a level further mathematics - the maths orchard · the pearson edexcel level 3 advanced gce in...

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

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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 42

Paper 3A – sample question paper and mark scheme 71

Paper 4A– sample question paper and mark scheme 101

Paper 3B/4B – sample question paper and mark scheme 143

Paper 4E– sample question paper and mark scheme 177

Paper 3C/4C – sample question paper and mark scheme 213

Paper 4F – sample question paper and mark scheme 249

Paper 3D/4D – sample question paper and mark scheme 285

Paper 2G – sample question paper and mark scheme 325Mathematical formulae and statistical tables

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.

Pearson Edexcel Level 3 Advanced GCE in Further Mathematics – Sample Assessment Materials (SAMs) – Issue 1 – Accredited pre-publication – May 2017 © Pearson Education Limited 2017

1


2

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 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.

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… dependent

or dep

indep independent

dp decimal places

sf significant figures

The answer is printed on

the paper or ag- answer

given


3

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.


4

Pearson Edexcel Level 3 GCE

Further Mathematics Advanced Paper 1: Core Pure Mathematics 1

Sample assessment material for first teaching September 2017 Time: 1 hour 30 minutes

Paper Reference(s)

9FM0/01

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.

Answer all questions and ensure that your answers to parts of questions are clearly labelled.

Answer the questions in the spaces provided – there may be more space than you need.

You should show sufficient working to make your methods clear. Answers 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 9 questions in this question paper. The total mark for this paper is 75.

The marks for each question are shown in brackets – 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.

Try to answer every question.

Check your answers if you have time at the end.


5

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

1. Prove that

1

1

1 3 12 2 3

n

r

n an b

r r n 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)


7

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

3 1 2 1f 2 3 5n nn

is divisible by 17

(6)

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9

3.

4 3 2f ( ) 6 65z z az z bz

where a and b are real constants.

Given that 3 2iz 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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11

4.

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 9

2

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 3p q

where p and q are rational numbers to be found.

(9)

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13

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,

d 450

d 200

x x

t 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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15

6.

2

2f

9

xx

x

(a) Show that

2f d ln 9 arctan3

xx x A x B 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

1 1ln 2

6 18

(3)

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

f(x) + lnk

where k is a positive constant, giving your answer in the form 1

ln6

p q , where p and q are

constants and q is in terms of k.

(2)

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17

7.

Figure 2

Figure 2 shows the image of a gold pendant which has height 2 cm. The pendant is

modelled by a solid of revolution of a curve C about the y-axis. The curve C has

parametric equations

1

cos sin 2 , 1 sin 0 22

x y

(a) Show that a Cartesian equation of the curve C is

2 4 32x y y

(4)

(b) Hence, using the model, find, in cm3, the volume of the pendant.

(4)

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21

8. The line l1 has equation 2 4 6

4 2 1

x y z

The plane Π has equation x – 2y + z = 6

The line l2 is the reflection of the line l1 in the plane Π.

Find a vector equation of the line l2

(7)

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25

9. A company plans to build a new fairground ride. The ride will consist of a capsule that

will hold the passengers and the capsule will be attached to a tall tower. The capsule is to

be released from rest from a point half way up the tower and then made to oscillate in a

vertical line.

The vertical displacement, x metres, of the top of the capsule below its initial position at

time t seconds is modelled by the differential equation,

2

2

d d4 200cos , 0

d d

x xm x t t

t t

where m is the mass of the capsule including its passengers, in thousands of kilograms.

The maximum permissible weight for the capsule, including its passengers, is 30 000N.

Taking the value of g to be 10 ms-2 and assuming the capsule is at its maximum

permissible weight,

(a) (i) explain why the value of m is 3

(ii) show that a particular solution to the differential equation is

40sin 20cosx t t

(iii) hence find the general solution of the differential equation.

(8)

(b) Using the model, find, to the nearest metre, the vertical distance of the top of the

capsule from its initial position, 9 seconds after it is released.

(4)

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TOTAL FOR PAPER IS 75 MARKS


29


30

Paper 1: Core Pure Mathematics 1 Mark Scheme

Question Scheme Marks AOs

1

1

..., ...1 3 1 3

A BA B

r r r r

M1 3.1a

1

1

1 3

1 1 1 1 1 1 1 1...

2 2 2 4 2 3 2 5 2 2 2 2 1 2 3

n

rr r

n n n n

M1 2.1

1 1 1 1

4 6 2 2 2 3n n

A1 2.2a

5 2 3 6 3 6 2

12 2 3

n n n n

n n

M1 1.1b

5 13

12 2 3

n n

n n

A1 1.1b

(5)

Alternative by Induction:

2 21 1 11 , 2

8 12 3 4 8 15 12 4 5

a ba bn n

18, 2 23 ..., ...a b a b a b

M1 3.1a

Assume true for n = k so

1

5 131

1 3 12 2 3

k

r

k k

r r k k

1

1

5 131 1

1 3 12 2 3 2 4

k

r

k k

r r k k k k

M1 2.1

5 13 5 13 4 12 31

12 2 3 2 4 12 2 3 4

k k k k k k

k k k k k k k

A1 2.2a

3 2 1 2 5 185 33 52 12 36

12 2 3 4 12 2 3 4

k k kk k k k

k k k k k k

M1 1.1b

1 5 1 13

12 1 2 1 3

k k

k k

So true for n = k + 1

So

1

5 131

1 3 12 2 3

n

r

n n

r r n n

A1 1.1b

(5)

(5 marks)


31

Notes:

(Main Scheme)

M1: Valid attempt at partial fractions

M1: Starts the process of differences to identify the relevant fractions at the start and end

A1: Correct fractions that do not cancel

M1: Attempt common denominator

A1: Correct answer

(Alternative by Induction)

M1: Uses n = 1 and n = 2 to identify values for a and b

M1: Starts the induction process by adding the (k + 1)th term to the sum of k terms

A1: Correct single fraction

M1: Attempt to factorise the numerator

A1: Correct answer and conclusion


32

Question Scheme

Marks AOs

2 When n = 1, 3 1 2 12 3 5 16 375 391n n

391 17 23 so the statement is true for n = 1 B1 2.2a

Assume true for n = k so 3 1 2 12 3 5k k is divisible by 17 M1 2.4

3 4 2 3 3 1 2 1f 1 f 2 3 5 2 3 5k k k kk k M1 2.1

3 1 2 1 2 17 2 7 3 5 17 3 5k k k

2 17f 17 3 5 kk A1 1.1b

2 1f 1 8f 17 3 5 kk k A1 1.1b

If the statement is true for n = k then it has been shown true for n

= k + 1 and as it is true for n = 1, the statement is true for all

positive integers n.

A1 2.4

(6)

(6 marks)


33


3 3 2iz is also a root B1 1.2

3 2i 3 2i ...z z

or

Sum of roots = 6, Product of roots = 13 ...

M1 3.1a

2 6 13z z A1 1.1b

4 3 2 2 26 65 6 13 5 ...z az z bz z z z cz c

M1 3.1a

2 2 5 0z z A1 1.1b

2 2 5 0 ...z z z M1 1.1a

1 2iz A1 1.1b

B1

3 ± 2i

Plotted

correctly

1.1b

B1ft

1 ± 2i

Plotted

correctly

1.1b

(9 marks)

Notes:

B1: Identifies the complex conjugate as another root

M1: Uses the conjugate pair and a correct method to find a quadratic factor

A1: Correct quadratic

M1: Uses the given quartic and their quadratic to identify the value of c

A1: Correct 3TQ

M1: Solves their second quadratic

A1: Correct second conjugate pair

B1: First conjugate pair plotted correctly and labelled

B1ft: Second conjugate pair plotted correctly and labelled (Follow through their second conjugate pair)

Re

Im

(3, 2)

(3, -2)

(-1, 2)

(-1, -2)


34


4 94 cos 2 ...

2 M1 3.1a

6

A1 1.1b

2 21 1

4 cos 2 d 16 8cos 2 cos 2 d2 2

M1 3.1a

2 1 1 1 1 1cos 2 cos 4 16 8cos 2 cos 4 d

2 2 2 2 2A

M1 3.1a

1 sin 416 4sin 2

2 8 2

A1 1.1b

1 33 3

Using limits 0 and their : 2 3 06 2 12 16

M1 1.1b

Area of triangle = 1 1 81 1 3

cos sin2 2 4 2 2

r r M1 3.1a

Area of R = 33 33 3 81 3

24 32 32

M1 1.1b

11 3 3 11 3,

8 2 8 2p q

A1 1.1b

(9 marks)

Notes:

M1: Realises the angle for A is required and attempts to find it.

A1: Correct angle

M1: Uses a correct area formula and squares r to achieve a 3TQ integrand in cos 2θ

M1: Use of the correct double angle identity on the integrand to achieve a suitable form for integration

A1: Correct integration

M1: Correct use of limits

M1: Identifies the need to subtract the area of a triangle and so finds the area of the triangle

M1: Complete method for the area of R

A1: Correct final answer


35


5(a) Pond contains 1000 + 5t litres after t days M1 3.3

If x is the amount of pollutant in the pond after t days

Rate of pollutant out = 201000 5

x

t

g per day

M1 3.3

Rate of pollutant in = 25 2 g = 50g per day B1 2.2a

d 450 *

d 200

x x

t t

A1* 1.1b

(4)

(b)

4

d 4 4 4200e 200 200 50 200 d

ttI t x t t t

M1 3.1b

4 5

200 10 200x t t c A1 1.1b

120, 0 3.2 10x t c M1 3.4

12

4

3.2 108 10 200 8

200 8t x

M1 1.1b

= 370g A1 2.2b

(5)

(c) Examples

The model should take into account the fact that the

pollutant does not dissolve throughout the pond upon

entry

The rate of leaking could be made to vary with the volume

of water in the pond

B1 3.5c

(1)

(10 marks)

Notes:

(a)

M1: Forms an expression of the form 1000 + kt for the volume of water in the pond at time t

M1: Expresses the amount of pollutant out in terms of x and t

B1: Correct interpretation for pollutant entering the pond

A1*: Puts all the components together to form the correct differential equation

(b)

M1: Uses the model to find the integrating factor and attempts solution of their differential equation

A1: Correct solution

M1: Interprets the initial conditions to find the constant of integration

M1: Uses their solution to the problem to find the amount of pollutant after 8 days

A1: Correct number of grams

(c)

B1: Suggests a suitable refinement to the model


36


6(a) 2 2 2

2 2f

9 9 9

x xx

x x x

B1 3.1a

2

2d ln 9

9

xx k x c

x

M1 1.1b

32

2d arctan

9xx k c

x

M1 1.1b

2

2

2 1 2d ln 9 arctan

9 2 3 3

x xx x c

x

A1 1.1b

(4)

(b)

3 3

2

00

1 2f d ln 9 arctan

2 3 3

1 2 3 1 2ln18 arctan ln 9 arctan 0

2 3 3 2 3

1 18 2 3ln arctan

2 9 3 3

xx x x

M1 1.1b

Mean value = 1 1

ln 23 0 2 6

M1 2.1

1 1ln 2 *

6 18 A1* 2.2a

(3)

(c) 1 1ln 2 ln

6 18k M1 2.2a

61 1ln 2

6 18k A1 1.1b

(2)

(9 marks)

Notes:

(a)

B1: Splits the fraction into two correct separate expressions

M1: Recognises the required form for the first integration

M1: Recognises the required form for the second integration

A1: Both expressions integrated correctly and added together with constant of integration included

(b)

M1: Uses limits correctly and combines logarithmic terms

M1: Correctly applies the method for the mean value for their integration

A1*: Correct work leading to the given answer

(c)

M1: Realises that the effect of the transformation is to increase the mean value by lnk

A1: Combines ln’s correctly to obtain the correct expression


37


7(a) cos sin cos cosx y M1 2.1

sin 1y M1 2.1

2

21 1

xy

y

M1 2.1

2 4 32 *x y y A1* 1.1b

(4)

(b)

2 4 3d 2 dV x y y y y M1 3.4

5 4

5 2

y y

A1 1.1b

5 4 5 4

0 0 2 2

5 2 5 2

M1 3.4

31.6 cm or awrt 5.03 3cm A1 1.1b

(4)

(8 marks)

Notes:

(a)

M1: Obtains x in terms of y and cos θ

M1: Obtains an equation connecting y and sin θ

M1: Uses Pythagoras to obtain an equation in x and y only

A1*: Obtains printed answer

(b)

M1: Uses the correct volume of revolution formula with the given expression


M1: Correct use of correct limits

A1: Correct volume


38


8

2 4 2 4 2 6 6 ... M1 1.1b

2 Required point is 2 2 4 , 4 2 2 , 6 2 1

10, 0, 4 A1 1.1b

2 2 4 2 6 6 ...t t t t M1 3.1a

3soreflection of 2,4, 6 is 2 6 1 ,4 6 2 , 6 6 1t M1 3.1a

8, 8, 0 A1 1.1b

10 8 2

0 8 8

4 0 4

M1 3.1a

10 1

0 4 or equivalent

4 2

k

r e.g.

10 1

0 4

4 2

r 0 A1 2.5

(7)

(7 marks)

Notes:

M1: Substitutes the parametric equation of the line into the equation of the plane and solves for λ

A1: Obtains the correct coordinates of the intersection of the line and the plane

M1: Substitutes the parametric form of the line perpendicular to the plane passing through (2, 4, 6) into

the equation of the plane to find t

M1: Find the reflection of (2, 4, 6) in the plane

A1: Correct coordinates

M1: Determines the direction of l by subtracting the appropriate vectors

A1: Correct vector equation using the correct notation.

(8,8, 0)

(2, 4,6)

(10, 0,4)

π


39


9(a)(i) Weight = massg

300003000m

g

But mass is in thousands of kg, so m = 3

M1 3.3

(ii) 2

2

d d40cos 20sin , 40sin 20cos

d d

x xt t t t

t t M1 1.1b

3 40sin 20cos 4 40cos 20sin

40sin 20cos ...

t t t t

t t

M1 1.1b

200cos t so PI is 40sin 20cosx t t A1* 2.1

or

Let cos sinx a t b t

2

2

d dsin cos , cos sin

d d

x xa t b t a t b t

t t

M1 1.1b

4 2 200, 2 4 0 ..., ...b a b a a b M1 2.1

40sin 20cosx t t A1* 1.1b

(iii) 2 13 4 1 0 1,

3 M1 1.1b

13e ettx A B

A1 1.1b

x PI CF M1 1.1b

13e e 40sin 20costtx A B t t

A1 1.1b

(8)

(b) 0, 0 20t x A B M1 3.4

13

d 10, 40cos 20sin 0

d 3

140

3

ttxx Ae Be t t

t

A B

M1 3.4

1350e 30e 40sin 20costtx t t

A1 1.1b

9 33mt x A1 3.4

(4)

(12 marks)

Notes:

(a)(i)

M1: Correct explanation that in the model, m = 3

(ii)

M1: Differentiates the given PI twice

M1: Substitutes into the given differential equation

A1*: Reaches 200cost and makes a conclusion

Or

M1: Uses the correct form for the PI and differentiates twice


40

M1: Substitutes into the given differential equation and attempts to solve

A1*: Correct PI

(iii)

M1: Uses the model to form and solve the auxiliary equation

A1: Correct complementary function

M1: Uses the correct notation for the general solution by combining PI and CF

A1: Correct General Solution for the model

(b)

M1: Uses the initial conditions of the model, t = 0 at x = 0, to form an equation in A and B

M1: Uses dx/dt = 0 at x = 0 in the model to form an equation in A and B

A1: Correct PS

A1: Obtains 33m using the assumptions made in the model.


41


42


Further Mathematics Advanced Paper 2: Core Pure Mathematics 2


Paper Reference(s)

9FM0/02



Instructions











Advice





43


1. The roots of the equation

3 28 28 32 0x x x

are , and

Without solving the equation, find the value of

(i) 1 1 1

(ii) 2 2 2

(iii) 2 2 2

(8)

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45

2. The plane Π1 has vector equation

. 3 4 2 5 r i j k

(a) Find the perpendicular distance from the point (6, 2, 12) to the plane Π1

(3)

The plane Π2 has vector equation

2 5 2 r = i j k i j k

where λ and µ are scalar parameters.

(b) Show that the vector 3 i j k is perpendicular to Π2

(2)

(c) Show that the acute angle between Π1 and Π2 is 52o to the nearest degree.

(3)

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47

3. (i)

2 4

1 1 1

1 2 1

a

M

where a is a constant.

(a) For which values of a does the matrix M have an inverse?

(2)

Given that M is non-singular,

(b) find 1

M in terms of a

(4)

(ii) Prove by induction that for all positive integers n,

3 03 0

6 1 3 3 1 1

nn

n

(6)

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49

4. A complex number z has modulus 1 and argument θ.

(a) Show that

zn + 1nz

= 2cos nθ, n

(2)

(b) Hence, show that

cos4 θ = 1

8(cos 4θ + 4cos 2θ + 3)

(5)

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51

5.

sin sinhy x x

(a) Show that 4

4

d4

d

yy

x

(4)

(b) Hence find the first three non-zero terms of the Maclaurin series for y, giving each

coefficient in its simplest form.

(4)

(c) Find an expression for the nth non-zero term of the Maclaurin series for y.

(2)

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53

6. (a) (i) Show on an Argand diagram the locus of points given by the values of z satisfying

4 3i 5z

Taking the initial line as the positive real axis with the pole at the origin and given that

, , where 43

arctan ,

(ii) show that this locus of points can be represented by the polar curve with equation

8cos 6sinr

(6)

The set of points A is defined by

: 0 arg : 4 3i 53

A z z z z

(b) (i) Show, by shading on your Argand diagram, the set of points A.

(ii) Find the exact area of the region defined by A, giving your answer in simplest form.

(7)

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55

7. At the start of the year 2000, a survey began of the number of foxes and rabbits on an

island.

At time t years after the survey began, the number of foxes, f, and the number of rabbits,

r, on the island are modelled by the differential equations

d0.2 0.1

d

d0.2 0.4

d

ff r

t

rf r

t

(a) Show that 2

2

d d0.6 0.1 0

d d

f ff

t t

(3)

(b) Find a general solution for the number of foxes on the island at time t years.

(4)

(c) Hence find a general solution for the number of rabbits on the island at time t years.

(3)

At the start of the year 2000 there were 6 foxes and 20 rabbits on the island.

(d) (i) According to this model, in which year are the rabbits predicted to die out?

(ii) According to this model, how many foxes will be on the island when the rabbits

die out?

(iii) Use your answers to parts (i) and (ii) to comment on the model.

(7)

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Total for Question 7 is 17 marks)



59


60

Paper 2: Core Pure Mathematics 2 Mark Scheme


1(i) 8, 28, 32 B1 3.1a

1 1 1

M1 1.1b

7

8 A1ft 1.1b

(3)

(ii) 2 2 2 2 2 4 2 M1 1.1b

2 4 8 A1 1.1b

32 2 28 4 8 8 128 A1 1.1b

(3)

Alternative for part (ii)

3 2

2 8 2 28 2 32 0x x x M1 1.1b

... 8 ... 32 ... 56 32 128 A1 1.1b

2 2 2 128 A1 1.1b

(3)

(iii) 22 2 2 2 M1 3.1a

28 2 28 8 A1ft 1.1b

(2)

(8 marks)

Notes

(i)

B1: Identifies the correct values for all 3 expressions (can score anywhere)

M1: Uses a correct identity

A1ft: Correct value (follow through their 8, 28 and 32)

(ii)

M1: Attempts to expand

A1: Correct expansion

A1: Correct value

Alternative:

M1: Substitutes x – 2 for x in the given cubic

A1: Calculates the correct constant term

A1: Changes sign and so obtains the correct value

(iii)

M1: Establishes the correct identity

A1ft: Correct value (follow through their 8, 28 and 32)


61

Question Scheme

Marks AOs

2(a) 3 6

4 2 18 8 24

2 12

M1 3.1a

2 2 2

18 8 24 5

3 4 2d

M1 1.1b

29 A1 1.1b

(3)

(b) 1 2 1 1

3 1 ... and 3 1 ...

