a level further mathematics sample assessment materials - draft

A Level Further Mathematics

Sample Assessment MaterialsPearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0)First teaching from September 2017First certifi cation from 2019 Issue 1

Edexcel, BTEC and LCCI qualifications

Edexcel, BTEC and LCCI qualifications are awarded by Pearson, the UKs largest awarding body offering academic and vocational qualifications that are globally recognised and benchmarked. For further information, please visit our qualification website at qualifications.pearson.com. Alternatively, you can get in touch with us using the details on our contact us page at qualifications.pearson.com/contactus

About Pearson

Pearson is the world's leading learning company, with 35,000 employees in more than 70 countries working to help people of all ages to make measurable progress in their lives through learning. We put the learner at the centre of everything we do, because wherever learning flourishes, so do people. Find out more about how we can help you and your learners at qualifications.pearson.com

References to third party material made in this sample assessment materials are made in good faith. Pearson does not endorse, approve or accept responsibility for the content of materials, which may be subject to change, or any opinions expressed therein. (Material may include textbooks, journals, magazines and other publications and websites.)

All information in this document is correct at time of publication.

Original origami artwork: Mark Bolitho Origami photography: Pearson Education Ltd/Naki Kouyioumtzis

ISBN 978 1 4469 3352 7

All the material in this publication is copyright Pearson Education Limited 2017

Contents

Introduction 1

General marking guidance 3

Paper 1 sample question paper and mark scheme 5

Paper 2 sample question paper and mark scheme 43

Paper 3A sample question paper and mark scheme 73

Paper 4A sample question paper and mark scheme 105

Paper 3B/4B sample question paper and mark scheme 145

Paper 4E sample question paper and mark scheme 179

Paper 3C/4C sample question paper and mark scheme 215

Paper 4F sample question paper and mark scheme 251

Paper 3D/4D sample question paper and mark scheme 287

Paper 4G sample question paper and mark schemeg 325

Edexcel, BTEC and LCCI qualifications

Edexcel, BTEC and LCCI qualifications are awarded by Pearson, the UKs largest awarding body offering academic and vocational qualifications that are globally recognised and benchmarked. For further information, please visit our qualification website at qualifications.pearson.com. Alternatively, you can get in touch with us using the details on our contact us page at qualifications.pearson.com/contactus

About Pearson

Pearson is the world's leading learning company, with 35,000 employees in more than 70 countries working to help people of all ages to make measurable progress in their lives through learning. We put the learner at the centre of everything we do, because wherever learning flourishes, so do people. Find out more about how we can help you and your learners at qualifications.pearson.com

References to third party material made in this sample assessment materials are made in good faith. Pearson does not endorse, approve or accept responsibility for the content of materials, which may be subject to change, or any opinions expressed therein. (Material may include textbooks, journals, magazines and other publications and websites.)

All information in this document is correct at time of publication.

Original origami artwork: Mark Bolitho Origami photography: Pearson Education Ltd/Naki Kouyioumtzis

ISBN 978 1 4469 3352 7

All the material in this publication is copyright Pearson Education Limited 2017

Introduction

The Pearson Edexcel Level 3 Advanced GCE in Further Mathematics is designed for use in schools and colleges. It is part of a suite of AS/A Level qualifications offered by Pearson.

These sample assessment materials have been developed to support this qualification and will be used as the benchmark to develop the assessment students will take.

The booklet Mathematical Formulae and Statistical Tables will be provided for use with these assessments and can be downloaded from our website, qualifications.pearson.com.

1Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials Issue 1 July 2017 Pearson Education Limited 2017

2 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials Issue 1 July 2017 Pearson Education Limited 2017

General marking guidance

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

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

Examiners should mark according to the mark scheme not according to their perception of where the grade boundaries may lie.

All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidates 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/indicative content will not be exhaustive. However different examples of responses will be provided at standardisation.

When examiners are in doubt regarding the application of the mark scheme to a candidates response, a senior examiner must be consulted before a mark is given.

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

Specific guidance for mathematics

1. These mark schemes use the following types of marks:

M marks: Method marks are awarded for knowing a method and attempting to apply it, unless otherwise indicated.

