= 28

°

BDE

= 56

°

BED

= 5

6°

EBF

= 56

°

y =

56° –

32°

OR BA

D =

28°

BDE

= 56

°

DBE

= 6

8°

ECB

= 32

°

y =

180°

– 2

8° – 28

° – 6

8° – 3

2°

24o

4B1

BAD

= 2

8°

M1

BDE

= 56

°

M1

EBF

= 56

°

A1 c

ao

OR

B1BA

D =

28°

M1

BDE

= 56

°

M1

ECB

= 32

°

A1 c

ao

Tota

l for

Que

stio

n: 4

mar

ks


5MM

2FQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 18

.Ci

rcle

:×

6 ×

6 =

113.

097.

. 14

4 –

113.

097…

= 3

0.90

3 30

.903

÷ 4

7.73

4

M1

for

× 6

× 6

B1

for

144

M

1 14

4 –

‘113

.097

’…

A1 a

nsw

ers

in r

ange

7.7

– 7

.8

Tota

l for

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stio

n: 4

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ks

19.

(a)

5y =

– 2

y

= –

522

M1

corr

ect

proc

ess

of is

olat

ing

the

term

s in

y

A1–

52oe

(b)

252

126

−=

+−

xx

x12

252

6+

−=

−+

xx

x

513−

=x

513−

=x

4B1

2512

6−

=−

xx

M1

corr

ect

proc

ess

of is

olat

ing

the

term

s in

xfr

om a

4 o

r 5

term

eq

uati

onM

1 co

llect

ter

ms

in x

on

one

side

A1513

−=

x o

e

Tota

l for

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stio

n: 6

mar

ks

20.

(a)

800

100

48×

= 38

438

4 2

M1

80

010

048×

A1 c

ao

(b)

100

800

176

×22

2M

110

080

017

6×

A1 c

ao

(c)

384

÷6 =

64

1005.

12)

384

800

(×

− =

52

800

– 64

– 5

2 =

684

684

4M

1 38

4 ÷6

M1

1005.

12)'

384

'80

0(

×−

M1

800

– ‘6

4’ –

‘52

’A1

cao

Tota

l for

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stio

n: 8

mar

ks


5MM

2FQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 21

.Ex

t an

gle

of a

pen

tago

n is

72

o

Ext

angl

e of

oct

agon

is 4

5o

APQ

= 72

+ 4

5= 1

17o

PAR

= 1

80 –

117

= 6

3

OR

APE

= 1

08o

QPE

= 1

35o

APQ

= 36

0 –

108

– 13

5 =

117o

PAR

= 18

0 –

117

= 63

63o

7M

1 fo

r 5360

or

8360

A1,

A1 f

or 7

2 an

d 45

M

1 fo

r ‘7

2’ +

’45

’ A1

for

117

M

1 fo

r 18

0 –

‘117

’ A1

cao

O

R

M1

905

410

×−

or

908

416

×−

A1,

A1 f

or 1

08 a

nd 1

35

M1

for

360

– 10

8 –

135

A1 f

or 1

17

M1

for

180

– ‘1

17’

A1 c

ao

Tota

l for

Que

stio

n: 7

mar

ks

= = =




Total Marks

Paper Reference

Turn over

*S38372A0122*

Edexcel GCSE

Methods in MathematicsUnit 2: Methods 2 For Approved Pilot Centres ONLY

Higher Tier


You must have: Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.

5MM2H/01


Instructions• Use black ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answer all questions.• Answer the questions in the spaces provided – there may be more space than you need.• Calculators may be used.• If your calculator does not have a π button, take the value of π to be

3.142 unless the question instructs otherwise.