1 5 1 2

M1 2.1

1 2 1 1

3 1 0 and 3 1 0

1 5 1 2

3 i j k is perpendicular to Π2

A1 2.2a

(2)

(c) 1 3

3 4 3 12 2

1 2

M1 1.1b

2 2 2 22 2

2 2 2 22 2

1 3 1 3 4 2 cos 11

11cos

1 3 1 3 4 2

M1 2.1

So angle between planes o52 * A1* 2.4

(3)

(8 marks)

Notes

(a)

M1: Realises the need to and so attempts the scalar product between the normal and the position

vector

M1: Correct method for the perpendicular distance

A1: Correct distance

(b)

M1: Recognises the need to calculate the scalar product between the given vector and both

direction vectors

A1: Obtains zero both times and makes a conclusion

(c)

M1: Calculates the scalar product between the two normal vectors

M1: Applies the scalar product formula with their 11 to find a value for cos θ

A1*: Identifies the correct angle by linking the angle between the normal and the angle between

the planes.


62


3(i)(a) 2 1 2 1 1 4 2 1 0 ...a a M M1 2.3

The matrix M has an inverse when 5a A1 1.1b

(2)

(b)

3 2 1

Minors : 8 2 4

4 6 2

a a

a a

or

3 2 1

Cofactors : 8 2 4

4 6 2

a a

a a

B1 1.1b

1 1adj M M

M M1 1.1b

1

3 8 41

2 2 62 10

1 4 2

a a

aa a

M

2 correct rows or columns.

Follow through their detM. A1ft 1.1b

All correct.

Follow through their detM. A1ft 1.1b

(4)

(ii) When n = 1, lhs =

3 0

6 1

, rhs =

1

1

3 0 3 0

6 13 3 1 1

So the statement is true for n = 1

B1 2.2a

Assume true for n = k so

3 03 0

6 1 3 3 1 1

kk

k

M1 2.4

1 3 03 0 3 0

6 1 6 13 3 1 1

kk

k

M1 2.1

3 3 0

3 3 3 1 6 1

k

k

A1 1.1b

1

1

3 0

3 3 1 1

k

k

A1 1.1b

If the statement is true for n = k then it has been shown true for n = k

+ 1 and as it is true for n = 1, the statement is true for all positive

integers n.

A1 2.4

(6)

(12 marks)


63

Notes:

(i)(a)

M1: Attempts determinant, equates to zero and attempts to solve for a in order to establish the restriction for

a

A1: Provides the correct condition for a if M has an inverse

(i)(b)

B1: A correct matrix of minors or cofactors

M1: For a complete method for the inverse

A1ft: Two correct rows following through their determinant

A1ft: Fully correct inverse following through their determinant

(ii)

B1: Shows the statement is true for n = 1

M1: Assumes the statement is true for n = k

M1: Attempts to multiply the correct matrices

A1: Correct matrix in terms of k

A1: Correct matrix in terms of k + 1

A1: Correct complete conclusion


64

Question Scheme

Marks AOs

4(a) cos isin cos isinn nz z n n n n M1 2.1

2cos *n A1* 1.1b

(2)

(b) 4

1 416cosz z B1 2.1

4

1 4 2 2 44 6 4z z z z z z M1 2.1

4 4 2 24 6z z z z A1 1.1b

2cos4 4 2cos2 6 M1 2.1

cos4 θ = 1

8(cos 4θ + 4cos 2θ + 3)* A1* 1.1b

(5)

(7 marks)

Notes

(a)

M1: Identifies the correct form for zn and z-n and adds to progress to the printed answer

A1*: Achieves printed answer with no errors

(b)

B1: Begins the argument by using the correct index with the result from part (a)

M1: Realises the need to find the expansion of 4

1z z

A1: Terms correctly combined

M1: Links the expansion with the result in part (a)

A1*: Achieves printed answer with no errors


65

Question Scheme

Marks AOs

5(a) dsin cosh cos sinh

d

yx x x x

x M1 1.1a

2

2

dcos cosh sin sinh cos cosh sin sinh

d

2cos cosh

yx x x x x x x x

x

x x

M1 1.1b

3

3

d2cos sinh 2sin cosh

d

yx x x x

x M1 1.1b

4

4

d4sinh sin 4 *

d

yx x y

x A1* 2.1

(4)

(b) 2 6 10

2 6 10

0 0 0

d d d2, 8, 32

d d d

y y y

x x x

B1 3.1a

Uses 2 3

0 0 0 0 ...2! 3!

x xy y xy y y with their values M1 1.1b

2 6 10

2 8 322! 6! 10!

x x x A1 1.1b

6 102

90 113400

x xx A1 1.1b

(4)

(c)

4 21

2 44 2 !

nn x

n

M1 A1 3.1a

2.2a

(2)

(10 marks)

Notes

(a)

M1: Realises the need to use the product rule and attempts first derivative

M1: Realises the need to use a second application of the product rule and attempts the second

derivative

M1: Correct method for the third derivative

A1*: Obtains the correct 4th derivative and links this back to y

(b)

B1: Makes the connection with part (a) to establish the general pattern of derivatives and finds

the correct non-zero values

M1: Correct attempt at Maclaurin series with their values

A1: Correct expression un-simplified

A1: Correct expression and simplified

(c)

M1: Generalising, dealing with signs, powers and factorials.

A1: Correct expression.


66


6(a)(i)

M1 1.1b

A1 1.1b

(a)(ii) 2 2

4 3i 5 i 4 3i 5 4 3 ...z x y x y M1 2.1

2 2

4 3 25 or anycorrect formx y A1 1.1b

2 2

2 2 2 2

2

cos 4 sin 3 25

cos 8 cos 16 sin 6 sin 9 25

8 cos 6 sin 0

r r

r r r r

r r r

M1 2.1

8cos 6sin *r A1* 2.2a

(6)

(b)(i)

B1 1.1b

B1ft 1.1b

(b)(ii)

22

2 2

1 1d 8cos 6sin d

2 2

164cos 96sin cos 36sin d

2

A r

M1 3.1a

1

32 cos 2 1 96sin cos 18 1 cos 2 d2

M1 1.1b

1

14cos 2 50 48sin 2 d2

A1 1.1b

3

0

1 1 7 3 507sin 2 50 24cos 2 12 24

2 2 2 3

M1 2.1

7 3 2518

4 3

A1 1.1b

(7)

Im

Re

Im

Re


67

A

Alternative:

Candidates may take a geometric approach

E.g. by finding sector + 2 triangles

Angle ACB = 2

3

so area sector ACB = 21 2

52 3

Area of triangle OCB 1

8 32

M1 3.1a

Sector area ACB + triangle area OCB = 25

3

+ 12 A1 1.1b

Area of triangle OAC:

Angle ACO = 2 2 2

12 5 5 82 cos

3 2 5 5

so area OAC = 2 11 4 7

5 sin cos2 3 25

M1 1.1b

1 125 4 7 4 7sin cos cos cos sin cos

2 3 25 3 25

225 7 3 1 7 7 3

1 62 50 2 25 4

Total area = 25 1 7 3

8 3 63 2 4

M1 2.1

7 3 2518

4 3

A1 1.1b

(13 marks)

C

B O


68

Notes

(a)(i)

M1: Draws a circle which passes through the origin

A1: Fully correct diagram.

(a)(ii)

M1: Uses z = x + iy in the given equation and uses modulus to find equation in x and y only

A1: Correct equation in terms of x and y in any form – may be in terms of r and θ

M1: Introduces polar form, expands and uses 2 2cos sin 1 leading to a polar equation

A1*: Deduces the given equation (ignore any reference to r = 0 which gives a point on the curve)

(b)(i)

B1: Correct pair of rays added to their diagram

B1ft: Area between their pair of rays and inside their circle from (a) shaded, as long as there is an

intersection.

(b)(ii)

M1: Selects an appropriate method by linking the diagram to the polar curve in (a), evidenced by use

of the polar area formula

M1: Uses double angle identities

A1: Correct integral

M1: Integrates and applies limits

A1: Correct area

(b)(ii) Alternative:

M1: Selects an appropriate method by finding angle ACB and area of sector ACB and finds area of

triangle OCB to make progress towards finding the required area

A1: Correct combined area of sector ACB + triangle OCB

M1: Starts the process of finding the area of triangle OAC by calculating angle ACO and attempts

area of triangle OAC

M1: Uses the addition formula to find the exact area of triangle OAC and employs a full correct

method to find the area of the shaded region

A1: Correct area


69


7(a) 2

2

d d d d10 2 10 2

d d d d

f r f fr f

t t t t M1 2.1

2

2

d d d10 2 0.2 0.4 10 2

d d d

f f ff f

t t t

M1 2.1

2

2

d d0.6 0.1 0*

d d

f ff

t t A1* 1.1b

(3)

(b) 22 0.6 0.6 4 0.1

0.6 0.1 02

m m m

M1 3.4

0.3 0.1im A1 1.1b

e cos sintf A t B t M1 3.4

0.3e cos0.1 sin 0.1tf A t B t A1 1.1b

(4)

(c) 0.3 0.3d

0.3e cos0.1 sin 0.1 0.1e cos0.1 sin 0.1d

t tfA t B t B t A t

t M1 3.4

0.3 0.3

d10 2

d

e 3 cos0.1 3 sin 0.1 2e cos0.1 sin 0.1t t

fr f

t

A B t B A t A t B t

M1 3.4

0.3e cos0.1 sin 0.1tr A B t B A t A1 1.1b

(3)

(d)(i) 0, 6 6t f A M1 3.1b

0, 20 14t r B M1 3.3

0.3e 20cos0.1 8sin 0.1 0tr t t M1 3.1b

tan0.1 2.5t A1 1.1b

2019 A1 3.2a

(d)(ii) 3750 foxes B1 3.4

(d)(iii) E.g.

The model predicts a large number of foxes are on the island

when the rabbits have died out and this may not be sensible.

B1 3.5a

(7)

(17 marks)


70

Notes

(a)

M1: Attempts to differentiate the first equation with respect to t

M1: Proceeds to the printed answer by substituting into the second equation

A1*: Achieves the printed answer with no errors

(b)

M1: Uses the model to form and solve the auxiliary equation

A1: Correct values for m

M1: Uses the model to form the CF

A1: Correct CF

(c)

M1: Differentiates the expression for the number of foxes

M1: Uses this result to find an expression for the number of rabbits

A1: Correct equation

(d)(i)

M1: Realises the need to use the initial conditions in the model for the number of foxes

M1: Realises the need to use the initial conditions in the model for the number of rabbits to find

both unknown constants

M1: Obtains an expression for r in terms of t and sets = 0

A1: Rearranges and obtains a correct value for tan

A1: Identifies the correct year

(d)(ii)

B1: Correct number of foxes

(d)(iii)

B1: Makes a suitable comment on the outcome of the model


71


72


Further Mathematics Advanced Paper 3: Further Pure Mathematics 1


Paper Reference(s)

9FM0/3A

You must have: Mathematical Formulae Calculator


Instructions



Fill in the boxes at the top of this page with your name, centrenumber and candidate number.

Answer all questions and ensure that your answers to parts ofquestions are clearly labelled.

Answer the questions in the spaces provided– there may be more space than you need.

You should show sufficient working to make your methods clear.Answers without working may not gain full credit.

Inexact answers should be given to three significant figures unlessotherwise stated.

Information


There are 8 questions in this question paper. The total mark for thispaper is 75.

The marks for each question are shown in brackets– use this as a guide as to how much time to spend on each question.

Advice





73


1. Use Simpson’s Rule with 6 intervals to estimate

(5)

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75

2. Given k is a constant and that

y = x3 ek x

use Leibnitz theorem to show that

dn y

dxn= k n-3ek x k3x3 + 3nk 2x2 + 3n(n -1)k x + n(n -1)(n - 2)( )

(4)

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77

3. A vibrating spring, fixed at one end, has an external force acting on it such that the

centre of the spring moves in a straight line. At time t seconds, the displacement

of the centre C of the spring from a fixed point O is x micrometres.

The displacement of C from O is modelled by the differential equation

t2 d 2x

dt2- 2t

dx

dt+ (2 + t2 )x = t4 (I)

(a) Show that the transformation x = t v transforms equation (I) into the equation

d 2v

dt2+ v = t (II)

(5)

(b) Hence find the general equation for the displacement of C from O at time t seconds.

(7)

(c) (i) State what happens to the displacement of C from O as t becomes large.

(ii) Comment on the model with reference to this long term behaviour.

(2)

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79

4.

d 2 y

dx2- 2x

dy

dx+ y = 0 (I)

(a) Show that

d 5 y

dx5= ax

d 4 y

dx4+ b

d3y

dx3

where a and b are integers to be found.

(4)

(b) Hence find a series solution, in ascending powers of x, as far as the term in x5,

of the differential equation (I) where y = 0 and d

1d

y

x at x = 0

(5)

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81

5. The normal to the parabola y2 = 4ax at the point

P ap2 , 2ap( ) passes through the

parabola again at the point Q aq2 , 2aq( ).

The line OP is perpendicular to the line OQ, where O is the origin.

Prove that p2 = 2

(9)

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83

6. A tetrahedron has vertices A(1, 2, 1), B(0, 1, 0), C (2, 1, 3) and

D(10, 5, 5).

Find

(a) a Cartesian equation of the plane ABC.

(3)

(b) the volume of the tetrahedron ABCD.

(3)

The plane P has equation 2x - 3y + 3 = 0

The point E lies on the line AC and the point F lies on the line AD.

Given that P contains the point B, the point E and the point F,

(c) find the value of k such that

(3)

Given that

(d) show that the volume of the tetrahedron ABCD is 45 times the volume of the

tetrahedron ABEF.

(2)

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85

7. P and Q are two distinct points on the ellipse described by the equation x2 + 4y2 = 4

The line l passes through the point P and the point Q.

The tangent to the ellipse at P and the tangent to the ellipse at Q intersect at the

point (r , s).

Show that an equation of the line l is

4s y + r x = 4

(8)

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87

8.

Figure 1

Figure 1 shows the graph of the function h(x) with equation

h( ) 45 15sin 21sin 25cos [0, 40]2 2

x xx x x

(a) Show that

dh

dx=

t2 - 6t - 17( ) 9t2 + 4t - 3( )2 1 + t2( )

2

where tan .4

xt

(6)

Figure 2

Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from

08:00 on the 3rd January 2017 to the end of the 4th January 20171.

The graph of k h(x), where k is a constant and x is the number of hours after 08:00 on 3rd

of January, can be used to model the predicted tide heights, in metres, for this period of

time.

(b) (i) Suggest a value of k that could be used for the graph of h( )k x to form a suitable

model.

(ii) Why may such a model be suitable to predict the times when the tide heights are

at their peaks, but not to predict the heights of these peaks?

(3)

(c) Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of

the highest tide height on the 4th January 2017.

(6) 1Data taken on 29th December 2016 from http://www.ukho.gov.uk/easytide/EasyTide


88


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90

AL Paper 3: Further Pure Mathematics 1 Mark Scheme


1. Step 0.5 B1 1.1b

1 1.5 2 2.5 3 3.5 4 3

M1 1.1b

0 1 2 3 4 5 64 2 4 2 4 "77.23"y y y y y y y M1 1.1b

1 x3 dx1

4

0.5

3 "77.23" M1 1.1b

12.9 A1 1.1b

(5) (5 marks)

Notes B1 Use of step length 0.5 M1 Attempt to find y values with at least 2 correct. M1 Use of formula" y

0 4y

1 2y

2 4y

3 2y

4 4y

5 y

6" with correct coefficients

A1 0.5

3 their 77.23

A1 awrt 12.9


2. y x3ekx so u x3 and

2 3 4

22 3 4

d d d d3 and 6 and 6 and 0

d d d d

u u u ux x

x x x x

M1 1.1b

1 21 2

1 2

d d de and e and e and e

d d d(and...)

n n nkx n kx n kx n kx

n n n

v v vv k k k

x x x

M1 2.1

( 1) ( 1)( 2)3 2 1 2 3

2 3!

de 3 e 6 e 6 e

d

nn n n n nn kx n kx n kx n kx

n

yx k n x k xk k

x

and remaining terms disappear. M1 2.1

So dn y

dxn k n3ekx k3x3 3nk 2x2 3n(n1)kx n(n1)(n 2) * A1* 1.1b

(4) (4 marks)

Notes M1 Differentiate 3u x three times. M1 Use u ekx and establish the form of the derivatives, with at least the three shown. M1 Uses correct formula, with 2 and 3! (or 6) and with terms shown to disappear after the

fourth term. A1* Correct solution leading to the given answer stated. No errors seen.

0y 1y 2y 3y 4y 5y 6y

xy 2 4.375 16.625 28 43.875 65


91


3. (a) Use of x tv to give

dx

dt v t

dv

dt M1 1.1b

Hence

2 2

2 2

d d d d

d d d d

x v v vt

t t t t

M1 2.1 A1 1.1b Uses

t2 their 2ndderivative 2t their 1stderivative (2 t2 )x t4

and simplifies LHS.

M1 2.1

t3 d2v

dt2 t3v t4 leading to

2

2

d*

d

vv t

t A1* 1.1b

(5)

(b) Solve 2 1 0 to give 2 1 M1 1.1b

Acos t Bsin t A1ft 1.1b Particular Integral is kt l B1 2.2a

2

2

d dand 0 and solve 0 to give 1, 0

d d

v vk kt l t k l

t t M1 1.1b

Solution: Acos t Bsin t t A1 1.1b

Displacement of C from O is given by ...x tv M1 3.4

cos sinx t A t B t t A1 2.2a

(7) (c)(i) For large t, the displacement gets very large (and positive). B1 3.2a (ii) Model suggests midpoint of spring moving relative to fixed point

has large displacement when t is large, which is unrealistic. The spring may reach elastic limit / will break.

B1 3.5a

(2) (14 marks)

Notes 3. (a) M1 Uses product rule to obtain first derivative. M1 Continues to differentiate again, with product rule and chain rule as appropriate, in order

to establish the second derivative. A1 Correct second derivative. Accept equivalent expressions. M1 Shows clearly the substitution into the given equation in order to form the new equation

and gathers like terms. A1* Fully correct solution leading to the given answer. (b) Accept variations on symbols for constants throughout. M1 Form and solve a quadratic Auxiliary Equation. A1ft Correct form of the Complementary Function for their solutions to the AE. B1 Deduces the correct form for the Particular Integral (note v mt2 kt l is fine). M1 Differentiates their Particular Integral and substitutes their derivatives into the equation to

find the constants (m = 0 if used). A1 Correct general solution for equation (II). M1 Links the solution to equation (II) to the solution of the model equation correctly to find

the displacement equation. A1 Deduces the correct general solution for the displacement.


92

Notes continued

3. (c)(i)

B1 States that for large t the displacement is large o.e. Accept e.g. as t , x. (c)(ii) B1 Reflect on the context of the original problem. Accept ‘model unrealistic’ / ‘spring will

break’.


4. (a) '' 2 ' ''' 2 '' 2 ' 'y xy y y xy y y M1 1.1b A1 1.1b

''' 2 '' ' '''' 2 ''' 2 '' ''y xy y y xy y y M1 2.1 y '''' 2xy ''' 3y '' y ''''' 2xy '''' 5y ''' A1 2.1 (4)

(b) 0, 0, ' 1 ''(0) 0x y y y from equation B1 2.2a

y '''(0) 2 0 y ''(0)1 1; y ''''(0) 2 01 3 0 0; x 0, y '''(0) 1, y ''''(0) 0 y '''''(0) 5

M1 1.1b A1 1.1b

2 3 4 5''(0) '''(0) ''''(0) '''''(0)(0) '(0) ...

2 6 24 120

y y y yy y y x x x x x M1 2.5

Series solution: y x

1

6x3

1

24x5 ... A1ft 1.1b

(5) (9 marks)

Notes 4. (a) M1 Attempts to differentiate equation with use of the product rule. A1 cao. Accept if terms all on one side. M1 Continues the process of differentiating to progress towards the goal. Terms may be kept

on one side, but an expression in the fourth derivative should be obtained. A1 Completes the process to reach the fifth derivative and rearranges to the correct form to

obtain the correct answer by correct solution only. (b) B1 Deduces the correct value for '' (0)y from the information in the question. M1 Finds the values of the derivatives at the given point. A1 All correct M1 Correct mathematical language required with given denominators. Can be in factorial

form. A1ft Correct series, must start y = …. Follow through the values of their derivatives at 0.