A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned.

B marks are unconditional accuracy marks (independent of M marks)

Marks should not be subdivided.

2. Abbreviations

These are some of the traditional marking abbreviations that may appear in the mark schemes.

bod benefit of doubt

ft follow through

this symbol is used for correct ft

cao correct answer only

cso correct solution only.There must be no errors in this part of the question to obtain this mark

isw ignore subsequent working

awrt answers which round to

SC: special case

o.e. or equivalent (and appropriate)

d dependentor dep

indep independent

dp decimal places

sf significant figures

The answer is printed on the paper or ag- answer given


or d The second mark is dependent on gaining the first mark

3. All M marks are follow through.

All A marks are correct answer only (cao.), unless shown, for example, as A1 ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but answers that dont logically make sense e.g. if an answer given for a probability is >1 or

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Answer ALL questions. Write your answers in the spaces provided.

1. Prove that

11 3 12 2 31 ( )( )

( )( )( )r rn an bn nr

n

+ += +

+ +=

where a and b are constants to be found.(5)

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Question 1 continued

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(Total for Question 1 is 5 marks)


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Answer ALL questions. Write your answers in the spaces provided.

1. Prove that

11 3 12 2 31 ( )( )

( )( )( )r rn an bn nr

n

+ += +

+ +=

where a and b are constants to be found.(5)

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2. Prove by induction that for all positive integers n,

f (n) = 23n +1 + 3(52n +1)

is divisible by 17(6)

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3. f (z) = z4 + az3 + 6z2 + bz + 65

where a and b are real constants.

Given that z = 3 + 2i is a root of the equation f (z) = 0, show the roots of f (z) = 0 on a single Argand diagram.

(9)

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4. =

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A

Initial line

Figure 1

The curve C shown in Figure 1 has polar equation

r = 4 + cos 2 0 2

At the point A on C, the value of r is 92

The point N lies on the initial line and AN is perpendicular to the initial line.

The finite region R, shown shaded in Figure 1, is bounded by the curve C, the initial line and the line AN.

Find the exact area of the shaded region R, giving your answer in the form p + q 3 where p and q are rational numbers to be found.

(9)

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5. A pond initially contains 1000 litres of unpolluted water.

The pond is leaking at a constant rate of 20 litres per day.

It is suspected that contaminated water flows into the pond at a constant rate of 25 litres per day and that the contaminated water contains 2 grams of pollutant in every litre of water.

It is assumed that the pollutant instantly dissolves throughout the pond upon entry.

Given that there are x grams of the pollutant in the pond after t days,

(a) show that the situation can be modelled by the differential equation,

ddxt

= 50 4

200x

t+(4)

(b) Hence find the number of grams of pollutant in the pond after 8 days.(5)

(c) Explain how the model could be refined.(1)

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6. f (x) =

xx

++

292

(a) Show that

f (x)d x = A ln(x2 + 9) + B arctanx3

+ c

where c is an arbitrary constant and A and B are constants to be found.(4)

(b) Hence show that the mean value of f (x) over the interval [0, 3] is

16

ln 2 + 1

18

(3)

(c) Use the answer to part (b) to find the mean value, over the interval [0, 3], of

f (x) + ln k

where k is a positive constant, giving your answer in the form p + 16

ln q, where p and q are constants and q is in terms of k.

(2)

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6. f (x) =

xx

++

292

(a) Show that

f (x)d x = A ln(x2 + 9) + B arctanx3

+ c

where c is an arbitrary constant and A and B are constants to be found.(4)

(b) Hence show that the mean value of f (x) over the interval [0, 3] is

16

ln 2 + 1

18

(3)

(c) Use the answer to part (b) to find the mean value, over the interval [0, 3], of

f (x) + ln k

where k is a positive constant, giving your answer in the form p + 16

ln q, where p and q are constants and q is in terms of k.

(2)

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OT

WRI

TE IN

TH

IS A

REA

D

O N

OT

WRI

TE IN

TH

IS A

REA


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a level further mathematics sample assessment materials - draft

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