Formulae – Higher Tier


Volume of a prism = area of cross section × length Area of trapezium = (a + b)h

Volume of sphere = r3 Volume of cone = r2h

Surface area of sphere = 4 r2 Curved surface area of cone = rl

In any triangle ABC The Quadratic Equation The solutions of ax2+ bx + c = 0 where a 0, are given by

Sine Rule

Cosine Rule a2= b2+ c2– 2bc cos A

Area of triangle = ab sin C

length

crosssection

rh

r

l

C

ab

c BA

13

a b csin A sin B sin C

xb b ac

a=

− ± −( )2 42

43

12

b

a

h

12





1 Divide £60 in the ratio 2:3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


2 The diameter of a circle is 15 cm.

Find its circumference. Give the units of your answer. Give your answer correct to 1 decimal place.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



3 Here is a triangular prism

(a) Calculate the volume of the triangular prism.(2)


8 cm

9 cm

14 cm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm3

A cube has the same volume as this triangular prism.

(b) Calculate the surface area of the cube. Give your answer correct to the nearest whole number.

(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2



4 (a) n is an integer and –3 < n 2

Write down all the possible values of n(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Here is a number line.

(b) Display the inequality –3 < x 2 on the number line.(2)


1 2 3 4–1–2 –3–4 0 x


5

d

l

The diagram is a shape made up of a rectangle of width d and length l together with a semicircle of diameter d.

The perimeter P of the shape is given by P d l d= + +22

π

(a) Work out the perimeter of the shape when the width is 8.4 cm and the length is 11.7 cm.

(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm

(b) Rearrange the formula to make d the subject. (3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



6

A D E C

B F

32°28° y


In the diagram, AD = BD = BE

ADEC is a straight line. Angle DBA = 28° Angle CBF = 32° BF is parallel to AC.

Find the size of the angle marked y. You must show clearly how you obtained your answer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °



7

A

R

B

D E

P

Q

X


ABDEP is a regular pentagon. QPEX is part of a regular octagon. PARQ is a parallelogram.

Calculate the size of angle PAR.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °



8 The quadratic curve with equation y = x2 + bx + c passes throughthe points (2, – 4) and (–1, 2)

Find the values of b and c

b = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

c = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



9 Here are four graphs.

x

y

x

y

x

y

x

y

A B

C D

Here are five equations of graphs.

P y = sin x Q y = cos x

R y = 2xS y =

x1

T y = x2

Match each graph with an equation.

A with equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B with equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C with equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D with equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



10 y is directly proportional to the square root of x and y = 8 when x = 25

(a) Find a formula for y in terms of x(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

y = 36

(b) Find the value of x(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



11


5 cm

34 cm

30 cm

28 cm

AB

C D

E

Here is the design for a logo, ABCD. The design is made from two right-angled triangles.

(a) Work out the perimeter of the logo.(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm



5 cm

34 cm

30 cm

28 cm

A

B

C

D

E

The logo is turned to rest on the ground at the corners A and D.

(b) Calculate the size of angle BAD.(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °



12 On a farm, 38

of the land area is used to keep sheep.

Half of the rest of the land area on the farm is used to grow crops.

The land area of the farm used to grow crops is 600 000 m2.

Work out the land area of the farm used to keep sheep.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m2



13 (a) Prove that the recurring decimal 0 124 41330

• •=

(3)

(b) Hence or otherwise, express as a fraction, the recurring decimal 0 0621.• •

(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



14

2

x + 4

x


x + 3

The diagram shows a hexagon. All the angles are right angles. All the measurements are in centimetres.

The area of the hexagon is 50 cm2.

(a) Show that x2 + 4x – 44 = 0(3)

(b) (i) Solve the equation x2 + 4x – 44 = 0(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) Write down the length of the longest side.(1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .cm



15 (a) Find the set of values of x for which 3x – 5 < x + 8(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(b) Hence or otherwise, find the set of values of x for which

3x – 5 < x + 8 and 5x > x – 8(3)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



16 Find two numbers which have a difference of 3 and have a product of 88(5)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



17 Solve x2 + y2 = 34

y – x = 8(5)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



18

2

4

6

8

10

12

14

16

1 2 3 4 5–1–2 0 x

y

Here is the graph of y = f(x) = x2 − 4x + 7 for values of x from −2 to 4

(a) On the grid draw the graph of y = f(x+1) for values of x from −2 to 3(2)

The algebraic equation of another graph is y = g(x) = x2 − 3

y = f(x) can be mapped onto y = g(x) by a transformation.