93


5. 2 d4 2 4

d

yy ax y a

x M1 2.1

d 2

d

y a

x y

Gradient of normal is

2

yp

a

A1 1.1b

Equation of normal is : 22 ( )y ap p x ap M1 1.1b

Normal passes through 2( ,2 )Q aq aq so 2 32 2aq apq ap ap M1 3.1a

Grad OP Grad OQ = 12 2

2 21

ap aq

ap aq M1 2.1

4

qp

A1 1.1b

3

2

4 162 2a ap ap ap

p p

p4 2 p2 8 0 M1 2.1

( p2 2)( p2 4) 0 p2 ... M1 1.1b

Hence (as p2 4 0), p2 2* A1* 1.1b

(9)

5. ALT 1

First three marks as above and then as follows. M1 2.1 A1 1.1b M1 1.1b

Solves y2 4ax and their normal simultaneously to find, in terms

of a and p, either 22

44Q

ax ap a

p

or 4

2Q

ay ap

p

M1 3.1a

Finds the second coordinate of Q in terms of a and p M1 1.1b

Both 22

44Q

ax ap a

p and

42Q

ay ap

p A1 1.1b

Grad OP Grad OQ = 12

4

2 2 4

221

4

ap

ap

apap

ap ap a

M1 2.1

Simplifies expression and solves: 4 p2 8 p4 4 p2 4

p4 4 0 ( p2 2)( p2 2) 0 p2 ...M1 2.1

Hence (as p2 2 0), p2 2* A1* 1.1b

(9)


94


5. ALT 2

First three marks as above and then as follows. M1 2.1 A1 1.1b M1 1.1b

Solves y2 4ax and their normal simultaneously to find, in terms

of a and p, either 22

44Q

ax ap a

p

or 4

2Q

ay ap

p

M1 3.1a

Forms a relationship between p and q from their first coordinate:

either 2 2

2Qy a p q pp p

or xQ a p

2

p

2

q p 2

p

M1 2.1

2

q pp

(if x coordinate used the correct root must be clearly

identified before this mark is awarded). A1 1.1b

Grad OP Grad OQ = 1 2 2

2 21

ap aq

ap aq

4.q

p

M1 2.1

Sets 2 4

q pp p

and solves to give p2 ... M1 1.1b

2 4

Hence as = gives no solution ,q pp p

p2 2 (only)* A1* 1.1b

(9) (9 marks)

Notes M1 Begins proof by differentiating and using the perpendicularity condition at point P in

order to find the equation of the normal. A1 Correct gradient of normal, p only.

M1 Use of 1 1( )y y m x x . Accept use of y mx c and then substitute to find c. M1 Substitute coordinates of Q into their equation to find an equation relating p and q. M1 Use of 1 2 1m m with OP and OQ to form a second equation relating p and q.

A1 4

qp

only.

M1 Solves the simultaneous equations and cancels a from their results to obtain a quadratic equation in p2 only.

M1 Attempts to solve their quadratic in p2. Usual rules. A1* Correct solution leading to given answer stated. No errors seen.


95

Notes continued

5. ALT 1 M1A1M1 As main scheme. M1 Solves y2 4ax and their normal simultaneously to find one of the coordinates for

Q in terms of a and p as shown. M1 Finds the second coordinate of Q in terms of a and p. A1 Both coordinates correct in terms of a and p. M1

Use of m1m

2 1 with OP and OQ. i.e.

2

their21

theirQ

Q

yap

ap x with coordinates of

P and their expressions for xQ

and yQ

.

M1 Cancels the a’s, simplifies to a quadratic in p2 and solves the quadratic. Usual rules. A1* Correct solution leading to the given answer stated. No errors seen. ALT 2 M1A1M1 As main scheme. M1 Solves y2 4ax and their normal simultaneously to find one of the coordinates for

Q in terms of a and p as shown. M1 Uses their coordinate to form a relationship between p and q. Allow

2q p

p

for this mark. A1 For

2.q p

p If the x coordinate was used to find q then consideration of the

negative root is needed for this mark. Allow for 2

.q pp

M1 Use of 1 2 1m m with OP and OQ to form a second equation relating p and q only.

M1 Equates expressions for q and attempts to solve to give p2 .... A1* Correct solution leading to the given answer stated. No errors seen. If x coordinate

used, invalid solution must be rejected.


96


6. (a) AB AC

1

1

1

1

12

211 2

11

31

2

M1 1.1b

r.31

2

0

1

0

.

3

1

2

1 M1 1.1b

Hence 3x y 2z 1 A1 1.1b (3)

(b) Volume of Tetrahedron = 1

.( )6

n AD M1 3.1a

1

6

31

2

.105

5

1

2

1

M1 1.1b

1

6(27 3 8)

8

3 A1 1.1b

(3)

(c) kAE AC so E is (1 k , 2 k , 1 2k) M1 3.1a E lies on plane so 2(1 k) 3(2 k) 3 0, leading to k ... M1 3.1a

Hence

k

1

5 A1 1.1b

(3)

(d) Volume ABEF = 1

6AB AE .AF

1

6AB

1

5AC

.1

9AD M1 3.1a

1 1

45 6

AB AC . AD and hence result. * A1* 2.2a

(2) (11 marks)

Notes 6. (a) M1 Attempting a suitable cross product. Accept use of unit vectors. M1 Complete method that would lead to finding the Cartesian equation of plane. A1 Accept any equivalent form. (b) M1 Identifies suitable vectors and attempts to substitute into a correct formula. Accept use of

unit vectors. M1 Correct form of scalar triple product using their n from part (a).

A1 8

3 or exact equivalent form.


97

Notes continued

6. (c) M1 Uses that E is on AC in order to find an expression for E. M1 Uses that E is in the plane to form and solve an expression in k.

A1 1

5 o.e. only.

(b) M1 Uses formula for volume of tetrahedron and substitutes for AE and AF

A1* Deduces result: Use of

1

6AB AC .AD is required and no errors seen in solution.


7. x2 4y2 4 2x 8y

dy

dx 0

dy

dx ... M1 3.1a

Equation of tangent at 1 1( , )P x y is ( y y1)

x1

4y1

(x x1) M1 3.1a

xx1 4yy

1 x

12 4y

12 4 and at Q (x

2, y

2) : xx

2 4yy

2 4 A1 2.2a

Intersect at (r , s) gives rx1 4sy

1 4 and rx

2 4sy

2 4 B1 2.1

Uses their previous results to find the gradient of the line l. M1 3.1a

y

2 y

1

x2 x

1

r

4s A1 1.1b

Equation of l is y y

1 r

4s(x x

1) M1 2.1

4sy rx 4sy1 rx

1 4* A1* 2.2a

(8) (8 marks)

Notes

M1 Attempts to solve the problem by using differentiation to obtain an expression for dy

dx.

M1 Realise the need to form a general equation of the tangent at 1 1( , ).x y May use alternative

variables. A1 Deduces 2 2

1 14 4x y to obtain a correct equation and deduces a correct second equation.

B1 Uses (r, s) in both equations to form the two given equations or exact equivalents.

M1 Uses their previous results to find the gradient of the line l.

A1

r

4s

M1 Formulates the line l with their

r

4s. Use of 1 1( )y y m x x or y mx c with their

gradient and an attempt to find c. A1* Correct solution leading to 1 14 4sy rx sy rx with deduction that this equals 4 as

1 1( , )x y is on the ellipse. No errors seen.


98


8. h( ) 45 15sin 21sin 25cos2 2

x xx x

(a)

dh

dx

21 2515cos cos sin

2 2 2 2

x xx

M1 1.1b

dh

dx

2

2 2

1 2... .. ..

1 1

t t

t t

M1 1.1a

E.g.

dh

dx

22

2

1.. 2 1 ... or

1

t

t

dh

dx

2

2

2

2

21

1.. ...

21

1

tt

tt

M1 3.1a

E.g.

dh

dx

22 2

2 2 2

1 21 1 25 215 2 1

1 2 1 2 1

t t t

t t t

A1 1.1b

…2 2 2 2 2 2 2

2 2

15[4(1 ) 2(1 ) ] 21(1 )(1 ) 50 (1 )

2(1 )

t t t t t t

t

M1 2.1

…4 3 2 2 2

2 2 2 2

9 50 180 50 51 ( 6 17)(9 4 3)*

2(1 ) 2(1 )

t t t t t t t t

t t

A1* 2.1

(6)

(a) ALT 1

2

2 2

2 1h( ) ... 21 25

1 1

t tx

t t

M1 1.1a

2

2 2

2 1... 15 2 ...

1 1

t t

t t

2

2

2

22

1or ... 15 ...

21

1

tt

tt

M1 2.1

2 2 4

2 2

15(4 (1 )) 42 (1 ) 25(1 )h( ) 45

(1 )

t t t t tx

t

M1 1.1b

4 3 4 3 2

2 2 2 2

25 18 102 25 20 18 90 102 70h( ) 45 or

(1 ) (1 )

t t t t t t tx

t t

A1 1.1b

dh

dx

dh

dt

dt

dx

('u ')(1 t2 )2 ('u ')(4t(1 t2 ))

(1 t2 )4 M1 3.1a

4 3 2 2 2

2 2 2 2

9 50 180 50 51 ( 6 17)(9 4 3)... *

2(1 ) 2(1 )

t t t t t t t t

t t

A1* 2.1

(6)

(b)(i) Accept any value between 1

40 0.025and

1

60 0.167inclusive. B1 3.3

(ii) Suitable for times since the graphs both oscillate bi-modally with about the same periodicity.

B1 3.4

Not suitable for predicting heights since the heights of the peaks vary over time, but the graph of h (x) has fixed peak height.

B1 3.5b

(3)

1

4(1 t2 )


99


8. (c) Solves at least one of the quadratics

M1 1.1b

Finds corresponding x values, for at least one value

of t from the factor.M1 1.1b

One correct value for these x. A1 1.1b

Maximum peak height occurs at smallest positive value of x, from first graph, but the third of these peaks needed, so is the required time.

M1 3.4

corresponds to 26 hours and 39 minutes

(nearest minute) after 08:00 on 3rd January. (Allow if a different greatest peak height used.)

M1 3.4

Time of greatest tide height is approximately 10:39 (am) (also allow 10:38 or 10:40)

A1 3.2a

(6) (15 marks)

Notes 8. (a) M1 Differentiates h(x).

M1 Applies t-substitution to both terms with their coefficients.

M1 Forms a correct expression in t for the cos x term, using double angle formula and t-substitution, or double ‘t’-substitution.

A1 Fully correct expression in t for dh

dx.

M1 Gets all terms over the correct common factor. Numerators must be appropriate for their terms.

A1* Achieves the correct answer via expression with correct quartic numerator before factorisation.

t 6 36 4117

2 3 26

or t 4 16 4 9 (3)

182 31

9x 4tan1(t)

29 4 3t t e.g. x awrt 2.797 or 9.770,1.510

t 1.509...8 26.642...26.642....x

x

2


100

Notes Continued

8. (a) ALT 1

M1 Applies t-substitution to both terms.

M1 Forms a correct expression in t for the sin x term, using double angle formula and t-substitution, or double ‘t’-substitution.

M1 Gets all terms in t over the correct common factor. Numerators must be appropriate for their terms. May include the constant term too.

A1 Fully correct expression in t for . M1 Differentiates, using both chain rule and quotient rule with their ‘u’. A1* Achieves the correct answer via expression with correct quartic numerator before

factorisation. NOTE The individual terms may be differentiated before putting over a common denominator.

In this case score the third M for differentiating with chain rule and quotient rule, then return to the original scheme.

(b) (i) B1 Any value between

1

40 (e.g. taking h(0) as reference point) or

1

60 (taking lower peaks

as reference). NB: Taking high peak as reference gives 1

50.

(b)(ii) B1 Should mention both the bimodal nature and periodicity for the actual data match the

graph of h. B1 Mentions that the heights of peaks vary in each oscillation. (c) M1 Solves (at least) one of the quadratic equations in the numerator. M1 Must be attempting to solve the quadratic factor from which the solution comes

and using to find a corresponding value for x.

A1 At least one correct x value from solving the requisite quadratic: awrt any of – 2.797, 1.510, 9.770, 14.076, 22.336, 26.642, 34.902 or 39.208.

M1 Uses graph of h to pick out their x = 26.642 as the time corresponding to the third of the higher peaks, which is the highest of the peaks on 4th January on the tide height graph. As per scheme or allow if all times listed and correct one picked.

M1 Finds the time for one of the values of t corresponding to the highest peaks. E.g. 1.5096...~ 09:31 (3rd January) or 14.076... ~ 22:05 (3rd January) or 26.642…~ 10:39 (4th January) or 39.208…~ 23:13 (4th January). (Only follow through on use of the smallest positive t solution 4k .)

A1 Time of greatest tide height on 4th January is approximately 10:39. Also allow 10:38 or 10:40.

x

2

h( )x

2(9 4 3)t t tan4

xt


101


102


Further Mathematics Advanced Paper 4: Further Mathematics Option 2 Option 4A: Further Pure Mathematics 2


Paper Reference(s)

9FM0/4A

You must have: Mathematical Formulae Calculator


Instructions







Inexact answers should be given to three significant figures unless otherwise stated.

Information




Advice





103


1. (i) Use the Euclidean algorithm to find the highest common factor of 602 and 161.

Show each step of the algorithm.

(3)

(ii) The digits which can be used in a security code are 1, 2, 3, 4, 5, 6, 7, 8 and 9

Originally the code used consisted of two distinct odd digits, followed by three

distinct even digits.

To enable more codes to be generated, a new system is devised. This uses two

distinct even digits, followed by any three other distinct digits. No digits are

repeated.

Find the increase in the number of possible codes which results from using the new

system.

(4)

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105

2. A transformation from the z-plane to the w-plane is given by

2w z

(a) Show that the line with equation Im(z) = 1 in the z-plane is mapped to a parabola in

the w-plane, giving an equation for this parabola.

(4)

(b) Sketch the parabola on an Argand diagram.

(2)

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107

3. The matrix M is given by

M =

2 1 0

1 2 0

1 0 4

(a) Show that 4 is an eigenvalue of M, and find the other two eigenvalues.

(4)

(b) For each of the eigenvalues find a corresponding eigenvector.

(4)

(c) Find a matrix P such that 1P MP is a diagonal matrix.

(2)

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111

4. (i) A group G contains distinct elements a, b and e where e is the identity element and

the group operation is multiplication.

Given 2

a b ba , prove ab ba

(4)

(ii) The set H = {1, 2, 4, 7, 8, 11, 13, 14} forms a group under the operation of

multiplication modulo 15

(a) Find the order of each element of H.

(3)

(b) Find three subgroups of H each of order 4, and describe each of these subgroups.

(4)

The elements of another group J are the matrices

4 4

4 4

cos sin

sin cos

k k

k k

where k =1, 2, 3, 4, 5, 6, 7, 8 and the group operation is matrix multiplication.

(c) Determine whether H and J are isomorphic, giving a reason for your answer.

(2)

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115

5.

Figure 1

An engineering student makes a miniature arch as part of the design for a piece of

coursework.

The cross-section of this arch is modelled by the curve with equation

12cosh 2y A x , ln lna x a

where a >1 and A is a positive constant. The curve begins and ends on the x-axis, as

shown in Figure 1.

(a) Show that the length of this curve is 2

2

1k a

a

, stating the value of the constant k.

(5)

The length of the curved cross-section of the miniature arch is required to be 2 m long.

(b) Find the height of the arch, according to this model, giving your answer to

2 significant figures.

(4)

(c) Find also the width of the base of the arch giving your answer to 2 significant

figures.

(1)

(d) Give the equation of another curve that could be used as a suitable model for the

cross-section of an arch, with approximately the same height and width as you found

using the first model.

(You do not need to consider the arc length of your curve)

(2)


116


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119

6. A curve has equation

6 2 6z z z

(a) Show that the curve is a circle with equation 2 2 20 36 0x y x

(2)

(b) Sketch the curve on an Argand diagram.

(2)

The line l has equation * * 0az a z , where a and z

Given that the line l is a tangent to the curve and that arg a = θ

(c) find the possible values of tan

(5)

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123

7.

2

0sin dn

nI x x

, n 0

(a) Prove that, for n 2,

2( 1)n nnI n I

(4)

(b)

Figure 2

A designer is asked to produce a poster to completely cover the curved surface area of a

solid cylinder which has diameter 1m and height 0.7 m.

He uses a large sheet of paper with height 0.7 m and width of π m.

Figure 2 shows the first stage of the design, where the poster is divided into two sections

by a curve.

The curve is given by the equation

2 10sin (4 ) sin (4 )y x x

relative to axes taken along the bottom and left hand edge of the paper.

The region of the poster below the curve is shaded and the region above the curve

remains unshaded, as shown in Figure 2.

Find the exact area of the poster which is shaded.

(5)

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127

8. A staircase has n steps. A tourist moves from the bottom (step zero) to the top (step n).

At each move up the staircase she can go up either one step or two steps, and her overall

climb up the staircase is a combination of such moves.

If nu is the number of ways that the tourist can climb up a staircase with n steps,

(a) explain why nu satisfies the recurrence relation

1 2n n nu u u , with 1 1u and 2 2u

(3)

(b) Find the number of ways in which she can climb up a staircase when there are eight

steps.

(1)

A staircase at a certain tourist attraction has 400 steps.

(c) Show that the number of ways in which she could climb up to the top of this staircase

is given by

401 401

1 1 5 1 5

2 25

(5)

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131


132

Paper 4: Further Pure Mathematics 2 Mark Scheme


1(i) 602 3 161 119 M1 1.1b

161 119 42, 119 2 42 35 M1 1.1b

42 35 7, 35 5 7, hcf 7 A1 1.1b

(3)

(ii) Number of codes under old system = 5 4 4 3 2 ( 480) B1 3.1b

Number of codes under new system = 4 3 7 6 5 ( 2520) B1 3.1b

Subtracts first answer from second M1 1.1b

Increase in number of codes is 2040 A1 1.1b

(4)

(7 marks)

Notes:

(i)

M1: Attempts Euclid’s algorithm – (there may be an arithmetic slip finding 119)

M1: Uses Euclid’s algorithm a further two times with 161 and “their 119” and then with “their 119”

and “their 42”

A1: This should be accurate with all the steps shown

(ii)

B1: Correctly interprets the problem and uses the five odd digits and four even digits to form a correct

product

B1: Interprets the new situation using the four even digits, then the seven digits which have not been

used, to form a correct product

M1: Subtracts one answer from the other

A1: Correct answer


133


2(a) Let z = x + i M1 2.1

2 2( i) ( 1) 2 iw x x x A1 1.1b

Let w = u + iv, then 2

( 1) and 2u x v x M1 2.1

2

4( 1)v u , which therefore represents a parabola A1ft 2.2a

(4)

(b)

M1

1.1b

A1 1.1b

(2)

(6 marks)

Notes:

(a)

M1: Translates the information that Im(z) = 1 into a cartesian form; e.g. z = x + i

A1: Obtains a correct expression for w

M1: Separates the real and imaginary parts and equates to u and v respectively

A1ft: Obtains a quadratic equation and states that their quadratic equation represents a parabola

(b)

M1: Sketches a parabola with symmetry about the real axis

A1: Accurate sketch

M1: Sketches a

parabola with

symmetry about

the real axis

A1: Accurate

sketch


134


3(a) Finds the characteristic equation

2(2 ) (4 ) (4 ) 0 M1 2.1

so 2

(4 )( 4 3) 0 so 4 * A1* 2.2a

Solves quadratic equation to give M1 1.1b

1 and 3 A1 1.1b

(4)

(b) Uses a correct method to find an eigenvector M1 1.1b

Obtains a vector parallel to one of

0

0

1

or

1

1

1

or

3

3

1

A1 1.1b

Obtains two correct vectors A1 1.1b

Obtains all three correct vectors A1 1.1b

(4)

(c) Uses their three vectors to form a matrix M1 1.2

0 1 3

0 1 3

1 1 1

or other correct answer with

columns in a different order.

