as mathematics pdf

Original Origami Artwork designed by Beth Johnson and folded by Mark Bolitho Origami photography: Pearson Education Ltd/Naki Kouyioumtzis ISBN 978 1 4469 3346 6 All the material in thi

Sample Assessment Materials

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8MA0)

First teaching from September 2017

First certifi cation from 2018

AS

Mathematics

Issue 1

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 designed by Beth Johnson and folded by Mark Bolitho

Origami photography: Pearson Education Ltd/Naki Kouyioumtzis

ISBN 978 1 4469 3346 6

All the material in this publication is copyright

© Pearson Education Limited 2017

Introduction 1

Edexcel, BTEC and LCCI qualifications

Edexcel, BTEC and LCCI qualifications are awarded by Pearson, the UK’s 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 designed by Beth Johnson and folded by Mark Bolitho

Origami photography: Pearson Education Ltd/Naki Kouyioumtzis

ISBN 978 1 4469 3346 6

All the material in this publication is copyright

© Pearson Education Limited 2017

The Pearson Edexcel Level 3 Advanced Subsidiary GCE in 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

 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 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/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

candidate’s 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

 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

  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 don’t

logically make sense e.g if an answer given for a probability is >1 or <0, should

never be awarded A marks

4 For misreading which does not alter the character of a question or materially simplify

it, deduct two from any A or B marks gained, in that part of the question affected

5 Where a candidate has made multiple responses and indicates which response they

wish to submit, examiners should mark this response If there are several attempts at

a question which have not been crossed out, examiners should mark the final answer

which is the answer that is the most complete

6 Ignore wrong working or incorrect statements following a correct answer

7 Mark schemes will firstly show the solution judged to be the most common response

expected from candidates Where appropriate, alternative answers are provided in the

notes If examiners are not sure if an answer is acceptable, they will check the mark

scheme to see if an alternative answer is given for the method used If no such

alternative answer is provided but deemed to be valid, examiners must escalate the

response to a senior examiner to review

Centre Number Candidate Number

Total Marks Paper Reference

S54257A

Mathematics Advanced Subsidiary

Paper 1: Pure Mathematics

Sample Assessment Material for first teaching September 2017

You must have:

Mathematical Formulae and Statistical Tables, calculator

Candidates may use any calculator permitted by Pearson regulations

Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them

Instructions

• Use black ink or ball-point pen.

• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B)

• Fill in the boxes at the top of this page with your name,

centre number and candidate number

clearly labelled

– there may be more space than you need.

without working may not gain full credit

• Answers should be given to three significant figures unless otherwise stated

Information

• A booklet ‘Mathematical Formulae and Statistical Tables’ is provided

• There are 17 questions in this question paper The total mark for this paper is 100

– use this as a guide as to how much time to spend on each question.

Advice

• Read each question carefully before you start to answer it

new answer and any working underneath

Pearson Edexcel Level 3 GCE

Turn over

 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 don’t

logically make sense e.g if an answer given for a probability is >1 or <0, should

never be awarded A marks

4 For misreading which does not alter the character of a question or materially simplify

it, deduct two from any A or B marks gained, in that part of the question affected

5 Where a candidate has made multiple responses and indicates which response they

wish to submit, examiners should mark this response If there are several attempts at

a question which have not been crossed out, examiners should mark the final answer

which is the answer that is the most complete

6 Ignore wrong working or incorrect statements following a correct answer

7 Mark schemes will firstly show the solution judged to be the most common response

expected from candidates Where appropriate, alternative answers are provided in the

notes If examiners are not sure if an answer is acceptable, they will check the mark

scheme to see if an alternative answer is given for the method used If no such

alternative answer is provided but deemed to be valid, examiners must escalate the

response to a senior examiner to review

Centre Number Candidate Number

Total Marks Paper Reference

S54257A

Mathematics Advanced Subsidiary

Paper 1: Pure Mathematics

Sample Assessment Material for first teaching September 2017

You must have:

Mathematical Formulae and Statistical Tables, calculator

Candidates may use any calculator permitted by Pearson regulations

Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them

Instructions

• Use black ink or ball-point pen.

• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B)

• Fill in the boxes at the top of this page with your name,

centre number and candidate number

clearly labelled

– there may be more space than you need.

without working may not gain full credit

• Answers should be given to three significant figures unless otherwise stated

Information

• A booklet ‘Mathematical Formulae and Statistical Tables’ is provided

• There are 17 questions in this question paper The total mark for this paper is 100

– use this as a guide as to how much time to spend on each question.

Advice

• Read each question carefully before you start to answer it

new answer and any working underneath

Pearson Edexcel Level 3 GCE

Turn over

Answer ALL questions Write your answers in the spaces provided.

1 The line l passes through the points A (3, 1) and B (4, −2).

Find an equation for l.

Find the gradient of the curve at the point P (5, 6).

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(4)

(Total for Question 2 is 4 marks)

Answer ALL questions Write your answers in the spaces provided.

1 The line l passes through the points A (3, 1) and B (4, −2).

Find an equation for l.

Find the gradient of the curve at the point P (5, 6).

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(4)

(Total for Question 2 is 4 marks)

3 Given that the point A has position vector 3i − 7j and the point B has position vector 8i + 3j,

(a) find the vector AB

(Total for Question 3 is 4 marks)

3 (a) Find the first 3 terms, in ascending

powers of x, of the binomial expansion of

(b) Explain how you would use your

expansion to give an estimate for the value

(2)

(b) Hence show that 3 is the only real root of the equation f ( x ) = 0

(4)

(Total for Question 4 is 6 marks)

3 Given that the point A has position vector 3i − 7j and the point B has position vector 8i + 3j,

(a) find the vector AB

(Total for Question 3 is 4 marks)

3 (a) Find the first 3 terms, in ascending

powers of x, of the binomial expansion of

(b) Explain how you would use your

expansion to give an estimate for the value

(2)

(b) Hence show that 3 is the only real root of the equation f ( x ) = 0

(4)

(Total for Question 4 is 6 marks)

(Total for Question 6 is 4 marks)

(Total for Question 6 is 4 marks)

(Total for Question 7 is 5 marks)

(Total for Question 7 is 5 marks)

Given that angle BAC = 70° and angle ABC = 60°,

(a) calculate the area of the lawn to 3 significant figures

(Total for Question 8 is 5 marks)

Given that angle BAC = 70° and angle ABC = 60°,

(a) calculate the area of the lawn to 3 significant figures

(Total for Question 8 is 5 marks)

Give your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

_

(Total for Question 9 is 5 marks)

Give your answers to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

_

(Total for Question 9 is 5 marks)

(Total for Question 11 is 3 marks)

(Total for Question 11 is 3 marks)

22x + 4 −  9(2 x) = 0

22x + 24 −  9(2 x) = 0Let 2x = y

y2 − 9y + 8 = 0 (y −  8)(y  −  1) = 0

(Total for Question 12 is 4 marks)

22x + 4 −  9(2 x) = 0

22x + 24 −  9(2 x) = 0Let 2x = y

y2 − 9y + 8 = 0 (y −  8)(y  −  1) = 0

(Total for Question 12 is 4 marks)

The point with coordinates (−3, 0) lies on the curve with equation

y = (x + a) 3 + 10(x + a) 2 + 25(x + a) where a is a constant.

(c) Find the two possible values of a.

(Total for Question 13 is 7 marks)

The point with coordinates (−3, 0) lies on the curve with equation

y = (x + a) 3 + 10(x + a) 2 + 25(x + a) where a is a constant.

(c) Find the two possible values of a.

(Total for Question 13 is 7 marks)

A town’s population, P, is modelled by the equation P = a b  t , where a and b are constants

and t is the number of years since the population was first recorded.

The line l shown in Figure 2 illustrates the linear relationship between t and log10 P for

the population over a period of 100 years

The line l meets the vertical axis at (0, 5) as shown The gradient of l is 1

(c) With reference to the model interpret

(i) the value of the constant a,

(ii) the value of the constant b.