* (b) Describe fully this transformation. You must show clearly how you obtained your answer.

(4)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



19 (a) Show that (mn – 1)2 + (m + n)2 = (m2 + 1)(n2 + 1)(3)

The two shorter sides of a right-angled triangle are 2m and m2 – 1 where m is a positive integer.

(b) Show that the hypotenuse is always a positive integer.(2)

(c) Find two square numbers which have a sum equal to 50005(2)


*


20

70o

A

B

C

D


9 cm

10.5 cm

7.5 cm

80o

In the quadrilateral ABCD, AD = 10.5 cm AB = 9 cm CD = 7.5 cm Angle BAD = 70° Angle BCD = 80°

Calculate the size of angle DBC.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . °


TOTAL FOR PAPER = 100 MARKSEND


Uni

t 2

Hig

her:

Met

hods

2

5MM

2HQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 1.

)32(

60+

÷ =

12

12×

3

12×

4

24,

36

3 M

1 )3

2(60

+÷

A1 c

ao

A1 c

ao

Tota

l for

Que

stio

n: 3

mar

ks

2.

15

π×

=47.

1(2…

) 47

.1 c

m

3 M

1 15

π×

=47.

1(2…

) A1

cao

B1

(in

dep)

cm

To

tal f

or Q

uest

ion:

3 m

arks

3.

(a

) 8

936

2×=

3614

504

×=

504

2 M

1 8

936

2×=

,'3

6'14×

or

214

98

÷×

×

A1 c

ao

(b

) 3

504

x=

x =

7.95

8(1.

.)

Surf

ace

area

= 6

× 7

.958

(1..

)2

380

3 M

1 x

=3

504

M1

6 ×

‘7.9

58(1

..)2 ’

A1 c

ao

Tota

l for

Que

stio

n: 5

mar

ks

4.

(a)

–2

,–1

, 0,

1,

2

2 B2

cao

(B

1–1

, 0,

1,

2 or

–2,

–1,

0, 1

)

(b)

Co

rrec

t nu

mbe

rlin

e

2M

1 lin

e fr

om (

–3 t

o +2

) A1

wit

h op

en c

ircl

e at

−3

and

clos

ed c

ircl

e at

2

Tota

l for

Que

stio

n: 4

mar

ks


=5M

M2H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

5.

(a)

P =

8.4

+ 2

× 11

.7 +

×

8.4

÷ 2

45

.0

2 M

1 fo

r 8.

4 +

2 ×

11.7

+

× 8.

4 ÷

2 A1

44.

9 –

45.0

(b

) 2

2d

dl

Pπ

+=

−

)2

1(2

π+

=−

dl

P2

1

2 π+−

=l

Pd

3M

1 fo

r 2

2d

dl

Pπ

+=

−

M1

for

)

21(

2π

+=

−d

lP

A1

21

2 π+−

=l

Pd

oe

SC If

is

tak

en t

o be

3.1

4 th

en a

llow

1 m

ark

for

57.22l

Pd

−=

oe

Tota

l for

Que

stio

n: 5

mar

ks

6.

BA

D =

28°

BDE

= 56

°

BED

= 5

6°

EBF

= 56

°

y =

56°

– 32

°

OR

BAD

= 2

8°

BDE

= 56

°

DBE

= 6

8°

ECB

= 32

°

y =

180°

– 2

8°–

28°

– 68

°–

32°

24°

4 B1

BA

D =

28°

M1

BDE

= 56

°

M1

EBF

= 56

°

A1 c

ao

OR

B1BA

D =

28°

M1

BDE

= 56

°

M1

ECB

= 32

°

A1 c

ao

Tota

l for

Que

stio

n: 4

mar

ks


=5M

M2H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

7.