A1 1.1b

(2)

(10 marks)

Notes:

(a)

M1: Attempts to find the characteristic equation (there may be one slip)

A1*: Deduces that 4 is a solution by the method shown or by checking that 4 satisfies the

characteristic equation

M1: Solves their quadratic equation

A1: Obtains the two correct answers as shown above

(b)

M1: Uses a correct method to find an eigenvector

A1: Obtains one correct vector (may be a multiple of the given vectors)

A1: Obtains two correct vectors (may be multiples of the given vectors)

A1: Obtains all three correct vectors (may be multiples of the given vectors)


135

Notes: (continued)

(c)

M1: Forms a matrix with their vectors as columns

A1:

0 1 3

0 1 3

1 1 1

or

1 0 3

1 0 3

1 1 1

or

3 1 0

3 1 0

1 1 1

or other correct alternative


4(i) 2 2If we assume ; as thenab ba a b ba ab a b M1 2.1

1 1 1 2 1So a abb a a bb M1 2.1

So e a A1 2.2a

But this is a contradiction, as the elements e and a are distinct so

ab ba A1 2.4

(4)

(ii)(a) 2 has order 4 and 4 has order 2 M1 1.1b

7, 8 and 13 have order 4 A1 1.1b

11 and 14 have order 2 and 1 has order 1 A1 1.1b

(3)

(ii)(b) Finds the subgroup {1, 2, 4, 8} or the subgroup {1, 7, 4, 13} M1 1.1b

Finds both and refers to them as cyclic groups, or gives generator 2 and

generator 7 A1 2.4

Finds {1, 4, 11, 14} B1 2.2a

States each element has order 2 or refers to it as Klein Group B1 2.5

(4)

(ii)(c) J has an element of order 8, (H does not) or J is a cyclic group (H is

not) or other valid reason

M1 2.4

They are not isomorphic

A1 2.2a

(2)

(13 marks)


136

Notes:

(i)

M1: Proof begins with assumption that 2and deduces that this impliesab ba ab a b

M1: A correct proof with working shown follows, and may be done in two stages

A1: Concludes that assumption implies that e a

A1: Explains clearly that this is a contradiction, as the elements e and a are distinct so ab ba

(ii)(a)

M1: Obtains two correct orders (usually the two in the scheme)

A1: Finds another three correctly

A1: Finds the final three so that all eight are correct

(ii)(b)

M1: Finds one of the cyclic subgroups

A1: Finds both subgroups and explains that they are cyclic groups, or gives generators 2 and 7

B1: Finds the non cyclic group

B1: Uses correct terms that each element has order 2 or refers to it as Klein Group

(ii)(c)

M1: Clearly explains how J differs from H

A1: Correct deduction


137


5(a)

dsinh 2

d

yx

x

B1 2.1

so S = 21 sinh 2 dx x M1 2.1

s cosh 2 dx x A1 1.1b

= ln

12 lnsinh 2

a

ax

or

ln

0sinh 2

ax M1 2.1

= 2ln -2ln1

2sinh 2ln [e e ]a a

a = 21

2 2

1a

a

(so k = ½) A1 1.1b

(5)

(b) 21

2 2

1a

a

= 2 so

4 24 1 0a a M1 1.1b

2 2 5a (and a = 2.06 (approx.) ) M1 1.1b

When x = lna, y = 0 so 1

2cosh 2lnA a M1 3.4

Height = A – 0.5 = awrt 0.62m A1 1.1b

(4)

(c) The width of the base = 2lna = 1.4m B1 3.4

(1)

(d) A parabola of the form 20.62 1.19y x , or other symmetric curve

with its equation e.g. 0.62cos(2.2x) M1A1

3.3

3.3

(2)

(12 marks)

Notes:

(a) B1: Starts explanation by finding the correct derivative

M1: Uses their derivative in the formula for arc length

A1: Uses suitable identity to simplify the integrand and to obtain the expression in scheme

M1: Integrates and uses appropriate limits to find the required arc length

A1: Uses the definition of sinh to complete the proof and identifies the value for k

(b)

M1: Uses the formula obtained from the model and the length of the arch to create a quartic equation

M1: Continues to use this model to obtain a quadratic and to obtain values for a

M1: Attempts to find a value for A in order to find h

A1: Finds a value for the height correct to 2sf (or accept exact answer)


138

Notes: (continued)

(c) B1: Finds width to 2 sf i.e. 1.4m

(d)

M1: Chooses or describes an even function with maximum point on the y axis

A1: Gives suitable equation passing through (0, 0.62) and (0.7, 0) and (- 0.7, 0)


139


6(a) 2 2 2 2( 6) 4[( 6) ]x y x y M1 2.1

2 2

20 36 0x y x which is the equation of a circle* A1* 2.2a

(2)

(b)

M1 1.1b

A1 1.1b

(2)

(c)

Let a = c + id and a* = c − id then (c + id)(x − iy) + (c − id)(x + iy) = 0

M1 3.1a

Soc

y xd

A1 1.1b

The gradients of the tangents (from geometry) are 4

3

B1 3.1a

So 4

3

c

d and

3

4

d

c M1 3.1a

So 3

tan4

A1 1.1b

(5)

(9 marks)


140

Notes:

(a)

M1: Obtains an equation in terms of x and y using the given information

A1*: Expands and simplifies the algebra, collecting terms and obtains a circle equation correctly,

deducing that this is a circle

(b)

M1: Draws a circle with centre at (10, 0)

A1: (Radius is 8) so circle does not cross the y axis

(c)

M1: Attempts to convert line equation into a cartesian form

A1: Obtains a simplified line equation

B1: Uses geometry to deduce the gradients of the tangents

M1: Understands the connection between arg a and the gradient of the tangents and uses this

connection

A1: Correct answers


141


7(a) 2 1

0sin sin dn

nI x x x

M1 2.1

221 2 2

0 0cos sin ( ) cos ( 1)sin dn nx x x n x x

A1 1.1b

Obtains 2 2 2

00 ( ) (1 sin )( 1)sin dnx n x x

M1 1.1b

So 2( 1) ( 1)

n n nI n I n I

and hence 2

( 1)n n

nI n I

* A1* 2.1

(4)

(b) uses

2

( 1)n n

nI I

n

to give

10 8

9

10I I or

2 0

1

2I I M1 3.1b

So 10 0

9 7 5 3 1

10 8 6 4 2I I

M1 2.1

0

2I

B1 1.1b

Required area is 2 102( )I I or

1

2 1048 ( )I I M1 3.1b

263 65

2 m4 512 256

A1 1.1b

(5)

(9 marks)

Notes:

(a) M1: Splits the integrand into the product shown and begins process of integration by parts (there may

be sign errors)

A1: Correct work

M1: Uses limits on the first term and expresses cos2 term in terms of sin2

A1*: Completes the proof collecting nI terms correctly with all stages shown

(b)

M1: Attempts to find 10I and/or 2

I

M1: Finds 10I in terms of 0

I

B1: Finds 0I correctly

M1: States the expression needed to find the required area

A1: Completes the calculation to give this exact answer


142


8(a) 1 1u as there is only one way to go up one step B1 2.4

2 2u as there are two ways: one step then one step or two steps B1 2.4

If first move is one step then can climb the other (n1) steps in 1nu

ways

If first move is two steps can climb the other (n2) steps in 2nu

ways

So 1 2n n nu u u

B1 2.4

(3)

(b) Sequence begins 1, 2, 3, 5, 8, 13, 21, 34,… so 34 ways of climbing 8

steps B1 1.1b

(1)

(c) To find general term use

2

1 2gives 1

n n nu u u

M1 2.1

This has roots 1 5

2

A1 1.1b

So general form is 1 5 1 5

2 2

n n

A B

M1 2.2a

Uses initial conditions to find A and B reaching two equations in A and B M1 1.1b

Obtains 1 5

2 5A

and 1 5

2 5B

and so when n = 400

obtains

401 401

1 1 5 1 5

2 25

*

A1* 1.1b

(5)

(9 marks)

Notes:

(a)

B1: Need to see explanation for 1 1u

B1: Need to see explanation for 2 2u with the two ways spelled out

B1: Need to see the first move can be one step or can be two steps and clear explanation of the

Iterative expression as in the scheme

(b)

B1: The answer is enough for this mark


143

Notes: (continued)

(c)

M1: Obtains this characteristic equation

A1: Solves quadratic – giving exact answers

M1: Obtains a general form

M1: Use initial conditions to obtains two equations which should be 1 5 1 5 2A B o.e.

and 3 5 3 5 4A B but allow slips here.

A1*: Must see exact correct values for A and B and conclusion given for n = 400


144


Further Mathematics Advanced

Further Mathematics Option 1 Paper 3: Further Statistics 1



Paper Reference(s)

9FM0/3B 9FM0/4B



Instructions




Answer all questions.







Advice





145


1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory

opens and a scientist claims it is polluting the river with bacteria. He takes a sample of

0.5 litres of water from the river near the factory and finds that it contains 7 bacteria.

Stating your hypotheses clearly test, at the 5% level of significance, whether there is

evidence that the level of pollution has increased.

(5)

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147

2. A call centre routes incoming telephone calls to agents who have specialist knowledge to

deal with the call. The probability of a caller, chosen at random, being connected to the

wrong agent is p.

The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent

is 0.049

The call centre receives 1000 calls each day.

(a) Find the mean and variance of the number of wrongly connected calls a day.

(7)

(b) Use a Poisson approximation to find, to 3 decimal places, the probability that more than

6 calls each day are connected to the wrong agent.

(2)

(c) Explain why the apporoximation used in part (b) is valid.

(2)

The probability that more than 6 calls each day are connected to the wrong agent using the

binomial distribution is 0.8711 to 4 decimal places.

(d) Comment on the accuracy of your answer in part (b)

(1)

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149


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151

3. Bags of £1 coins are paid into a bank. Each bag contains 20 coins.

The bank manager believes that 5% of the £1 coins paid into the bank are fakes. He

decides to use the distribution X ~ B(20, 0.05) to model the random variable X, the

number of fake £1 coins in each bag.

The bank manager checks a random sample of 150 bags of £1 coins and records the

number of fake coins found in each bag. His results are summarised in Table 1. He then

calculates some of the expected frequencies, correct to 1 decimal place.

Number of fake coins in each bag 0 1 2 3 4 or more

Observed frequency 43 62 26 13 6

Expected frequency 53.8 56.6

8.9

Table 1

(a) Carry out a hypothesis test, at the 5% significance level, to see if the data supports

the bank manager’s statistical model. State your hypotheses clearly.

(10)

The assistant manager thinks that a binomial distribution is a good model but

suggests that the proportion of fake coins is higher than 5%. She calculates the actual

proportion of fake coins in the sample and uses this value to carry out a new

hypothesis test on the data. Her expected frequencies are shown in Table 2.

Number of fake coins in each bag 0 1 2 3 4 or more

Observed frequency 43 62 26 13 6

Expected frequency 44.5 55.7 33.2 12.5 4.1

Table 2

(b) Explain why there are 2 degrees of freedom in this case.

(2)

(c) Given that she obtains a χ2 test statistic of 2.67, test the assistant manager’s

hypothesis that the binomial distribution is a good model for the number of fake

coins in each bag. Use a 5% level of significance and state your hypotheses clearly.

(2)

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152


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154


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155

4. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3

Estimate the probability that the mean of the sample is greater than 2.5

(4)

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157

5. The probability of Richard winning a prize in a game at the fair is 0.15

Richard plays a number of games.

(a) Find the probability of Richard winning his second prize on his 8th game,

(2)

(b) State two assumptions that have to be made, for the model used in part (a) to be valid.

(2)

Mary plays the same game, but has a different probability of winning a prize. She plays

until she has won r prizes. The random variable G represents the total number of games

Mary plays.

(c) Given that the mean and standard deviation of G are 18 and 6 respectively, determine

whether Richard or Mary has the greater probability of winning a prize in a game.

(4)

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159

6. The probability generating function of the discrete random variable X is given by

GX (t) = 2

23 2k t t

(a) Show that k = 1

36

(2)

(b) Find P(X = 3)

(2)

(c) Show that Var(X) = 29

18

(8)

(d) Find the probability generating function of 2X + 1

(2)

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163

7. Sam and Tessa are testing a spinner to see if the probability, p, of it landing on red is less

than 1

5. They both use a 10% significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

(a) Find the critical region for Sam’s test.

(2)

(b) Write down the size of Sam’s test.

(1)

Tessa decides to spin the spinner until it lands on red and she records the number of

spins.

(c) Find the critical region for Tessa’s test.

(6)

(d) Find the size of Tessa’s test.

(1)

(e) (i) Show that the power function for Sam’s test is given by

19

1 1 19p p

(ii) Find the power function for Tessa’s test.

(4)

(f) With reference to parts (b), (d) and (e), state, giving your reasons, whether you

would recommend Sam’s test or Tessa’s test when p = 0.15

(4)

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167


168

AL Further Statistics 1


Q1 Ho : = 5 (= 2.5) H1 : > 5 (> 2.5) B1 2.5

X ~ Po (2.5) B1 3.3

Method 1 Method 2

P 7 1 P )6 ( X X

= 1 − 0.9858

P(X 5) = 0.1088

P(X 6) = 0.042 M1 1.1b

= 0.0142 CR X 6 A1 1.1b

0.0142 < 0.05 7 6 or 7 is in critical region or 7 is significant

Reject H0. There is evidence at the 5% significance level that the

level of pollution has increased.

or

There is evidence to support the scientists claim is justified

A1cso 2.2b

(5 marks)

Notes

B1: Both hypotheses correct using or and 5 or 2.5

B1: Realising that the model Po(2.5) is to be used. This may be stated or used.

M1: Using or writing 1 P ( )6 X or ( ) 1 P < 7 X

a correct CR or P(X 5) = awrt 0.109 and P(X 6) = awrt 0.042

A1: awrt 0.0142 or CR X 6 or X > 5.

A1: A fully correct solution and drawing a correct inference in context


169

Question Scheme

Marks AOs

Q2(a) P(X 1) = 1 – P(X = 0)

1 – P(X = 0) = 0.049 B1 3.1b

P(X = 0) = 0.951 B1 1.1b

5 0.951x

0.99x M1 3.1b

p = 0.01 A1 1.1b

X ~B(1000, 0.01) M1 3.3

Mean = np = 10 A1ft 1.1b

Variance = np(1 – p) = 9.9 A1ft 1.1b

(7)

(b) X ~ Po(“10”) then require: P(X > 6) = 1 – P (X 6) M1 3.4

= 1 – 0.1301

= 0.870 A1 1.1b

(2)

(c) The approximation is valid as : the number of calls is large B1 2.4

the probability of connecting to the wrong agent is small B1 2.4

(2)

(d) The answer is accurate to 2 decimal place B1 3.2b

(1)

(12 marks)

Notes

(a) B1: Realising that the P(at least 1 call ) = 1 – P(X = 0)

B1: Calculating P(X = 0) = 0.951

M1: Forming the equation 5 "their 0.951"x may be implied by p = 0.01

A1: 0.01 only

M1: Realising the need to use the model B(1000, 0.01) This may be stated or used

A1: mean =10 or ft their p but only if 0 < p < 1

A1: Var = 9.9 or ft their p but only if 0 < p < 1

(b) M1: Using the model Po(“their 10”) ( this may be written or used) and 1 – P (X 6)

A1: awrt 0.870 Award M1 A1 for awrt 0.870 with no incorrect working

(c) B1:Explaining why approximation is valid - need the context of number and calls

B1: need the context connecting, wrong agent

(d) B1: Evaluating the accuracy of their answer in (b). Allow 2 significant figures


170

Question Scheme

Marks AOs

Q3(a) Expected value for 2 =150 P( 2)X M1 3.4

= 28.3015… A1 1.1b

Expected value for 4 or more = 150 – (53.8 + 56.6 + 28.3 + 8.9)

= 2.4 A1ft 1.1b

H0: Bin(20, 0.05) is a suitable model

H1: Bin(20, 0.05) is not a suitable model B1 2.5

Combining last two groups

3

Observed frequency 19

Expected frequency 11.3

M1 2.1

B1 1.1b

Critical value, χ2 (0.05) = 7.815 B1 1.1a

Test statistic =

2 243 53.8 62 56.6

...53.8 56.6

M1 1.1b

= 8.117 A1 1.1b

In critical region, sufficient evidence to reject H0, accept H1

Significant evidence at 5% level to reject the manager’s model A1 3.5a

(10)

(b) = 4 – 2 = 2

4 classes due to pooling B1 2.4

2 restrictions (equal total and mean/proportion) B1 2.4

(2)

(c) H0: Binomial distribution is a good model

H1: Binomial distribution is not a good model B1 3.4

Critical value, χ2 (0.05) = 5.991

Test statistic is not in critical region, insufficient evidence to

reject H0

There is evidence that the Binomial distribution is a good model.

B1 3.5a

(2)

(14 marks)

Notes

(a) M1: Using the binomial model 2 18150 (1 )p p may be implied by 28.3

A1: awrt 28.3

A1: awrt 2.4 or ft their “28.3”

B1: Both hypotheses correct using the correct notation or written out in full.

M1: For recognising the need to combine groups

B1: Number of degrees of freedom = 3 may be implied by a correct CV

B1: awrt 7.82

M1: Attempting to find

2

i i

i

O E

E or

2

i

i

ON

E may be implied by awrt 8.12

A1: awrt 8.12

A1: Evaluating the outcome of a model by drawing a correct inference in context

(b) B1: Explaining why there are 4 classes

B1: Explanation of why 2 is subtracted

(c) B1: Correct hypotheses for the refined model

B1: The CV awrt 5.99 and drawing the correct inference for the refined model


171

Question Scheme

Marks AOs

Q4. Po(2.3) n = 100 = 2.3 2 2.3

CLT 2.3

2.3,100

NX

M1

A1

3.1a

1.1b

2.5 2.3( 2.5)

0.023P PX Z

M1 3.4

= P(Z > 1.318..)

= 0.09632… A1 1.1b

(4)

(4 marks)

M1: For realising the need to use the CLT to set X normal with correct mean.

May be implied by using the correct normal distribution.

A1: For fully correct normal stated or used

M1: Use of the normal model to find ( 2.5)P X . Can be awarded for 2.5 2.3

0.023

or

awrt 1.32

A1: awrt 0.0963


172

Question Scheme

Marks AOs

Q5(a)

627

0.15 0.851

M1 3.3

= 0.05940… = awrt 0.0594 A1 1.1b

(2)

(b) The model is only valid if:

the games (trials) are independent B1 3.5b

the probability of winning a prize, 0.15, is constant for each

game B1 3.5b

(2)

(c) 2

2

118 and 6

r pr

p p

M1

A1

3.1b

1.1b

Solving: 2p = 1 – p M1 1.1b

p = 1

3 (> 0.15) so Mary has the greater chance of winning a

prize

A1 3.2a

(4)

(8 marks)

Notes

5(a) M1: For selecting an appropriate model negative binomial or B(7, 0.15) with an

extra success in 8th trial e.g.

67

0.15 0.85 0.151

Allow

670.85 0.15 0.85

1

may be implied by awrt

0.0594

A1: awrt 0.0594

(b) B1: Stating the first assumption that games are independent

B1: Stating the second assumption that the probability remains constant

(c) M1: Forming an equation for the mean or for the standard deviation.

A1: Both equations correct

M1: Solving the 2 equations leading to 2p = 1 – p

A1: For p = 1

3followed by a correct deduction


173


Q6(a) G (1) 1X gives M1 2.1

26 1k so k = 1

36 * A1*cso 1.1b

(2)

(b) P(X = 3) = coefficient of t3 so 3G ( ) ... 4 ...X t k t M1 1.1b

[ P(X = 3) =] 1

9 A1 1.1b

(2)

(c) 2G ( ) 2 3 2 1 4X t k t t t M1 2.1

E(X) = G (1) 2 3 1 2 1 4X k M1 1.1b

= 5

3 A1 1.1b

22G ( ) 2 3 2 4 1 4X t k t t t

M1

A1

2.1

1.1b

2G (1) 2 [6 4 5 ]X k 49

18

M1 1.1b

Var(X) = 2 49 5 25

G (1) G (1) G (1)18 3 9

X X X M1 2.1

= 29

18* A1*cso 1.1b

(8)

(d)

22

2 2

2 1G ( ) 3 236

X

tt t t [ t or sub t2 for t] M1 3.1a

= 2

2 4

2 1G ( ) 3 236

X

tt t t A1 1.1b

(2)

(14 marks)

Notes

6(a) M1: Stating G (1) 1X

A1*cso: Fully correct proof with no errors

(b) M1: Attempting to find the coefficient of t3. May be implied by obtaining

1

9 or awrt

0.11

A1: 1

9, allow awrt 0.111


174

Notes (continued)

(c) M1: Attempting to find G ( )X t . Allow Chain rule or multiplying out the brackets

and differentiating

M1: Substituting t = 1 into G ( )X t

A1: 5

3, allow awrt 1.67

M1: Attempting to find G ( )X t

A1: 222 3 2 4 1 4k t t t

or

2(48 24 26)k t t o.e.