(2)

(d) Find

(i) the population predicted by the model when t = 100, giving your answer to the

nearest hundred thousand,

(ii) the number of years it takes the population to reach 200 000, according to the model

(Total for Question 14 is 13 marks)

A town’s population, P, is modelled by the equation P = a b  t , where a and b are constants

and t is the number of years since the population was first recorded.

The line l shown in Figure 2 illustrates the linear relationship between t and log10 P for

the population over a period of 100 years

The line l meets the vertical axis at (0, 5) as shown The gradient of l is 1

(c) With reference to the model interpret

(i) the value of the constant a,

(ii) the value of the constant b.

(2)

(d) Find

(i) the population predicted by the model when t = 100, giving your answer to the

nearest hundred thousand,

(ii) the number of years it takes the population to reach 200 000, according to the model

(Total for Question 14 is 13 marks)

The curve C1, shown in Figure 3, has equation y = 4x2 – 6x + 4.

22,



  lies on C1

2x + ln(2x).

The normal to C1 at the point P meets C2 at the point Q.

Find the exact coordinates of Q.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(Total for Question 15 is 8 marks)

The curve C1, shown in Figure 3, has equation y = 4x2 – 6x + 4.

22

The normal to C1 at the point P meets C2 at the point Q.

Find the exact coordinates of Q.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(Total for Question 15 is 8 marks)

D

E

2xm

Figure 4 shows the plan view of the design for a swimming pool

The shape of this pool ABCDEA consists of a rectangular section ABDE joined to a

semicircular section BCD as shown in Figure 4.

(a) show that the perimeter, P metres, of the pool is given by

2502

(Total for Question 16 is 10 marks)

D

E

2xm

Figure 4 shows the plan view of the design for a swimming pool

The shape of this pool ABCDEA consists of a rectangular section ABDE joined to a

semicircular section BCD as shown in Figure 4.

(a) show that the perimeter, P metres, of the pool is given by

2502

(Total for Question 16 is 10 marks)

17 A circle C with centre at ( –2, 6) passes through the point (10, 11).

(a) Show that the circle C also passes through the point (10, 1).

(3)

The tangent to the circle C at the point (10, 11) meets the y axis at the point P and the

tangent to the circle C at the point (10, 1) meets the y axis at the point Q.

(b) Show that the distance PQ is 58 explaining your method clearly.

17 A circle C with centre at ( –2, 6) passes through the point (10, 11).

(a) Show that the circle C also passes through the point (10, 1).

(3)

The tangent to the circle C at the point (10, 11) meets the y axis at the point P and the

tangent to the circle C at the point (10, 1) meets the y axis at the point Q.

(b) Show that the distance PQ is 58 explaining your method clearly.

Paper 1: Pure Mathematics Mark Scheme

Question Scheme Marks AOs

1 Way 1

Uses y = mx + c with both (3, 1) and (4,  2) and attempt to find m

(3)

Or Way 3

Uses ax + by + k = 0 and substitutes both x = 3 when y = 1 and x =

4 when y = 2 with attempt to solve to find a, b or k in terms of one

Notes:

M1: Need correct use of the given coordinates A1: Need fractions simplified to 3 (in ways 1 and 2) A1: Need constants combined accurately

N.B Answer left in the form (y – 1) = – 3(x – 3) or (y – (– 2)) = – 3(x – 4) is awarded

M1A1A0 as answers should be simplified by constants being collected

Note that a correct answer implies all three marks in this question

Paper 1: Pure Mathematics Mark Scheme

Question Scheme Marks AOs

1 Way 1

Uses y = mx + c with both (3, 1) and (4,  2) and attempt to find m

(3)

Or Way 3

Uses ax + by + k = 0 and substitutes both x = 3 when y = 1 and x =

4 when y = 2 with attempt to solve to find a, b or k in terms of one

Notes:

M1: Need correct use of the given coordinates A1: Need fractions simplified to 3 (in ways 1 and 2) A1: Need constants combined accurately

N.B Answer left in the form (y – 1) = – 3(x – 3) or (y – (– 2)) = – 3(x – 4) is awarded

M1A1A0 as answers should be simplified by constants being collected

Note that a correct answer implies all three marks in this question

Attempt to differentiate M1 1.1a

d 4 12d

d 8

d

y x

(4 marks)

Notes:

M1: Differentiation implied by one correct term

A1: Correct differentiation

M1: Attempts to substitute x = 5 into their derived function

A1ft: Substitutes x = 5 into their derived function correctly i.e Correct calculation of their

f ′(5) so follow through slips in differentiation

Question Scheme Marks AOs

3(a) Attempts AB OB OA  or similar M1 1.1b

Notes:

(a)

M1: Attempts subtraction but may omit brackets

A1: cao (allow column vector notation)

(b)

M1: Correct use of Pythagoras theorem or modulus formula using their answer to (a)

A1ft: AB 5 5 ft from their answer to (a)

Note that the correct answer implies M1A1 in each part of this question

4(3)3 – 12(3)2 + 2(3) – 6 = 108 – 108 + 6 – 6 = 0 and so (x – 3) is a

(2) (b) Begins division or factorisation so x

4x3 – 12x2 + 2x – 6 = (x −3)(4x2 +…) M1 2.1

4x3 – 12x2 + 2x – 6 = (x −3)(4x2 + 2) A1 1.1b Considers the roots of their quadratic function using completion of

Notes:

(a) M1: States or uses f (+3) = 0 A1: See correct work evaluating and achieving zero, together with correct conclusion (b)

M1: Needs to have (x – 3) and first term of quadratic correct A1: Must be correct – may further factorise to 2(x − 3)(2x2 + 1)

M1: Considers their quadratic for no real roots by use of completion of the square or

consideration of discriminant then

A1*: A correct explanation

Attempt to differentiate M1 1.1a

d 4 12d

d 8

d

y x

(4 marks)

Notes:

M1: Differentiation implied by one correct term

A1: Correct differentiation

M1: Attempts to substitute x = 5 into their derived function

A1ft: Substitutes x = 5 into their derived function correctly i.e Correct calculation of their

f ′(5) so follow through slips in differentiation

Question Scheme Marks AOs

3(a) Attempts AB OB OA  or similar M1 1.1b

Notes:

(a)

M1: Attempts subtraction but may omit brackets

A1: cao (allow column vector notation)

(b)

M1: Correct use of Pythagoras theorem or modulus formula using their answer to (a)

A1ft: AB 5 5 ft from their answer to (a)

Note that the correct answer implies M1A1 in each part of this question

4(3)3 – 12(3)2 + 2(3) – 6 = 108 – 108 + 6 – 6 = 0 and so (x – 3) is a

(2) (b) Begins division or factorisation so x

4x3 – 12x2 + 2x – 6 = (x −3)(4x2 +…) M1 2.1

4x3 – 12x2 + 2x – 6 = (x −3)(4x2 + 2) A1 1.1b Considers the roots of their quadratic function using completion of

Notes:

(a) M1: States or uses f (+3) = 0 A1: See correct work evaluating and achieving zero, together with correct conclusion (b)

M1: Needs to have (x – 3) and first term of quadratic correct A1: Must be correct – may further factorise to 2(x − 3)(2x2 + 1)

M1: Considers their quadratic for no real roots by use of completion of the square or

consideration of discriminant then

A1*: A correct explanation

B1: Correct function with numerical powers

M1: Allow for raising power by one x nx n 1

A1: Correct three terms

M1: Substitutes limits and rationalises denominator

A1*: Completely correct, no errors seen

M1: Expands the bracket as above or 3(xx) 32  x26x x 3( )x 2

A1: Substitutes correctly into earlier fraction and simplifies A1*: Uses Completes the proof, as above ( may use  x 0), considers the limit and states a

conclusion with no errors