Ext

angl

e of

a p

enta

gon

is 7

2o

Ext

angl

e of

oct

agon

is 4

5o

APQ

= 72

+ 4

5 =

117o

PAR

= 1

80 –

117

= 6

3

OR

APE

= 1

08o

QPE

= 1

35o

APQ

= 36

0 –

108

– 13

5 =

117o

PAR

= 18

0 –

117

= 63

63o

7M

1 fo

r 5360

or

8360

A1,

A1 f

or 7

2 an

d 45

M

1 fo

r ‘7

2’+’

45’

A1 f

or 1

17

M1

for

180

– ‘1

17’

A1 c

ao

OR

M1

905

410

×−

or

908

416

×−

A1,

A1 f

or 1

08 a

nd 1

35

M1

for

360

– 10

8 –

135

A1 f

or 1

17

M1

for

180

– ‘1

17’

A1 c

aoTo

tal f

or Q

uest

ion:

7 m

arks

8.

42

22−

=+

+c

b8

2−

=+

cb

2)1

()1

(2

=+

−+

−c

b 3=

+−

cb

3−=

b2−

=c

5C1

42

22−

=+

+c

bM

1 3

=+

−c

bM

1 co

rrec

t m

etho

d to

elim

inat

e b

or c

A13−

=b

A12−

=c

Tota

l for

Que

stio

n: 5

mar

ks

9.

AS

BP

C R

D T

3 B3

all

corr

ect

(B2

3 co

rrec

t)

(B1

2 co

rrec

t)

Tota

l for

Que

stio

n: 3

mar

ks


=5M

M2H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

10

a x

ky

=25

8×

=k 58

=k

xy

6.1=

3M

1 x

ky

=M

1 25

8×

=k

A158

=k

b

x×

=6.1

36

2

6.136=

x

506.

25

2 M

1 x

×=

6.136

A1 c

ao

Tota

l for

Que

stio

n: 5

mar

ks


=5M

M2H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

11.

(a)

256

900

1156

3034

22

=−

=−

1625

6=

EC=

28

– 16

= 1

2

169

144

2512

52

2=

+=

+13

169

=

30 +

28

+ 5

+ 13

+ 3

4

110

4 M

1 2

230

34−

M1

(dep

) EC

= 2

8 –

‘16’

M1

16

914

425

'12'

52

2=

+=

+A1

110

OR

M1

for

34si

nBE

A=

,BE

= 3

4 ×

sin

28.

07

A1 f

or B

E =

16,

CE =

26

–BE

= 1

2, D

E =

13)

125(

22

=+

M1

for

22

125

+=

169

A1 1

10

(b

)

285

30ta

n−

=BA

D

12.1ta

n1−

=BA

D

48.2

°4

M1

reco

gnit

ion

of a

rig

ht-a

ngle

d tr

iang

le w

ith

side

s 30

– 5

and

28

M1

285

30ta

n−

=BA

D

M1

12.1ta

n1−

=BA

DA1

48.

2 or

bet

ter

O

R

Angl

e CE

D =

125

tan

1−=

22.6

2º

Angl

e BE

A =

1630ta

n1−

= 61

.93º

Angl

e D

EA =

180

º –

22.6

2º –

61.9

3º

AD =

54.

3745.

95co

s34.

13.213

342

2=

−+

Angl

e D

AE =

20.

17º

Angl

e BA

E =

28.0

7º

Angl

e BA

D =

48.

2º

Tota

l for

Que

stio

n: 8

mar

ks


=5M

M2H

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

12.