A1: 22 [6 4 5 ]k o.e.

M1: Using 2

G (1) G (1) G (1)X X X to find the Variance

A1*cso: 29

18

(d) M1:Realising the need to t or sub t2 for t

A1: 2

2 43 236

tt t , or 2 4 6 89 6 13 4 4

36

tt t t t o.e.


175

Question Scheme

Marks AOs

Q7(a) X ~ B(20, 0.2) and seek c such that P(X c) < 0.10 M1 3.3

[P(X 1) = 0.0692] CR is X 1 A1 1.1b

(2)

(b) Size = 0.0692 B1ft 1.2

(1)

(c) Y = no. of spins until red obtained so Y~ Geo(0.2) M1 3.3

1

p so if p < 0.2 then mean is larger so seek d so that

P(Y d) < 0.10

M1 2.4

1P( ) 0.8 dY d M1 3.4

1 log(0.1)0.8 0.10 1

log(0.8)

d d M1 1.1b

d > 11.3.. A1 1.1b

CR is Y 12 A1 2.2b

(6)

(d) Size = [ 110.8 = 0.085899…] = 0.0859 B1 1.1b

(1)

(e)(i) Power = P(reject H0 when it is false) = P(X 1 | X ~B(20, p)) M1 2.1

= 20 19

1 20 1p p p M1 1.1b

= 19

1 1 19p p * A1*cso 1.1b

(ii) Power = 11

1 p B1 1.1b

(4)

(f) Sam’s test has smaller P(Type I error) (or size) so is better B1 2.2a

Power of Sam’s test = 0.1755… B1 1.1b

Power of Tessa’s test = 110.85 = 0.1673… B1 1.1b

So for p = 0.15 Sam’s test is recommended B1 2.2b

(4)

(18 marks)

Notes

7(a) M1: Realising the need to use the model Using B(20,0.2) with method for finding

the CR or implied by a correct CR

A1: X 1 or X < 2

(b) B1: awrt 0.0692


176

Notes (continued)

(c) M1: Realising that the model Geo(0.2)is needed. This may be written or used

M1: Realising the key step that they need to find P(Y d) < 0.10

M1: Using the model 10.8 d

M1: Using the model 10.8 0.10d and finding a method to solve leading to a

value/range of values for d

A1: For d > 11.3..

A1: For Y 12 or Y > 11 (a correct inference)

(d) B1ft: awrt 0.0692. ft their answer to part (c)

(e)(i) M1: Using B(20, p) and realizing they need to find P(X 1) o.e. This may be used

or written

M1: Using P(X = 0) + P(X = 1)

A1*cso: Fully correct proof ( no errors)

(ii) B1: For 11

1 p

(f) B1: Making a deduction about the tests using the answers to part(b) and (d)

B1: awrt 0.0176

B1: awrt 0.167

B1: A correct inference about which test is recommended


177


178





Paper Reference(s)

9FM0/4E



Instructions




Answer all questions.



Answers should be given correct to 3 significant figures unless otherwise stated. Information




Advice





179


1. The three independent random variables A, B and C each have a continuous uniform

distribution over the interval [0, 5].

(a) Find the probability that A, B and C are all greater than 3

(3)

The random variable Y represents the maximum value of A, B and C.

The cumulative distribution function of Y is

F(y) = 3

0 0

0 5125

1 5

y

yy

y

(b) Using algebraic integration, show that Var (Y) = 0.9375

(4)

(c) Find the mode of Y, giving a reason for your answer.

(2)

(d) Describe the skewness of the distribution of Y. Give a reason for your answer.

(1)

(e) Find the value of k such that P(k < Y < 2k) = 0.189

(3)

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183

2. A researcher claims that, at a river bend, the water gradually gets deeper as the distance

from the inner bank increases. He measures the distance from the inner bank, b cm, and

the depth of a river, s cm, at 7 positions. The results are shown in the table below.

Position A B C D E F G

Distance from

inner bank b cm 100 200 300 400 500 600 700

Depth s cm 60 75 85 76 110 120 104

The Spearman’s rank correlation coefficient between b and s is 6

7

(a) Stating your hypotheses clearly, test whether or not the data provides support for the

researcher’s claim. Use a 1% level of significance.

(4)

(b) Without re-calculating the correlation coefficient, explain how the Spearman’s rank

correlation coefficient would change if

(i) the depth for G is 109 instead of 104

(ii) an extra value H with distance from the inner bank of 800 cm and depth 130 cm

is included.

(3)

The researcher decided to collect extra data and found that there were now many tied ranks.

(c) Describe how you would find the correlation with many tied ranks.

(2)

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185

3. A nutritionist studied the levels of cholesterol, X mg/cm3, of male students at a large

college. She assumed that X was distributed N(, 2) and examined a random sample of

25 male students. Using this sample she obtained unbiased estimates of and 2 as

and 2

A 95% confidence interval for was found to be ( 1.128, 2.232 )

(a) Show that 2 = 1.79 (correct to 3 significant figures)

(4)

(b) Obtain a 95% confidence interval for 2

(3)

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187

4. The times, x seconds, taken by the competitors in the 100 m freestyle events at a school

swimming gala are recorded. The following statistics are obtained from the data.

No. of competitors Sample mean x

2x

Girls 8 83.1 55 746

Boys 7 88.9 56 130

Following the gala, a mother claims that girls are faster swimmers than boys. Assuming

that the times taken by the competitors are two independent random samples from

normal distributions,

(a) test, at the 10% level of significance, whether or not the variances of the two

distributions are the same. State your hypotheses clearly.

(7)

(b) Stating your hypotheses clearly, test the mother’s claim. Use a 5% level of

significance.

(6)

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191

5. Scaffolding poles come in two sizes, long and short. The length L of a long pole has the

normal distribution N(19.6, 0.62). The length S of a short pole has the normal distribution

N(4.8, 0.32). The random variables L and S are independent.

A long pole and a short pole are selected at random.

(a) Find the probability that the length of the long pole is more than 4 times the length

of the short pole. Show your working clearly.

(6)

Four short poles are selected at random and placed end to end in a row. The random

variable T represents the length of the row.

(b) Find the distribution of T.

(3)

(c) Find P(L – T < 0.2)

(4)

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195

6. A random sample of 10 female pigs was taken. The number of piglets, x, born to each

female pig and their average weight at birth, m kg, was recorded. The results were as

follows:

Number of piglets, x 4 5 6 7 8 9 10 11 12 13

Average weight at

birth, m kg

1.50 1.20 1.40 1.40 1.23 1.30 1.20 1.15 1.25 1.15

(You may use Sxx = 82.5 and Smm = 0.12756 and Sxm = 2.29 )

(a) Find the equation of the regression line of m on x in the form m = a + bx as a model

for these results.

(2)

(b) Show that the residual sum of squares (RSS) is 0.064 to 3 decimal places.

(2)

(c) Calculate the residual values.

(2)

(d) Write down the outlier.

(1)

(e) (i) Comment on the validity of ignoring this outlier .

(ii) Ignoring the outlier, produce another model.

(iii) Use this model to estimate the average weight at birth if x = 15

(iv) Comment, giving a reason, on the reliability of your estimate.

(5)

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199

7. Over a period of time, researchers took 10 blood samples from one patient with a blood

disease. For each sample, they measured the levels of serum magnesium, s mg/dl, in the

blood and the corresponding level of the disease protein, d mg/dl. One of the researchers

coded the data for each sample using x = 10s and 10( 9) y d but spilt ink over his

work.

The following summary statistics and unfinished scatter diagram are the only remaining

information.

2 1081.74d Sds = 59.524

and

64y Sxx = 2658.9

(a) Use the formula for Sxx to show that Sss = 26.589

(3)

(b) Find the value of the product moment correlation coefficient between s and d.

(4)

(c) With reference to the unfinished scatter diagram, comment on your result in part (b).

(1)

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200


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203


204

AL Further Statistics 2


Q1(a) P(A > 3) =

2

5 B1 1.1b

32 8

5 125

M1

A1

1.1a

1.1b

(3)

(b)

23f ( )

125

yy M1 2.1

35

0

3E( ) d

125

yY y

54

0

3=

500

y

15

= 4

M1 1.1b

25 4

0

3 15Var( ) d

125 4

yY y

M1 1.1b

= 0.9375* A1*cso 1.1b

(4)

(c) Mode = 5 B1 1.2

Or reason based on

( )

0df

d

y

y

B1

2.4

(2)

(d) From a sketch or mode > mean therefore it has negative skew B1ft 2.4

(1)

(e)

3 320.189

125 125

k k M1 3.1a

37

0.189125

k A1 1.1b

k = 1.5 A1 1.1b

(3)

(13 marks)


205

Notes

1(a) B1:

2

5 o.e. may be implied by a correct answer

M1:

32

"their "5

may be implied by a correct answer

A1: 8

125 o.e.

(b) M1: realising that firstly need to find pdf f(y) and attempt to differentiate F(y)

M1: Continuing the argument with an attempt to integrate " ( )"y y their f

1n ny y

M1: Integrating 2 " ( )"y y their f - [“their E(Y)”]2 1n ny y

A1*: Complete correct solution no errors.

(c) B1: 5 only

B1: Explain their reason by either an accurate sketch or ( )

0df

d

y

y therefore an

increasing function oe

(d) B1ft: Explaining the reason for their answer. Follow through their part(b) or mean

from(d) and mode from(c). A correct sketch of “their f(y)” – may be seen anywhere

in question or ft their mean and mode plus a correct conclusion

NB: Watch for gaming. A student who writes both negative skew with a reason and

positive skew with a reason. Please send these to your Team Leader

(e) M1: Attempting to translate the problem into an equation using 2k and k. Allow if

the brackets are missing e.g.3 32

125 125

k k . No need for the 0.189

A1: A correct equation in any form

A1: a correct answer only.


206


Q2(a) 0 1H : 0,H : 0 B1 2.5

Critical value at 1% level is 0.8929 B1 1.1b

sr < 0.8929 so not significant evidence to reject 0H , M1 2.1

The researcher's claim is not correct (at 1% level).

or insufficient evidence for researcher’s claim

or there is insufficient evidence that water gets deeper further from

inner bank.

or no (positive) correlation between depth of water and distance from

inner bank

A1ft

2.2b

(4)

(b)(i) The ranks will remain the same therefore there will be no change to

the spearman’s rank correlation coefficient

B1 2.4

(ii) Spearman’s rank correlation coefficient will increase since B1 2.2a

The ranks are the same for both distance and depth therefore

d = 0 however, n has increased or the new position follows the

pattern that large b is assosciated with large s and so rs will increase

B1 2.4

(3)

(c) The mean of the tied ranks is given to each… B1 2.4

… then use PMCC B1 2.4

(2)

(9 marks)

Notes

(a) B1: Both hypotheses correct written using the notation

B1: awrt 0.893

M1: Drawing a correct inference using their answer to part(a) and their CV

A1ft: Drawing a correct inference in context using their answer to part(a) and their

CV

(b)(i) B1: Stating no change and an explanation including ranks remain unchanged oe

and no change oe

(b)(ii) B1: Interpreted the outcome of adding a point as increased oe

B1: Explaining why. Need to mention the ranks are the same for both oe and n has

increased oe

(c) B1: Explaining that The mean of the values for the tied ranks is given to both values

B1: Explaining that the PMCC must be used


207


Q3(a) 95% CI for uses t value of 2.064 B1 3.3

ˆ 1

"2.064" 2.232 1.128225

or

ˆ1

2.232 1.128 "2.064" 2.2322 25

(oe)

M1 2.1

2.76

ˆ"2.064"

or 1.3372… M1 1.1b

2ˆ 1.788... [=1.79 (3sf)] * A1*cso 1.1b

(4)

(b) 12.401, <

2

79.124

<, 39.364

B1

M1

1.1b

1.1a

1.09 < 2 < 3.46 A1 1.1b

(3)

(7 marks)

Notes

(a) B1: Realising that the t-distribution must be used as a model and finding the correct

value awrt 2.06

M1: Using the correct formula with a t-value, ˆ 1

" value" 2.232 1.128225

t

or ˆ1

2.232 1.128 " value" 2.2322 25

t

or ˆ1

2.232 1.128 " value" 1.1282 25

t

M1: Rearranging one of these formula accurately to find a value of

A1cso*: A correct solution only using awrt 1.79

(b) B1: awrt 12.4 or 39.4 May be implied by a correct confidence interval

M1: 2

79.124

May be implied by a correct confidence interval

A1: awrt 1.09 and awrt 3.46


208


Q4(a) H0: 2

G = 2

B , H1: 2

G 2

B , B1 2.5

2

Bs = 61 (56130 – 7 88.92) =

6

53.807= 134.6

M1

A1

2.1

1.1b

2

Gs = 71 (55746 – 8 83.12) =

7

12.501= 71.58 A1 1.1b

2

2

G

B

s

s= 1.880… M1 3.4

critical value F6, 7 = 3.87 B1 1.1b

not significant, variances can be treated as the same A1 ft 2.2b

(7)

(b) H0: B = G , H1: B > G B1 2.5

pooled estimate of variance s2 = 13

58.7176.1346 = 100.6653… M1 3.1b

test statistic t =

81

71

1.839.88

s = awrt 1.12

M1

A1

1.1b

1.1b

critical value t13(5%) = 1.771 B1 1.1b

Insufficient evidence to support mother’s claim A1 ft 2.2b

(6)

(13 marks)

Notes

4(a) B1: Both hypotheses correct using the notation2 . Allow rather than

2 .

M1: Using a correct Method for either 2

Bs or 2

Gs May be implied by a correct value

A1: awrt 135

A1: awrt 71.6

M1: Using the F-distribution as the model eg2

2

G

B

s

s

B1: awrt 3.87

A1ft: Drawing a correct inference following through their CV and value for 2

2

G

B

s

s

(b) B1: Both hypotheses correct using the notation .

M1: For realising the need to find the pooled estimate for the test require from a

correct interpretation of the question.

M1: Correct method for test statistic t = 1 17 8

88.9 83.1

""their s

May be implied by a

correct awrt 1.12

A1: awrt 1.12

B1: awrt 1.77

A1ft Drawing a correct inference following through their CV and test statistic


209


Q5(a) Let X = L – 4S then E(X) = 19.6 – 4 4.8 M1 2.3

= 0.4 A1 1.1b

Var(X) = 2 2 2Var( ) 4 Var( ) 0.6 16 0.3L S M1 2.1

= 1.8 A1 1.1b

P(X > 0) = [P(Z > 0 0.4

0.298...1.8

…)] M1 2.1

= 0.617202.. awrt 0.617 A1 1.1b

(6)

(b) 1 2 3 4T S S S S (May be implied by 0.36) M1 3.3

T ~ N(19.2, 0.36) E(T) = 19.2 B1 1.1b

Var(T) = 0.36 or 20.6 A1 1.1b

(3)

(c) Let Y = L – T E(Y ) = E(L) – E(T) = [ 0.4] M1 3.3

Var(Y) = Var(L) + Var(T) = [ 0.72] M1 1.1b

Require P( – 0.2 < Y < 0.2) M1 3.1a

= 0.16708… awrt 0.167 A1 1.1b

(4)

(13 marks)

Notes

(a) M1: Selecting and using an appropriate model i.e – 4 L S . May be implied by

0.4

A1: 0.4 oe

M1: For realising the need to use Var(L) + 42Var(S). Allow use of 0.6 for Var(L)

instead of 20.6 and/or 0.3 for Var(S) instead of

20.3 may be implied by 1.8

A1: 1.8 only

M1: For realising P(X > 0) is required and an attempt to find it e.g.

0 0.4

"their Var( )"X

but do not allow a negative Var(X)

A1: awrt 0.617

(b) M1: Selecting and using an appropriate model ie 1 2 3 4S S S S : may be implied

by 0.36

B1: 19.2 only

A1: 0.36

(c) M1: Setting up and using the model Y = L – T. May be implied by

–E Y E L E T

M1: Using Var(Y) = Var(L) + Var(T)

M1: Dealing with the modulus and realising they need to find P( – 0.2 < Y < 0.2)

A1: awrt 0.167


210


Q6(a) – 0.02775

S76

Sxm

xx

b

M1 3.3

[a = m bx = 1.278 + 0.0277576× 8.5 = 1.5139]

m = 1.5139 – 0.02775…x A1 1.1b

(2)

(b) RSS =

2

2.290.12756

82.5

M1 1.1b

= 0.06399* A1* 1.1b

(2)

(c)

x m m = a + bx

4 1.50 1.4029 +0.0971

5 1.20 1.3752 – 0.1752

6 1.40 1.3474 +0.0526

7 1.40 1.3196 +0.0804

8 1.23 1.2919 – 0.0619

9 1.30 1.2641 +0.0359

10 1.20 1.2364 – 0.0364

11 1.15 1.2086 – 0.0586

12 1.25 1.1808 +0.0692

13 1.15 1.1531 – 0.0031

M1

A1

3.4

1.1b

(2)

(d) The point (5, 1.2) is an outlier B1ft 2.2b

(1)

(e)(i)

It is a valid piece of data so should be used

or It does not follow the pattern according to the residuals so may

contain an error making the result invalid so should be removed

B1 2.4

(ii)

a m bx =1.28667 +0.03765 × 8.88889 = 1.6213 M1 3.3

m =1.6213 – 0.03765x A1 1.1b

(iii)

m =1.6213 – 0.03765 15

= 1.056 or awrt 1.06 B1ft 3.4

(iv) The model is only reliable if the values are limited to those in the

given range so probably not reliable B1 3.5b

(5)

(12 marks)


211

Notes

6(a) M1: Realsing the need to use b =

S

S

xm

xx

and a = m bx

A1: m = awrt 1.51) – (awrt 0.0278) x. Award M1A1 for correct equation

(b) M1: Using

2

xm

mm

xx

SS

S

A1*: awrt 0.064

(c) M1: Using the model in part (a) i.e. m – (“1.5139” – “0.02775”x) implied by a

correct value

A1: All correct.

Award M1A1 for a list of correct residuals

(d) B1: Inferring from the residuals that the outlier is (5, 1.2) ft their residuals.

(e)(i) B1: Explaining why the outlier should be removed or not.

(ii) M1: Removing the outlier and refining the model by finding a new regression line.

A1: ( 1.62) ( 0.0377)awrt awrt m x

(iii) B1ft: using their model in e(i) with x = 15. awrt 1.06 or ft their e(ii)

(iv) B1: Realising the limitations of the model by stating it is not reliable and giving the

reason why ie extrapolation/out of range o.e.


212


Q7(a)

Sxx =

2

2 1010

10

ss

M1 2.1

2658.9 =

2

2 100100

10

ss

M1 1.1b

2658.9 = 100 Sss

Sss = 26.589 * A1*cso 1.1b

(3)

(b) 10

1

64 10( 9) id M1 3.1a

10

1

64 10 900id

10

1

96.4id A1 1.1b

Sdd =

2"96.4"

1081.74 -10

M1 1.1b

= 152.444

r = 0.935 A1ft 1.1b

(4)

(c) Linear correlation is significant but scatter diagram suggests a

non-linear relationship between the level of serum magnesium,

and the level of the disease protein

B1 3.5a

(1)

(8 marks)

Notes

(a)

M1: Attempting to use

2

2

10xx

xS x

with x = 10s

M1: Substituting in 2658.9 and dealing with the 10 correctly

A1*: cso A complete solution with no errors leading to 26.589 only

(b) M1: Realising that either

10

1

64 10( 9) id or 10

1

64 10 900id o.e. must be used.

May be implied by seeing 96.4

A1: 96.4 only

M1: Attempting to use

2

2

10dd

dS d

may be implied by 0.935

A1ft: awrt 0.935 ft “their 96.4”

(c) B1: A correct comment comparing their value of r and the scatter diagram in context


213


214


Further Mathematics Advanced Paper 3: Further Mathematics Option 1 Option 3C: Further Mechanics 1

Paper 4: Further Mathematics Option 2 Option 4C: Further Mechanics 1


Paper Reference(s)

9FM0/3C 9FM0/4C



Instructions











Advice





215


Unless otherwise indicated, whenever a numerical value of g is required, take g = 9.8 m s-2 and

give your answer to either 2 significant figures or 3 significant figures.

1. A particle P of mass 0.5 kg is moving with velocity 4 i j m s-1 when it receives

an impulse 2 i j N s.

Show that the kinetic energy gained by P as a result of the impulse is 12 J.

(6)

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217

2. A parcel of mass 5 kg is projected with speed 8 m s-1 up a line of greatest slope of a

fixed rough inclined ramp.

The ramp is inclined at angle to the horizontal, where 1

sin7

The parcel is projected from the point A on the ramp and comes to instantaneous

rest at the point B on the ramp, where 14AB m.

The coefficient of friction between the parcel and the ramp is .

In a model of the parcel's motion, the parcel is treated as a particle.

(a) Use the work-energy principle to find the value of .

(5)

(b) Suggest one way in which the model could be refined to make it more realistic.

(1)

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219

3. A particle of mass m kg lies on a smooth horizontal surface.

Initially the particle is at rest at a point O between two fixed parallel vertical walls.

The point O is equidistant from the two walls and the walls are 4 m apart.

At time t = 0 the particle is projected from O with speed u m s-1 in a direction

perpendicular to the walls.

The coefficient of restitution between the particle and each wall is 3

4

The magnitude of the impulse on the particle due to the first impact with a wall is mu N s.

(a) Find the value of .

(3)

The particle returns to O, having bounced off each wall once, at time 7t seconds.

(b) Find the value of u.

(5)

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221

4.

Figure 1

Figure 1 represents the plan view of part of a horizontal floor, where AB and BC

are perpendicular vertical walls.

The floor and the walls are modelled as smooth.

A ball is projected along the floor towards AB with speed u m s-1 on a path at an

angle of 60 to AB. The ball hits AB and then hits BC.

The ball is modelled as a particle.

The coefficient of restitution between the ball and wall AB is 1

3

The coefficient of restitution between the ball and wall BC is 2

5

(a) Show that, using this model, the final kinetic energy of the ball is 35% of the

initial kinetic energy of the ball.

(8)

(b) In reality the floor and the walls may not be smooth. What effect will the model

have had on the calculation of the percentage of kinetic energy remaining?

(1)

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60°

C

BA

u


222


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225

5. A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is v m s-1, the resistance to the motion of the car is

modelled as a force of magnitude 200 2v N.

The engine of the car is working at a constant rate of 12 kW.

(a) Find the acceleration of the car at the instant when 20v

(4)

Later on the car is moving up a straight road inclined at an angle to the horizontal, where

1sin

14

At the instant when the speed of the car is v m s-1, the resistance to the motion of the car from

non-gravitational forces is modelled as a force of magnitude 200 2v N.

The engine is again working at a constant rate of 12 kW.

At the instant when the car has speed w m s-1, the car is decelerating at 0.05 m s-2.

(b) Find the value of w.

(5)

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227

6. [In this question i and j are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere A has mass 2m kg and another smooth uniform sphere B,

with the same radius as A, has mass 3m kg.

The spheres are moving on a smooth horizontal plane when they collide obliquely.

Immediately before the collision the velocity of A is 3 3i j m s-1 and the velocity

of B is 5 2 i j m s-1.

At the instant of collision, the line joining the centres of the spheres is parallel to i.

The coefficient of restitution between the spheres is 1

4

(a) Find the velocity of B immediately after the collision.

(7)

(b) Find, to the nearest degree, the size of the angle through which the direction of

motion of B is deflected as a result of the collision.

(2)

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231

7. A particle P of mass m is attached to one end of a light elastic string of natural length a and

modulus of elasticity 3mg.

The other end of the string is attached to a fixed point O on a ceiling.

The particle hangs freely in equilibrium at a distance d vertically below O.

(a) Show that 4

3d a .

(3)

The point A is vertically below O such that 2OA a .

The particle is held at rest at A, then released and first comes to instantaneous rest at the point B.

(b) Find, in terms of g, the acceleration of P immediately after it is released from rest.

(3)

(c) Find, in terms of g and a, the maximum speed attained by P as it moves from A to B.

(5)

(d) Find, in terms of a, the distance OB.

(3)

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235

8. A particles P of mass 2m and a particle Q of mass 5m are moving along the same

straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.

Immediately before the collision the speed of P is 2u and the speed of Q is u.

The direction of motion of Q is reversed by the collision.

The coefficient of restitution between P and Q is e.

(a) Find the range of possible values of e.

(8)

Given that 1

3e

(b) show that the kinetic energy lost in the collision is 240

7

mu.

(5)

(c) Without doing any further calculation, state how the amount of kinetic energy

lost in the collision would change if 1

3e

(1)

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239


240

AL Further Mechanics 1 Mark Scheme


1

Use Impulse-momentum principle M1 2.1

2 0.5 0.5 4 i j v i j A1 1.1b

1 1

4 , 82 2

v i j v i j (m s-1) A1 1.1b

Use of KE = 2 21 1

2 2m mv u M1 2.1

1

0.5 64 1 16 12

A1 1.1b

1

48 124

(J) * A1* 1.1b

(6)

(6 marks)

Notes:

M1: Difference of terms & dimensionally correct

A1: Correct unsimplified equation

A1: C.A.O.

M1: Must be a difference of two terms.

Must be dimensionally correct.


A1*: Complete justification of given answer


241


2(a)

4 3

5 cos 5 48.497...7

R g g

M1 3.4

Force due to friction = 5 cosg M1 3.4

Work-Energy equation M1 3.4

1

5 64 5 9.8 14sin 142

R A1 1.1b

0.0913 or 0.091 A1 1.1b

(5)

(b) Appropriate refinement B1 3.5c

(1)

(6 marks)

Notes:

(a)

M1: Condone sin/cos confusion

M1: Use of their R

M1: Must be using work-energy. Requires all terms.

Condone sin/cos confusion, sign errors and their R

A1: Correct in and R .

A1: Accept 0.0913 or 0.091

(b)

B1: e.g.

- Do not model the parcel as a particle and therefore take air resistance into account.

- Take into account the dimensions/uniformity of the parcel.


242


3(a) Use NEL to find the speed of particle after the first impact

3

4eu u

B1 3.4

Impulse = 3

4mu mv mu mu mu

M1 3.1b

7

4 A1 1.1b

(3)

(b) Use NEL to find the speed of the particle after the second impact

3 3 9

4 4 16u u

B1 3.4

Use of s vt to find total time M1 3.1b

2 4 2 2 16 32

73 9 3 9

4 16

u u u uu u

A1 1.1b

Solve for u: 63 18 48 32u M1 1.1b

98 14

1.563 9

u A1 1.1b

(5)

(8 marks)

Notes:

(a)

B1: Using Newton's experimental law as a model to find the

speed after the first impact

M1: Must be a difference of two terms, taking account of the

change in direction of motion.

A1: cao

(b)

B1: Using NEL as a model to find the speed after the second

impact.

M1: Needs to be used for at least one stage of the journey

A1: or equivalent

M1: Solve their linear equation for u

A1: Accept 1.56 or better


243


4(a) Complete strategy to find the kinetic energy after the second impact M1 3.1b

Parallel to AB after collision: cos60u M1 3.1b

Perpendicular to AB after collision: 1

sin 603

u M1 3.4

Components of velocity after first impact: , 2 2

u u A1 1.1b

Parallel to BC after collision: 2

u

1sin 60

3u

M1 3.1b

Perpendicular to BC after collision: 2 1

5 2 10

uu

2

cos605

u

M1 3.4

Components of velocity after second impact: , 2 10

u u A1 1.1b

Final KE =

2 2 21 7

2 4 10 2 20

u u mum

Fraction of initial KE

2

2

7

2 20

2

mu

mu

7

35%20

* A1* 2.2a

(8)

(b) The answer is too large - rough surface means resistance so final

speed will be lower. B1 3.5a

(1)

(9 marks)

Notes:

(a)

M1: Use of CLM parallel to the wall. Condone sin/cos confusion

M1: Use NEL as a model to find the speed perpendicular to the wall. Condone sin/cos confusion

A1: Both components correct with trig substituted (seen or implied)

M1: Use of CLM parallel to the wall. Condone sin/cos confusion

M1: Use NEL as a model to find the speed perpendicular to the wall. Condone sin/cos confusion

A1: Both components correct with trig substituted (seen or implied)

M1: Correct expression for total KE using their components after 2nd collision.

A1*: Obtain given answer with sufficient working to justify it

(b)

Clear explanation of how the modelling assumption has affected the outcome


244


5(a)

Use of P Fv :

12000

20F B1 3.3

Equation of motion: 200 2 600F v a M1 3.4

600 240 600a A1ft 1.1b

360 600a , 0.6a (m s-2) A1 1.1b

(4)

(b) Equation of motion M1 3.3

12000

200 2 600 sin 600 0.05w gw

A1

A1

1.1b

1.1b

3 term quadratic and solve: 22 590 12000 0w w M1 1.1b

2590 590 96000

19.14

w

(m s-1) A1 1.1b

(5)

(9 marks)

Notes:

(a)

B1: 600 or equivalent

M1: Use the model to form the equation of motion.

Must include all terms.Condone sign errors.

A1ft: Correct for their F

A1: cao.

(b)

M1: Use the model to form the equation of motion.

All terms needed.

Condone sign errors and sin/cos confusion.

A1: All correct A1A1

One error A1A0

M1: Dependent on the preceding M1. Use the equation of motion to form a 3-term quadratic in w only

A1: Accept 19. Do not accept more than 3 s.f.


245


6(a)

Overall strategy to find Av M1 3.1a

Velocity of A perpendicular to loc after collision 3 j (m s-1) B1 3.4

CLM parallel to loc. M1 3.1a

2 3 3 5 3 2m m mw mv 9 3 2w v A1 1.1b

Correct use of impact law M1 3.1a

1

3 54

v w 2 A1 1.1b

Solve for w 3 2 9

2 2 4

w v

v w

2B v i j (m s-1), A1ft 1.1b

(7)

(b) ( 5 2 2cos

29 5

i j).( i j) M1 3.1a

41.63... 42 (nearest degree) A1 1.1b

Alternative method: 1 1 2tan 2 tan 41.63... 42

5

(nearest

degree)

(2)

(9 marks)

Notes:

(a)

M1: Correct overall strategy to form sufficient equations and solve for Av

B1: Use the model to find the component of Av perpendicular to the line of centres.

M1: Use CLM to form equation in v and w. Need all 4 terms, dimensionally correct

A1: Correct unsimplified

M1: Must be used the right way round

A1: Correct unsimplified

A1ft: vB correct. Follow their 2j

(b)

M1: Complete method for finding the required angle. Follow their vB

A1: cao

wivi

3j 2j

-5i+2j3i+3j

B(3m)A(2m)


246


7(a)

In equilibrium no resultant vertical force. M1 2.1

3mgx

mga

A1 1.1b

3

ax ,

4

3d a * A1* 2.2a

(3)

(b) Equation of motion. M1 3.1a

3mga

mg mxa

A1 1.1b

2x g A1 1.1b

(3)

(c) Max speed at equilibrium position B1 3.1a

Work energy & use of 2

EPE2

x

a

. M1 3.1a

2

22

33 1 23

2 2 2 3

amg

mga amv mg

a a

A1

A1

1.1b

1.1b

21 3 1 2 2

2 2 6 3 3v ga ga

,

4

3

gav A1 1.1b

(5)

(d) At max ht. KE = 0. EPE lost = GPE gained M1 3.1a

23

2

mgamgh

a A1 1.1b

2

aOB A1 1.1b

(3)

(14 marks)


247


Notes:

(a)

M1: Use x

Ta

to form equation for equilibrium


A1*: Requires sufficient working to justify given answer

plus a 'statement' that the required result has been achieved.

(b)

M1: Use x

Ta

to form equation of motion.

Need all 3 terms. Condone sign errors


A1: cao

(c)

B1: Seen or implied

M1: Form work-energy equation. All 4 terms needed.

Condone sign errors

A1: Correct unsimplified equation A1A1

One error in the equation A1A0

A1: cao

(d)

M1: Form energy equation


A1: cao


248


8(a)

Complete overall strategy to find v M1 3.1a

Use of CLM M1 3.1a

2 2 5 5 2m u m u m v m w ,

5 2u v w A1 1.1b

Use of Impact law M1 3.1a

2v w e u u A1 1.1b

Solve for v: 5 2

6 2 2

u v w

eu v w

7 6 1v u e 6 17

uv e

A1 1.1b

Direction of Q reversed: 0v M1 3.4

1

16

e A1 1.1b

(8)

(b) 1

3e

7

uv ,

6

7

uw B1 2.1

Equation for KE lost M1 2.1

2 22 21 36 1

2 4 52 49 2 49

u um u m u

A1

A1

1.1b

1.1b

2

21 72 5 408 5

2 49 49 7

mumu

* A1* 2.2a

(5)

(c) Increase e more elastic less energy lost B1 2.2a

(1)

(14 marks)

w v

u2u

Q

5m

P

2m


249


Notes:

(a)

M1: Complete strategy to form sufficient equations in v and w and solve for v.

M1: Use CLM to form equation in v and w.

Needs all 4 terms & dimensionally correct


M1: Use NEL as a model to form a second equation in v and w. Must be used the right way round


A1: for v or 7v correct

M1: Use the model to form a correct inequality for their v

A1: Both limits required

(b)

B1: or equivalent statements

M1: terms of correct structure combined correctly

A1: Fully correct unsimplified A1A1

One error on unsimplified expression A1A0

A1*: cso. plus a 'statement' that the required result has been achieved

(c)

B1: "less energy lost" or equivalent


250



Further Mathematics Option 2 Paper 4: Further Mechanics 2


Paper Reference(s)

9FM0/4F



Instructions











Advice



Check your answers if you have time at the end


251


Unless otherwise indicated, whenever a numerical value of g is required, take 2

9.8 m sg and give your answer to either 2 significant figures or 3 significant figures.

1. A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance x metres from its base, the mass per

unit length of the flag pole, m kg m-1 is given by the formula 10 125

xm

.

The flag pole is modelled as a rod.

(a) Show that the mass of the flag pole is 105 kg.

(3)

(b) Find the distance of the centre of mass of the flag pole from its base.

(4)

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253

2.

A hollow right circular cone, of base diameter 4a and height 4a is fixed with its axis

vertical and vertex V downwards, as shown in Figure 1.

A particle of mass m moves in a horizontal circle with centre C on the rough inner

surface of the cone with constant angular speed .

The height of C above V is 3a.

The coefficient of friction between the particle and the inner surface of the cone is 1

4

Find, in terms of a and g, the greatest possible value of .

(8)

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255

3.

A uniform solid cylinder has radius 2a and height h h a .

A solid hemisphere of radius a is removed from the cylinder to form the vessel V.

The plane face of the hemisphere coincides with the upper plane face of the cylinder.

The centre O of the hemisphere is also the centre of the upper plane face of the cylinder,

as shown in Figure 2.

(a) Show that the centre of mass of V is

2 23 8

8 6

h a

h a

from O.

(5)

The vessel V is placed on a rough plane which is inclined at an angle to the horizontal.

The lower plane circular face of V is in contact with the inclined plane.

Given that 5h a , the plane is sufficiently rough to prevent V from slipping and V is on

the point of toppling,

(b) find, to three significant figures, the size of the angle .

(4)


256


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257


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259

4. A car of mass 500 kg moves along a straight horizontal road.

The engine of the car produces a constant driving force of 1800 N.

The car accelerates from rest from the fixed point O at time 0t and at time t seconds

the car is x metres from O, moving with speed v m s-1.

When the speed of the car is v m s-1, the resistance to the motion of the car has

magnitude 22v N.

At time T seconds, the car is at the point A, moving with speed 10 m s-1.

(a) Show that 25

ln 26

T

(6)

(b) Show that the distance from O to A is 9

125ln8

m.

(5)

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260


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261

5.

Figure 3

A shop sign is modelled as a uniform rectangular lamina ABCD with a semicircular

lamina removed.

The semicircle has radius a, 4BC a and 2 .CD a

The centre of the semicircle is at the point E on AD such that AE d , as shown in

Figure 3.

(a) Show that the centre of mass of the sign is

44

3 16

a

from AD.

(4)

The sign is suspended using vertical ropes attached to the sign at A and at B and

hangs in equilibrium with AB horizontal.

The weight of the sign is W and the ropes are modelled as light inextensible strings.

(b) Find, in terms of W and , the tension in the rope attached at B.

(2)

The rope attached at B breaks and the sign hangs freely in equilibrium suspended

from A, with AD at an angle to the downward vertical.

Given that11

tan18

(c) find d in terms of a and .

(6)

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262


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264


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265

6. A small bead B of mass m is threaded on a circular hoop.

The hoop has centre O and radius a and is fixed in a vertical plane.

The bead is projected with speed 7

2ga from the lowest point of the hoop.

The hoop is modelled as being smooth.

When the angle between OB and the downward vertical is , the speed of B is v.

(a) Show that 2 3

2cos2

v ga

(3)

(b) Find the size of at the instant when the contact force between B and the hoop is

first zero.

(5)

(c) Give a reason why your answer to part (b) is not likely to be the actual value of .

(1)

(d) Find the magnitude and direction of the acceleration of B at the instant when B is

first at instantaneous rest.

(5)

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268


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(Total for Question 6 is 14marks)


269

7. Two points A and B are 6 m apart on a smooth horizontal surface.

A light elastic string of natural length 2 m and modulus of elasticity 20 N, has one end

attached to the point A.

A second light elastic string of natural length 2 m and modulus of elasticity 50 N, has

one end attached to the point B.

A particle P of mass 3.5 kg is attached to the free end of each string.

The particle P is held at the point on AB which is 2 m from B and then released from

rest.

In the subsequent motion both strings remain taut.

(a) Show that P moves with simple harmonic motion about its equilibrium position.

(7)

(b) Find the maximum speed of P.

(2)

(c) Find the length of time within each oscillation for which P is closer to A than to B.

(5)

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271


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273


274

AL Further Mechanics 2


1(a)

Total mass = 15

010 1 d

25

xx

M1 2.1

15

2

0

105

xx

A1 1.1b

225

150 1055

(kg) * A1* 1.1b

(3)

(b) Taking moments about the base :

15

010 1 d

25

xx x

M1 3.4

15

2 3

0

25 675

15x x

A1 1.1b

105 675d M1 3.4

6.43d (m) 3

67

(m) A1 1.1b

(4)

(7 marks)

Notes:

(a)

M1: Use integration (usual rules)


A1*: Use limits and show sufficient working to justify given answer.

(b)

M1: Use the model to find the moment about the base (usual rules for integration)


M1: Use the model to complete the moments equation.

Require 105 and their 675 used correctly

A1: 6.43 or better


275


2

Complete overall strategy M1 3.1b

Resolve vertically M1 3.3

cos sinmg F R A1 1.1b

Horizontal equation of motion M1 3.3

2 cos sinmr R F A1 1.1b

Use of limiting friction since maximum M1 3.3

Substitute for trig ratios:

23 9

2 2

a

g

M1 1.1b

Maximum 3g

a A1 1.1b

(8 marks)

Notes:

M1: Overall strategy to form equation in only e.g.

consider vertical and horizontal motrion and limiting friction

M1: needs all 3 terms. Condone sign errors and sin/cos confusion

A1: correct unsimplified equation

M1: needs all 3 terms. Condone sign errors and sin/cos confusion

A1: correct unsimplified equation

M1: seen or implied

M1: substitute to achieve equation in , and a g only

A1: or equivalent exact form

F

mg

R

θ

C

V

3a

4a

4a


276


3(a)

mass c of m from O

cylinder 24 a h

2

h

hemisphere 32

3a

3

8a

V 2 324

3a h a d

Mass ratios B1 1.2

Correct distances B1 1.2

Moments about a diameter through O. M1 2.1

2 3 22 3 14 2 2

2 3 8 3

ha h a a a h a d

A1 1.1b

22

2 23 88

8 62

3

ah h a

da h a

h

* A1* 2.2a

(5)

(b)

5 2.573...h a d a B1 1.1b

About to topple so c of m above tipping point M1 2.2a

2

tan5 2.573

a

a a

A1ft 1.1b

39.5 or 0.689 rads. A1 1.1b

(4)

(9 marks)


277

Notes:

(a)

B1: correct mass ratios

B1: correct distances

M1: All three terms & dimensionally correct.