8583

1=

− 5 16of

the

land

are

a =

6000

00

Land

are

a =

1660

0000

5×

= 19

2000

0

Shee

p ar

ea =

3

1920

000

8×

7200

00

3 M

1 5 16

of t

he la

nd a

rea

= 60

0000

M1

land

are

a =

16'6

0000

0'5

×

A1 c

ao

Tota

l for

Que

stio

n: 3

mar

ks

13.

(a)

x =

0.12

4242

4…

10x

= 1.

2424

…

1000

x =1

24.2

424…

99

0x =

123

(o

e)

123

4199

033

0x

==

(oe)

3 M

1 x

= 0.

1242

424…

M

1 10

xan

d 10

00x

wit

h su

btra

ct (

or x

and

100

x w

ith

subt

ract

)

A1 c

orre

ct

33041

990

123

=

(b

) 41

233

0÷

41 660

1 B1

cao

Tota

l for

Que

stio

n: 4

mar

ks


5MM

2HQ

uest

ion

Wor

king

Ans

wer

Mar k

Add

itio

nal G

uida

nce

14.

(a)

)4(

+xx

+2

3× =

50

506

42

=+

+x

x OR

50)2

()3

(2=

++

+x

xx

502

62

2=

++

+x

xx

AG

3 M

1 sp

lit a

rea

into

rec

tang

les

wit

h co

rrec

t ex

pres

sion

s fo

r ar

eas

A1 e

xpan

d co

rrec

tly

A1 s

et =

50

and

conc

lude

(b

)(i)

(ii)

2)

44(

14

44

2−

××

−±

−=

x

219

24

±−

=x OR

442

)2(

22

−−

+x48

)2(

±=

+x

4.93

or

–8.9

3

8.93

4M

1 )

)((

bx

ax

++

wit

h ab

= -

45

A1)5

)(9(

−+

xx

A1 c

ao

OR

M1

2)

44(

14

44

2−

××

−±

−=

x a

llow

sig

n er

rors

in b

and

c

M1

219

24

±−

=x

A1 OR

M1

442

)2(

22

−−

+x

M1

48)2

(±

=+x

A1x

= 4

.93

or

– 8.

93 (

both

‘or

bet

ter’

)

B1 c

ao

Tota

l for

Que

stio

n: 7

mar

ks

15.

(a)

3x–

x<

8 +

5 2x

< 5

x

<

213

or

6.5

x<

6.5

2 M

1

A1

(b

) 5x

> x

– 8

4x>

8

x>

2

Corr

ect

ineq

ualit

y di

agra

m

–2 <

x <

6.5

–2 <

x <

6.5

3

B1 f

or c

orre

ct s

olut

ion

of in

equa

lity

M1

for

corr

ect

ineq

ualit

y di

agra

m

A1 c

ao

Tota

l for

Que

stio

n: 5

mar

ks

=


5MM

2H/X

Q

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 16

.

88)3

(=

−xx

088

32

=−

−x

x0

)11

)(8(

=−

+x

x,8−

=x

11=

x

OR

88)3

(=

−xx

088

32

=−

−x

x

2)

88)(1(4

93

−−

±=

x

2193

±=

x

–8,

11

5 M

1 fo

r se

ttin

g up

equ

atio

n A1

for

cor

rect

sim

plif

icat

ion

M1

for

corr

ect

fact

oris

atio

n A1

cao

A1

cao

OR

M1

for

sett

ing

up e

quat

ion

A1 f

or c

orre

ct s

impl

ific

atio

n

M1

for

corr

ect

subs

titu

tion

into

for

mul

a A1

cao

A1

cao

Tota

l for

Que

stio

n: 5

mar

ks

17.

x

y+

=8

0)5

)(3(

030

162

034

)8(

34

2

222

=+

+=

++

+−

++

=+

xx

xx

xx

yx

3,5

5,3

=−

==

−=

yx

yx

5M

1 fo

r re

arra

ngin

g8

=−

xy

and

for

subs

titu

ting

into

342

2=

+y

xA1

for

cor

rect

sim

plif

icat

ion

M1

for

corr

ect

fact

oris

atio

n A2

cao

for

tw

o co

rrec

t pa

irs

of v

alue

s

A1 c

ao f

or e

ithe

r bo

th x

val

ues

corr

ect

or b

oth

y va

lues

cor

rect

Tota

l for

Que

stio

n: 5

mar

ks

=


=5M

M2H

/X

Que

stio

nW

orki

ngA

nsw

erM

ark

Add

itio

nal G

uida

nce

18.