Could use a parallel axis but final answer must be for the distance from O.


A1*: Deduce the given answer. Their working must make it clear how they reached

their answer.

(b)

B1: Distance of com from base

M1: Condone tan the wrong way up.

A1ft: Correct unsimplified expression for trig ratio for following their d

A1: 39.5 or 0.689 rads.


278


4(a)

Equation of motion 21800 2 500v a (when seen) B1 2.1

Select form for a: d

500d

v

t M1 2.5

2

2 1 1 1 1d d d

500 900 60 30 30t v v

v v v

M1 2.1

1 1

ln 30 ln 30 250 60 60

tv v C A1 1.1b

25 30 10 25

ln ln 26 30 10 6

T

* M1

A1*

2.1

2.2a

(6)

(b) Equation of motion: 2d500 1800 2

d

vv v

x M1 2.5

2

500d 1d

1800 2

vv x

v

M1 2.1

2125ln 1800 2 v x C A1 1.1b

Use boundary conditions: 125ln1600 125ln1800x M1 2.1

9

125ln8

x (m) * A1* 2.2a

(5)

(11 marks)

Notes:

(a)

B1: all three terms & dimensionally correct

M1: use of correct form for acceleration to give equation in v, t only

M1: Separate variables and integrate

A1: Condone missing C

M1: Use boundary conditions correctly

A1*: Show sufficient working to justify given answer and a 'statement' that the required

result has been achieved

(b)

M1: Correct form of acceleration in the equation of motion to give equation in v, x only

M1: Separate variables and integrate.

A1: Condone missing C

M1: Extract and use boundary conditions

A1*: Show sufficient working to justify given answer and a 'statement' that the required result

has been achieved


279


5(a)

Mass From AD

Rectangle 28a a

Semicircle 21

2a

4

3

a

Sign 2

82

a

h

Mass ratios B1 1.2

Moments about AD M1 2.1

2 2 2 3 3 31 4 2 228 8 8

2 2 3 3 3

aa h a a a a a a

A1 1.1b

22 448

3 2 3 16

ah a

* A1* 2.2a

(4)

(b) Moments about A

442

3 16

aaT W

M1 3.1b

22

2 3 16

hW WT

a

A1 1.1b

(2)


280


(c)

Take moments about AB to find distance of com from AB M1 3.1b

2 2 21 1

8 2 82 2

a a a d a v

A1 1.1b

32

16

a dv

A1 1.1b

Correct trig for the given angle M1 3.1b

11 44tan

18 3 32

h a

v a d

A1ft 1.1b

( 24 32a a d , 8a d ) 8a

d

A1 1.1b

(6)

(12 marks)

D

C

B

A

α

h

v

Notes:

(a)

B1: correct mass ratios

M1: need all three terms, must be dimensionally correct


A1*: Show sufficient working to justify the given answer and a 'statement' that the required result

has been achieved.

(b)

M1: Could also take moments about B or about the c.o.m. and use A BT T W

A1: cso

(c)

M1: all terms and dimensionally correct


A1: or equivalent

M1: Condone tan the wrong way up.

A1: Equation in a and d; follow through on their v

A1: cao.


281


6(a)

Conservation of energy M1 2.1

21 1 71 cos

2 2 2mv mga m ga

A1 1.1b

2 3

2cos *2

v ga

A1* 2.2a

(3)

(b)

Resolve parallel to OB and use

2mv

a M1 3.1b

2

cosmv

R mga

A1 1.1b

Use 0R

2

cosv

ga

M1 3.1b

Solve for 3

cos 2cos2

g g

M1 1.1b

120 A1 1.1b

(5)

(c) Any appropriate comment

e.g. The hoop is unlikely to be smooth B1 3.5b

(1)

Rv

7

2ga

B

O

θ

mg

a


282


(d) At rest 0 v M1 3.1b

3

cos4

A1 1.1b

Acceleration is tangential M1 3.1b

Magnitude cos 90 6.48g m s-2 or 7

4g A1 1.1b

At 1 3

cos 90 48.64

to the downward

vertical

A1 1.1b

(5)

(14 marks)

Notes:

(a)

M1: All terms required. Must be dimensionally correct


A1*: Show sufficient working to justify the given answer and a 'statement' that the required

result has been achieved.

(b)

M1: Resolve parallel to OB

A1: correct equation

M1: Use 0R seen or implied

M1: Solve for

A1: Accept 2

3

(c)

B1: Any appropriate comment e.g.

- hoop may not be smooth;

- air resistance could affect the motion

(d)

M1: 0v seen or implied

A1: correct equation in

M1: correct direction for acceleration

A1: Accept 6.48, 6.5 or exact in g

A1: Accept 0.848 (radians)


283


7(a)

20

2A

eT ,

50 2

2B

eT

M1 3.1a

In equilibrium A BT T , 10 25 2e e M1 3.1a

10

35 50 , 7

e e A1 1.1b

Equation of motion for P when distance x from equilibrium

position towards B: M1 3.1a

50 2 20

3.5 2 2

B A

e x e xx T T

A1

A1

1.1b

1.1b

4 1050 20

7 7

2 2

x x

3.5 35 , 10x x x x and hence SHM about

the equilibrium position A1 3.2a

(7)

(b) Amplitude 10 4

27 7

B1 ft 2.2a

Use of max speed a M1 1.1b

4

10 1.817

(m s-1) A1 ft 1.1b

(3)

P

50 N20 N

6m

BA


284


7(c) Nearer to A than to B: 3

7x B1 3.1a

Solve for 10t : 3

cos 104

t ,

10 2.418.............t

M1 3.1a

Length of time: 2

2.418...10

M1 1.1b

0.457 (seconds) A1 1.1b

Alternative: 3.864 2.419

0.45710

Alternative:

4 3

sin 107 7

x t 10 0.8481 or 10 2.29353t t

1 20.2682, 0.72527t t

time 0.457 (seconds)

(4)

(14 marks)

Notes:

(a)

M1: Use of x

Ta

M1: Dependent on the preceding M1. Equate their tensions

A1: cao

M1: Condone sign error

A1: Correct unsimplified equation in e and x A1A1

Equation with one error A1A0

A1: Full working to justify conclusion that it is SHM about the

equilibrium position.

(b)

B1ft: Seen or implied. Follow their e

M1: correct method for max. speed

A1ft: 1.81 or better. Follow their ,a

(c)

B1: Seen or implied

M1: Use of cosx a wt

M1: Correct strategy for the required interval

A1: 0.457 or better


285


286



Further Mathematics Option 1 Paper 3: Decision Mathematics 1

Further Mathematics Option 2 Paper 4: Decision Mathematics 1


Paper Reference(s)

9FM0/3D 9FM0/4D

You must have: Decision Mathematics Answer Book (enclosed), calculator

Instructions



Write your answers for this paper in the Decision Mathematics answer book provided.

Fill in the boxes at the top of the answer book with your name, centre number and candidate number.

Do not return the question paper with the answer book.







Advice





287

Answer ALL questions. Write your answers in the answer book provided.

1. A list of n numbers needs to be sorted into descending order starting at the left-hand end

of the list.

(a) Describe how to carry out the first pass of a bubble sort on the numbers in the list.

(2)

Bubble sort is a quadratic order algorithm.

A computer takes approximately 0.021 seconds to apply a bubble sort to a list of

2000 numbers.

(b) Estimate the time it would take the computer to apply a bubble sort to a list of 50 000

numbers. Make your method clear.

(2)



288

2.

Figure 1

(a) Define what is meant by a planar graph.

(2)

(b) Starting at A, find a Hamiltonian cycle for the graph in Figure 1.

(1)

Arc AG is added to Figure 1 to create the graph shown in Figure 2.

Figure 2

Taking ABCDEFGA as the Hamiltonian cycle,

(c) use the planarity algorithm to determine whether the graph shown in Figure 2 is

planar. You must make your working clear and justify your answer.

(4)



289

3.

(a) Explain clearly the difference between the classical travelling salesperson problem

and the practical travelling salesperson problem.

(2)

The table shows the least distances, in km, by road between seven towns, A, B, C, D,

E, F and G. The least distance between B and G is x km, where 25x

Preety needs to visit each town at least once, starting and finishing at A. She wishes

to minimise the total distance she travels.

(b) Starting by deleting B and all of its arcs, find a lower bound for Preety’s route.

(3)

Preety found the nearest neighbour routes from each of A and C. Given that the sum

of the lengths of these routes is 331 km,

(c) find ,x making your method clear.

(4)

(d) Write down the smallest interval that you can be confident contains the optimal

length of Preety’s route. Give your answer as an inequality.

(2)


A B C D E F G

A - 17 24 16 21 18 41

B 17 - 35 25 30 31 x

C 24 35 - 28 20 35 32

D 16 25 28 - 29 19 45

E 21 30 20 29 - 22 35

F 18 31 35 19 22 - 37

G 41 x 32 45 35 37 -


290

4.

Figure 3

The network in Figure 3 shows the roads linking a depot, D, and three collection points

A, B and C. The number on each arc represents the length, in miles, of the corresponding

road. The road from B to D is a one-way road, as indicated by the arrow.

(a) Explain clearly why Dijkstra’s algorithm cannot be used to find a route from D to A.

(1)

The initial distance and route tables for the network are given in the answer book.

(b) Use Floyd’s algorithm to find a table of least distances. You should show both the

distance table and the route table after each iteration.

(7)

(c) Explain how the final route table can be used to find the shortest route from D to B.

State this route.

(2)

There are items to collect at A, B and C. A van will leave D to make these

collections in any order and then return to D. A minimum route is required.

Using the final distance table and the Nearest Neighbour algorithm starting at D,

(d) find a minimum route and state its length.

(2)

Floyd’s algorithm and Dijkstra’s algorithm are applied to a network. Each will find

the shortest distance between vertices of the network.

(e) Describe how the results of these algorithms differ.

(2)



291

5. A garden centre makes hanging baskets to sell to its customers. Three types of hanging

basket are made, Sunshine, Drama and Peaceful. The plants used are categorised as

Impact, Flowering or Trailing.

Each Sunshine basket contains 2 Impact plants, 4 Flowering plants and 3 Trailing plants.

Each Drama basket contains 3 Impact plants, 2 Flowering plants and 4 Trailing plants.

Each Peaceful basket contains 1 Impact plant, 3 Flowering plants and 2 Trailing plants.

The garden centre can use at most 80 Impact plants, at most 140 Flowering plants and at

most 96 Trailing plants each day.

The profit on Sunshine, Drama and Peaceful baskets are £12, £20 and £16 respectively.

The garden centre wishes to maximise its profit.

Let x, y and z be the number of Sunshine, Drama and Peaceful baskets respectively,

produced each day.

(a) Formulate this situation as a linear programming problem, giving your constraints as

inequalities.

(5)

(b) State the further restriction that applies to the values of ,x y and z in this context.

(1)

The Simplex algorithm is used to solve this problem. After one iteration, the tableau

is

(c) State the variable that was increased in the first iteration. Justify your answer.

(2)

(d) Determine how many plants in total are being used after only one iteration of the

Simplex algorithm.

(1)

(e) Explain why for a second iteration of the Simplex algorithm the 2 in the z column is

the pivot value.

(2)

b.v. x y z r s t Value

r 1

4 0

1

2 1 0

3

4 8

s 5

2 0 2 0 1

1

2 92

y 3

4 1

1

2 0 0

1

4 24

P 3 0 6 0 0 5 480


292

After a second iteration, the tableau is

(f) Use algebra to explain why this tableau is optimal.

(1)

(g) State the optimal number of each type of basket that should be made.

(1)

The manager of the garden centre is able to increase the number of Impact plants

available each day from 80 to 100. She wants to know if this would increase her

profit.

(h) Use your final tableau to determine the effect of this increase. (You should not carry

out any further calculations.)

(2)


b.v. x y z r s t Value

r 3

8 0 0 1

1

4

7

8 31

z 5

4 0 1 0

1

2

1

4 46

y 1

8 1 0 0

1

4

3

8 1

P 21

2 0 0 0 3

7

2 756


293

6.

Figure 4

A project is modelled by the activity network shown in Figure 4. The activities are

represented by the arcs. The number in brackets on each arc gives the time, in days, to

complete that activity. Each activity requires one worker. The project is to be completed

in the shortest possible time.

(a) Calculate the early time and the late time for each event, using Diagram 1 in the

answer book.

(3)

(b) On Grid 1 in the answer book, complete the cascade (Gantt) chart for this project.

(3)

(c) On Grid 2 in the answer book, draw a resource histogram to show the number of

workers required each day when each activity begins at its earliest time.

(3)

The supervisor of the project states that only three workers are required to complete

the project in the minimum time.

(d) Use Grid 2 to determine if the project can be completed in the minimum time by only

three workers. Give reasons for your answer.

(3)



294

7. A linear programming problem in ,x y and z is described as follows.

Maximise 3 2 2P x y z

subject to 2 2 25x y z

4 15x y

3x

(a) Explain why the Simplex algorithm cannot be used to solve this linear programming

problem. (1)

The big-M method is to be used to solve this linear programming problem.

(b) Define, in this context, what M represents. You must use correct mathematical

language in your answer.

(1)

The initial tableau for a big-M solution to the problem is shown below.

b.v. x y z 1s 2s

3s 1t Value

1s 2 2 1 1 0 0 0 25

2s 1 4 0 0 1 0 0 15

1t 1 0 0 0 0 1 1 3

P (3 + M) 2 2 0 0 M 0 3M

(c) Explain clearly how the equation represented in the b.v. 1t row was derived.

(1)

(d) Show how the equation represented in the b.v. P row was derived.

(2)


295

The tableau obtained from the first iteration of the big-M method is shown below.

b.v. x y z 1s

2s 3s

1t Value

1s 0 2 1 1 0 2 2 19

2s 0 4 0 0 1 1 1 12

x 1 0 0 0 0 1 1 3

P 0 2 2 0 0 3 3 M 9

(e) Solve the linear programming problem, starting from this second tableau. You must

give a detailed explanation of your method by clearly stating the row operations

you use and

state the solution by deducing the final values of , , and . P x y z

(7)




296


Further Mathematics Advanced Further Mathematics Decision Mathematics 1

Sample assessment material for first teaching September 2017

Paper Reference(s)

9FM0/3D 9FM0/4D

Answer Book Do not return the question paper with the answer book.


297

1.

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298


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299

2.

Figure 1

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300


Figure 2

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301

3.

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A B C D E F G

A - 17 24 16 21 18 41

B 17 - 35 25 30 31 x

C 24 35 - 28 20 35 32

D 16 25 28 - 29 19 45

E 21 30 20 29 - 22 35

F 18 31 35 19 22 - 37

G 41 x 32 45 35 37 -


302


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A B C D E F G

A - 17 24 16 21 18 41

B 17 - 35 25 30 31 x

C 24 35 - 28 20 35 32

D 16 25 28 - 29 19 45

E 21 30 20 29 - 22 35

F 18 31 35 19 22 - 37

G 41 x 32 45 35 37 -


303

4.

Figure 3

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(b)

A B C D A B C D

A - 5 11 8 A A B C D

B 5 - 3 2 B A B C D

C 11 3 - 4 C A B C D

D 8 ∞ 4 - D A B C D

A B C D A B C D

A A

B B

C C

D D

A B C D A B C D

A A

B B

C C

D D


304


A B C D A B C D

A A

B B

C C

D D

A B C D A B C D

A A

B B

C C

D D

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305

5.

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307

6.

Diagram 1

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308


Grid 1

(There is a spare grid on the next page)

Grid 2


309


Copy of Grid 1

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310

7.

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b.v. x y z 1s 2s 3s 1t Value

1s 0 2 1 1 0 2 2 19

2s 0 4 0 0 1 1 1 12

x 1 0 0 0 0 1 1 3

P 0 2 2 0 0 3 3 M 9

b.v. x y z 1s 2s 3s 1t Value Row Ops

P


P


311



P


P

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312

AL Decision Mathematics 1 Mark Scheme


1(a)

In the first pass we compare the first value with the second

value and we swap these values if the second is larger than

the first.

B1 2.4

We then compare the value which is now second with the

third value and swap if the third is larger than the second.

We continue in this way until we reach the end of this list.

B1 2.4

(2)

(b) 250000

0.0212000

t

M1 1.1a

13.125t (seconds) A1 1.1b

(2)

(4 marks)

Notes:

(a)

B1: Comparing first value with second value, swap if second is larger (oe) – in their

reasoning it must be clear that the first value in the list is being compared with the second

value in the list and swapping if the second is larger than the first

B1: Compare second with third, (third with fourth), and so on until the end of the list –

must be clear in their reasoning that after the first comparison the second value in the

list is compared with the third value and so on until the end of the list

(b)

M1: Correct method seen – accept 25 for 50000/2000

A1: CAO


313


2(a)

A planar graph is a graph that can be drawn so that…….. B1 1.2

….. no arc meets another arc except at a vertex. B1 1.2

(2)

(b)

E.g. ABCDEGFA B1 1.1b

(1)

(c) Creates two lists of arcs, M1 2.1

e.g. BG AD

CG BD M1 1.1b

EG AE

CE AF A1 1.1b

Since no arc appears in both lists, the graph is planar

(or draws a planar version) A1 2.4

(4)

(7 marks)

Notes:

(a)

B1: A clear indication that a planar graph ‘can be drawn’ – allow this mark even if candidate

implies that arcs can cross each other

B1: CAO – no arc meets another arc except at a vertex – technical language must be correct

(b)

B1: Any correct Hamiltonian cycle (must start and finish at A) – must contain 8 vertices with

every vertex appearing only once (except A)

(c)

M1: Creates two list of arcs (with at least three arcs in each list) which contain no common

arcs

c2M1: Four arcs (in each list) and within each list there are no crossing arcs

c1A1: CAO

c2A1: Correct reasoning that no arc appears in both lists + so the graph is therefore planar


314


3(a) e.g. in the practical problem each vertex must be visited at least

once. In the classical problem each vertex must be visited just

once

B2,1,0 2.4

2.4

(2)

(b)

Prim’s algorithm on reduced network starting at A: AD, AF, AE,

CE, CG M1

1.1b

Lower bound = 107 + 17 + 25 = 149 (km) M1

A1

1.1b

1.1b

(3)

(c)

NNA from A: A – D – F – E – C – G – B – A = 126 x

NNA from C: C – E – A – D – F – B – G – C = 139 x

M1

A1

A1

1.1b

1.1b

1.1b

(126 ) (139 ) 331 33x x x A1 1.1b

(4)

(d) 149 optimal 159

M1

A1

2.2b

1.1b

(2)

(11 marks)

Notes:

(a)

B1: Understands the difference is connected to the number of times each vertex may be visited

(but maybe incorrectly attributed). Must be an attempt at a difference (so must refer to

both the classical and practical problems explicitly). Technical language (vertex/node)

must be correct. Need not imply each/every/all (oe) vertices for this first mark

B1: Correctly reasons which is classical and which is practical and correctly states the

difference. Must imply that each/every/all (oe) vertices are visited, so for example, ‘the

practical problem visits a vertex at least once while the classical visits a vertex only once’

is B1B0 (note that B0B1 is not possible in (a))

(b)

M1: Correctly applying Prim’s algorithm from node A for the first four arcs (or five nodes)

M1: Candidates weight of their RMST + 17 + 25 (the two smallest arcs incident to B)

A1: CAO (condone lack of units)

(c)

M1: Either one route, must return to A

A1: One correct route, must return to A and corresponding length correct (do not is in part

(c) if correct lengths seen but are then doubled)

A1: Both routes correct and their corresponding lengths correct

A1: CAO for x


315

Notes: (continued)

(d)

M1: Their numbers correctly used, accept any inequalities or any indication of interval from

their 149 to their 159 (so 149 – 159 can score this mark). This mark is dependent on two

routes seen in (c), however, neither of the two totals need to be correct. Please note that

UB > LB for this mark

A1: CAO (no follow through on their values) including correct inequalities or equivalent set

notation (but condone 149 optimal 159)


316


4(a) Dijkstra’s algorithm cannot be applied to directed

networks (in this problem there is a directed arc from

B to D)

B1 3.5b

(1)

(b)

Initial tables

5 11 8

5 3 2

11 3 4

8 4

A B C D

A B C D

A B C D

A B C D

1st iteration

3 2

3 4

4

5 11 8

5

11

8 13

A B C D

A B C D

B D

A CA

A C

D

M1

A1

1.1b

1.1b

2nd iteration

4

8

5

5 3 2

3

1 43

8 7

8

A B

A B C D

B C D

A A C

B

B

D

B

M1

A1ft

1.1b

1.1b

3rd iteration

8

3

5 7

5 2

8

8 3

47

4

A B B B

A B C D

B B C D

A C C D

M1

A1ft

1.1b

1.1b

4th iteration

5 8

5 3

7

2

8 4

8 3 4

7

A B B B

A B C D

B B C D

A C C D

no changes therefore optimal

A1 1.1b

(7)

(c) Start at D (4th) row and read across to the B (2nd) column,

there is a C there

so the route starts DC. Look at the C row, B column and you

see B B1 2.4

The route is therefore DCB B1 2.2a

(2)


317


(d)

D – C –B – A – B – D M1 2.2a

Length 19 (miles) A1 1.1b

(2)

(e) Dijkstra’s algorithm finds the shortest distances from one

vertex to all the others. Floyd’s algorithm finds the shortest

distance between every pair of vertices.