(a)

Co

rrec

t gr

aph

2 M

1 tr

ansl

atio

n pa

ralle

l to

the

x-ax

isA1

by

-1 u

nits

Q

WC

ii,

iii

(b)

7)

(4)

(2

++

−+

ax

ax

44

22

2+

−−

++

ax

aax

x2

=a

Tran

slat

ion

by

−−62

4 M

1 re

plac

e x

by x

+ a

in f(

x +

a)M

1 ex

pand

cor

rect

ly

A1a

= 2

A1 c

ao Q

WC:

All

calc

ulat

ions

are

att

ribu

tabl

e, w

ith

wor

king

cle

arly

la

id o

ut in

a c

lear

pro

gres

sion

of

proc

ess To

tal f

or Q

uest

ion:

6 m

arks

19

.Q

WC

ii,

iii

(a)

12

)1(

22

2+

−=

−m

nn

mm

n

mn

mn

m2

)(

22

2+

+=

+Ad

ding

2

22

21

nm

nm

++

+1

22

22

++

+n

mn

m)1

(1)1

(2

22

++

+n

nm

Proo

f3

B11

2)1

(2

22

+−

=−

mn

nm

mn

and

m

nn

mn

m2

)(

22

2+

+=

+B1

22

22

1n

mn

m+

++

B1 c

onvi

ncin

g fa

ctor

isat

ion

to g

ive

)1)(1

(2

2+

+n

mQ

WC:

All

calc

ulat

ions

are

att

ribu

tabl

e, w

ith

wor

king

cle

arly

laid

out

in a

cle

ar

prog

ress

ion

of p

roce

ss

(b

) Se

tm

=n

in t

he a

bove

an

d ge

t

=+

−2

22

)2(

)1(

mm

22

)1(

+m From

Pyt

hago

ras,

the

th

ird

side

is )1(

2+

m

Proo

f 2

M1

Set

m =

n in

the

abo

ve a

nd g

et:

=+

−2

22

)2(

)1(

mm

22

)1(

+m

C1 F

rom

Pyt

hago

ras,

the

thi

rd s

ide

is)1

(2

+m

(c

) 5

× 10

001

m=

2, n

= 1

00

1992 an

d 10

222

M1

set

m =

2,

n =

100

in (

a)A1

992 a

nd 1

022

Tota

l for

Que

stio

n: 7

mar

ks


5MM

2HQ

uest

ion

Wor

king

Ans

wer

Mar

kA

ddit

iona

l Gui

danc

e 20

.70

cos

5.10

92

5.10

92

2×

×−

+=

(BD

2 )19

1.25

– 6

4.64

= 1

26.6

'25.

11'80

sin

5.7si

n=

DBC

×=

−

'25.

11'80

sin

5.7si

n1

DBC

= 41

.0(3

…)

41.0

6

M1

70co

s5.

109

25.

109

22

××

−+

M1

corr

ect

orde

r of

eva

luat

ion

to g

et (

BD2 )

A1 1

1.3

or b

ette

r (1

1.25

2…)

M1

'25.

11'80

sin

5.7si

n=

DBC

M1

×=

−

'25.

11'80

sin

5.7si

n1

DBC

A1 4

1.0

or b

ette

r

Tota

l for

Que

stio

n: 6

mar

ks


= = = = = = OMK==

= = = = = =

OQSUNMNONQNS

OQ=

JOM

ñ

ó

================================================================================March 2010

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sample assessment material methods of mathematics unit 1

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