B1

B1

2.5

2.5

(2)

(14 marks)

Notes:

(a)

B1: CAO (must include mention of ‘directed’ network)

M1: No change in the first row and first column of both tables with at least one value in the

distance table reduced and one value in the route table changed

(b)

A1: CAO

M1: No change in the second row and second column of both tables with at least two values

in the distance table reduced and two values in the route table changed

A1ft: Correct second iteration follow through from the candidate’s first iteration

M1: No change in the third row and third column of both tables with at least one value in

the distance table reduced and one value in the route table changed

A1ft: Correct third iteration follow through from the candidate’s second iteration

A1: CSO (no change after the fourth iteration) – all previous marks must have been

awarded in this part

(c)

B1: Clear indication of how the final route table can be used to get from D to B (therefore

must mention the correct rows and columns in their reasoning)

B1: Completely correct argument + correct route (DCB)

(d)

M1: Deduce correctly their minimum route from their final distance table (dependent on all

M marks in (a)) must begin and end at D

A1: CAO (length of 19)

(e)

B1: CAO – must use correct language ‘one vertex to all other vertices’

B1: CAO – must use correct language ‘every pair of vertices’


318


5(a)

Maximise P = 12x + 20y + 16z B1 3.3

Subject to

2 3 80

4 2 3 140

3 4 2 96

, , 0

x y z

x y z

x y z

x y z

M1

A1

A1

B1

3.3

1.1b

1.1b

3.3

(5)

(b) The values must all be integers B1 3.3

(1)

(c)

Variable y entered the basic variable column…. M1 2.4

…so y was increased first A1 2.2a

(2)

(d) (80 140 96) (8 92) 216 plants B1 3.2a

(1)

(e)

The next pivot must come from a column which has a

negative value in the objective row so therefore the pivot

must come from column z.

M1 2.4

The pivot must be positive and the least of 92/2 = 46 and

24/0.5 = 48 so the pivot must be the 2 (from column z). A1 2.2a

(2)

(f) P + 10.5x + 3s + 3.5t = 756 so increasing x, s or t will

decrease profit B1 2.4

(1)

(g) Make 1 Drama basket and 46 Peaceful baskets B1 2.2a

(1)

(h) The slack variable, r, associated with this type of plant, is

currently at 31. Increasing the number of Impact plants by

a further 20 would have no effect.

M1

A1

3.1b

3.2a

(2)

(15 marks)


319

Notes:

(a)

B1: Correct objective function/expression (accept in pence rather than pounds e.g. 1200x +

2000y + 1600z)

M1: Correct coefficients and correct right-hand side for at least one inequality – accept any

inequality or equals

A1: Two correct (non-trivial) inequalities

A1: All three non-trivial inequalities correct

B1: , , 0x y z

(b)

B1: CAO

(c)

M1: Correct reasoning that y has become a basic variable

A1: Correct deduction that y was therefore increased first

(d)

B1: CAO

(e)

M1: Correct reasoning given that the pivot value must come from column z

A1: Correctly deduce (from correctly stated calculations) that the pivot value is the 2 in

column z

(f)

B1: States correct objective function and mention of increasing x, s or t will decrease profit

(g)

B1: CAO – in context so not in terms of y and z

(h)

M1: Identifies the slack variable r and its current value of 31

A1: Correct interpretation that increasing the number of Impact plants would have no effect


320


6(a)

See diagram on next page. Top and bottom boxes

Top boxes correct

M1

A1

2.1

1.1b

Bottom boxes correct A1 1.1b

(3)

(b)

See diagram below

At least 8 activities + 4 floats with clear distinction

between activity and their corresponding float M1 2.5

Correct critical activities + 4 correct non-critical activities A1 1.1b

All 13 correct A1 1.1b

(3)

(c)

M1 1.1b

Bars correct to time = 13 A1 1.1b

Bars correct from 14 to 24 A1 1.1b

(3)

(d) Until time 4 only A and B can happen.

After time 4, there are 6 worker-days to cover, but only 4

worker-days available.

Hence the project cannot be completed by time 24 with

three workers.

B1

M1

A1

3.1a

2.4

2.2a

(3)

(12 marks)


321

Notes:

(a)

M1: All top boxes and all bottom boxes completed. For the top boxes all values must be

increasing in the direction of the arrows for both the activities and the dummies. For

the bottom boxes all values must be decreasing in the opposite direction to the arrows

for both the activities and the dummies. While the values need not be correct each

value must be increasing or decreasing (as appropriate) in a logical and sequential

manner.

A1: CAO for top boxes

A1: CAO for bottom boxes

M1: At least 8 activities including 4 floats. Scheduling diagram scores M0 – clear

distinction must be shown between the notation used for an activity and its float

(b)

A1: Correct critical activities and 4 correct non-critical activities

A1: CAO (all 13 correct activities)

(c)

M1: Plausible histogram with no holes or overhangs (must go to at least 10 on the time axis)

A1: Histogram correct to time 13

A1: Histogram correct from time 14 to time 24

(d) B1: Considering an appropriate process to adjust Grid 2 so that no activity must be

completed by a 4th worker, for example, a correct argument that until time 4 only

activities A and B can happen (so no activity can use the spare worker before time 4)

M1: Uses their histogram to explain when the number of workers is greater or less than the

minimum number found in (b)

A1: Correctly deduces that the project cannot be completed by time 24 – this mark is

dependent on a correct histogram seen in (d)


322

Diagram for Question 6(a)


323


7(a) Simplex can only work with constraints B1 3.5b

(1)

(b) M is an arbitrary large real number B1 2.5

(1)

(c) 3 13 3x x s t where

3s is a surplus variable and 1t is an

artificial variable B1 2.4

(1)

(d) Let 13 2 2P x y z Mt (where M is an arbitrary large

number)

33 2 2 (3 )P x y z M x s

3(3 ) 2 2 3M x y z Ms M

3(3 ) 2 2 3P M x y z Ms M

M1

A1

2.1

1.1b

(2)

(e)

b.v. x y z 1s 2s

3s

1t Value Row Ops

3s 0 1 ½ ½ 0 1 1 19/2 1 1(1/ 2)r R

2s 0 3 1/2 1/2 1 0 0 5/2 2 1R r

x 1 1 ½ ½ 0 0 0 25/2 3 1R r

P 0 1 1/2 3/2 0 0 M 75/2 4 13R r

M1

A1

A1

1.1b

1.1b

1.1b

b.v. x y z 1s 2s

3s 1t Value Row Ops

z 0 2 1 1 0 2 2 19 1 12r R

2s 0 4 0 0 1 1 1 12 2 1(1/ 2)R r

x 1 0 0 0 0 1 1 3 3 1(1/ 2)R r

P 0 2 0 2 0 1 1M 47 4 1(1/ 2)R r

M1

A1

B1

1.1b

1.1b

2.4

47, 3, 0, 19P x y z B1ft

1.1b

(7)

(12 marks)


324

Notes:

(a)

B1: Correctly states the limitation of the Simplex model – Simplex involves iterations which

allow movement from one vertex in the feasible region to another vertex (in the feasible

region). If all constraints are of the form this means that the origin is always a feasible

solution and therefore can act as the initial starting point for the problem. However, the

constraint 3x means that the origin is not feasible and so the algorithm is unable to

begin.

(b)

B1: CAO including the correct mathematical language (must include ‘arbitrary’, ‘large’ and

‘real’)

(C)

(i)B1: Correctly states both the inequality 3x and the equation 3 1 3x s t together with an

explanation of the meaning behind the variables 3s and

1t

(ii)M1: 13 2 2P x y z Mt and substitutes their expression for

1t

(ii)A1: Correct mathematical argument including sufficient detail to allow the line of reasoning to

be followed to the correct conclusion – dependent on previous B mark in (c)

(d)

M1: Correct pivot located, attempt to divide row. If negative value used then no marks

A1: Pivot row correct (including change of b.v.) and row operations used at least once, one of

columns 1 1, , ,y z s t or Value correct

A1: CAO for values (ignore b.v. column and Row Ops)

M1: Pivot row consistent (following their previous table) including change of b.v. and row

operations used at least once, one of columns 1 3 1, , ,y s s t or Value correct

A1: CAO on final table (ignore Row Ops)

B1: The correct Row Operations explained either in terms of the ‘old’ or ‘new’ pivot rows

B1ft: Correctly states the final values of P, , and x y z from their correct corresponding rows of

the final table


325


326


Further Mathematics Advanced Decision Mathematics 2


Paper Reference(s)

9FM0/4G

You must have: Decision Mathematics Answer Book (enclosed), calculator

Instructions



Write your answers for this paper in the Decision Mathematics answer book provided.

Fill in the boxes at the top of the answer book with your name, centre number and candidate number.

Do not return the question paper with the answer book.







Advice





327

Answer ALL questions. Write your answers in the answer book provided.

1. (a) Find the general solution of the recurrence relation

2 1 , 1 n n nu u u n

(3)

Given that 1 1u and 2 1u

(b) find the particular solution of the recurrence relation.

(3)



328

2.

A company has three factories, A, B and C. It supplies mattresses to three shops, D, E and

F. The table shows the transportation cost, in pounds, of moving one mattress from each

factory to each shop. It also shows the number of mattresses available at each factory and

the number of mattresses required at each shop. A minimum cost solution is required.

(a) Use the north-west corner method to obtain an initial solution.

(1)

(b) Show how the transportation algorithm is used to solve this problem.

You must state, at each appropriate step, the

shadow costs,

improvement indices,

route,

entering cell and exiting cell,

and explain clearly how you know that your final solution is optimal.

(11)


D E F Available

A 15 19 9 25

B 11 18 10 55

C 11 12 18 20

Required 38 24 38


329

3. Four workers, A, B, C and D, are to be assigned to four tasks, P, Q, R and S.

Each worker must be assigned to at most one task and each task must be done by just one

worker.

The amount, in pounds, that each worker would earn while assigned to each task is

shown in the table below.

The Hungarian algorithm is to be used to find the maximum total amount which may be

earned by the four workers.

(a) Explain how the table should be modified.

(1)

(b) Reducing rows first, use the Hungarian algorithm to obtain an allocation which

maximises the total earnings, stating how each table was formed.

(7)

(c) Formulate the problem as a linear programming problem. You must define your

decision variables and make your objective function and constraints clear.

(5)


P Q R S

A 32 32 33 35

B 28 35 31 37

C 35 29 33 36

D 36 30 36 33


330

4. A game uses a standard pack of 52 playing cards.

A player gives 5 tokens to play and then picks a card. If they pick a 2, 3, 4, 5 or 6 then

they gain 15 tokens. If any other card is picked they lose.

If they lose, the card is replaced and they can choose to pick again for another 5 tokens.

This time if they pick either an ace or a king they gain 40 tokens. If any other card is picked

they lose.

Daniel is deciding whether to play this game.

(a) Draw a decision tree to model Daniel’s possible decisions and the possible outcomes.

(6)

(b) Calculate Daniel’s optimal EMV and state the optimal strategy indicated by the

decision tree.

(2)



331

5.

A two person zero-sum game is represented by the pay-off matrix for player A given

above.

(a) Explain, with justification, how this matrix may be reduced to a 3 3 matrix.

(2)

(b) Find the play-safe strategy for each player and verify that there is no stable solution

to this game.

(4)

The game is formulated as a linear programming problem for player A.

The objective is to maximise ,P V where V is the value of the game to player A.

One of the constraints is that 1 2 3 1,p p p where 1 2 3, ,p p p are the probabilities

that player A plays 1, 2, 3 respectively.

(c) Formulate the remaining constraints for this problem. Write these constraints as

inequalities.

(3)

The Simplex algorithm is used to solve the linear programming problem. The

solution obtained is 1 2 3

3 40, ,

7 7p p p

(d) Calculate the value of the game to player A.

(3)


B plays 1 B plays 2 B plays 3 B plays 4

A plays 1 4 2 3 2

A plays 2 3 1 2 0

A plays 3 1 2 0 3


332

6.

Figure 1

Figure 1 shows a capacitated, directed network. The number on each arc ( , )x y represents

the lower ( )x capacity and upper ( )y capacity of that arc.

(a) Calculate the value of the cut 1C and cut 2C

(2)

(b) Explain why the flow through the network must be at least 12 and at most 16

(1)

(c) Explain why arcs DG, AG, EG and FG must all be at their lower capacities.

(1)

(d) Determine a maximum flow pattern for this network and draw it on Diagram 1 in the

answer book. You do not need to use the labelling procedure.

(2)

(e) (i) State the value of the maximum flow through the network.

(ii) Explain why the value of the maximum flow is equal to the value of the

minimum flow through the network.

(3)

Node E becomes blocked and no flow can pass through it. To maintain the maximum

flow through the network the upper capacity of exactly one arc is increased.

(f) Explain how it is possible to maintain the maximum flow found in (d).

(3)



333

7. A company assembles boats.

They can assemble up to five boats in any one month, but if they assemble more than three

they will have to hire additional space at a cost of £800 per month.

The company can store up to two boats at a cost of £350 each per month.

The overhead costs are £1500 in any month in which work is done.

Boats are delivered at the end of each month. There are no boats in stock at the beginning

of January and there must be none in stock at the end of May.

The order book for boats is

Month January February March April May

Number ordered 3 2 6 3 4

Use dynamic programming to determine the production schedule which minimises the

costs to the company. Show your working in the table provided in the answer book and

state the minimum production cost.




334


Further Mathematics Advanced Decision Mathematics 2

Sample assessment material for first teaching September 2017

Paper Reference(s)

9FM0/4G

Answer Book Do not return the question paper with the answer book.


335

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2.

D E F Available

A 15 19 9 25

B 11 18 10 55

C 11 12 18 20

Required 38 24 38

(a)

D E F

A

B

C

(b)

D E F

A

B

C

D E F

A

B

C

D E F

A

B

C

D E F

A

B

C

D E F

A

B

C


338

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D E F

A

B

C

D E F

A

B

C

D E F

A

B

C

D E F

A

B

C

D E F

A

B

C


D E F

A

B

C

D E F

A

B

C


339

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3.

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

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P Q R S

A 32 32 33 35

B 28 35 31 37

C 35 29 33 36

D 36 30 36 33

P Q R S

A

B

C

D

P Q R S

A

B

C

D

P Q R S

A

B

C

D

P Q R S

A

B

C

D


340

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P Q R S

A

B

C

D

P Q R S

A

B

C

D


341

4.


342


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343

5.

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B plays 1 B plays 2 B plays 3 B plays 4

A plays 1 4 2 3 2

A plays 2 3 1 2 0

A plays 3 1 2 0 3


344


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345

6.

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Diagram 1

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347

7.

Stage State Action Dest Value


348


Number

assembled


349


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350

AL Decision Mathematics 2 Mark Scheme


1(a) Auxiliary equation: 2 1 0 and attempt to solve M1 1.1b

1 5

2

1 5 1 5

2 2

n n

nu A B

, where A and

B are arbitrary constants

M1

A1

1.1b

2.2a

(3)

(b) Use given conditions to obtain two equations in A and B M1 1.1b

Attempt to solve to obtain an A and B M1 1.1b

1 1 5 1 5

2 25

n n

nu

A1

1.1b

(3)

(6 marks)

Notes:

(a)

M1: writes down correct auxiliary equation and attempts to solve using either the formula or

completing the square

M1: writes down the general solution in the form 1 2( )n n

nu A B using their roots 1 2,

- dependent on the first M mark

A1: CAO – both lhs and rhs correct including defining A and B as (arbitrary) constants

(b)

M1: uses the correct initial conditions to write down two equations in A and B – for reference

these equations are 1 5 1 5 2A B and 2 2

1 5 1 5 4A B

M1: Attempts to solve these two equations (using a correct method but condone sign slips) to

achieve a value for A and B

A1: CAO


351


2(a)

D E F Available

A 25 25

B 13 24 18 55

C 20 20

Required 38 24 38

B1 1.1b

(1)

(b) Shadow

costs

15 22 14

D E F

0 A X -3 -5

-4 B X X X

4 C -8 -14 X

M1

A1

2.1

1.1b

D E F

A

B

24

18

C 20

Entering CE, exiting CF

D E F

A 25

B

13 4 38

C 20

M1

A1

2.1

1.1b

Shadow

costs

15 22 14

D E F

0 A X -3 -5

-4 B X X X

-10 C 6 X 14

M1

A1

1.1b

1.1b

D E F

A 25

B

13 38

C

Entering AF, exiting AD

D E F

A 25

B

38 4 13

C 20

M1

A1

1.1b

2.2a

Shadow

costs

10 17 9

D E F

0 A 5 2 X

1 B X X X

-5 C 6 X 14

No negative IIs so optimal solution of £1085

M1

A1

A1

2.1

1.1b

2.4

(11)

(12 marks)


352

Notes:

B1: CAO

M1: Finding all 6 shadow costs and the 4 improvement indices for the correct 4 entries –

candidates must clearly identify these two sets of results

A1: Shadow costs and II CAO

M1: A valid route, their most negative II chosen, only one empty square used, ’s balance

A1: CAO

M1: Finding all 6 shadow costs and the 4 improvement indices for the correct 4 entries


M1: A valid route, their most negative II chosen, only one empty square used, ’s balance

A1: CAO – including the deduction of all entering and exiting cells

M1: Finding all 6 shadow costs and the 4 improvement indices for the correct 4 entries – this

mark is depedent on all previous M marks which will therefore indicate a correct

mathematical argument leading from the initial solution to the confirmation of the optimal

solution


A1: CSO including the correct reasoning that the solution is optimal because there are no

negative IIs


353


3(a) Subtract each entry from a constant (eg 40) B1 2.4

(1)

(b)

e.g.

P Q R S

A 5 5 4 2

B 9 2 6 0

C 2 8 4 1

D 1 7 1 4

B1

1.1b

Reducing row A by 2, no reduction for row B, reduce row C by 1

and row D by 1. No reduction of columns P, R and S, reduce

column Q by 2.

P Q R S

A 3 3 2 0

B 9 2 6 0

C 1 7 3 0

D 0 6 0 3

then

P Q R S

A 3 1 2 0

B 9 0 6 0

C 1 5 3 0

D 0 4 0 3

B1

M1

A1

2.4

2.1

1.1b

Three lines required to cover the zeros hence solution is not

optimal – augment by 1 B1 2.4

P Q R S

A 2 1 1 0

B 8 0 5 0

C 0 5 2 0

D 0 5 0 4

M1 2.1

A – S, B – Q, C – P, D – R A1 2.2a

(7)

(c) 1 if worker does task

0 otherwise ij

i jx

B1 3.3

Where A,B,C,Di and P,Q,R,Sj B1 3.3

e.g. Minimise AP AQ AR AS BP BQ BR5 5 4 2 9 2 6x x x x x x x

CP CQ CR CS DP DQ DR DS2 8 4 7 4x x x x x x x x B1 3.3

Subject to:

P Q R S1, 1, 1, 1i i i ix x x x M1 3.3

A B C D1, 1, 1, 1j j j jx x x x A1 3.3

(5)

(13 marks)


354

Notes

(a)

B1: valid statement regarding converting a max. problem to a min. problem

(b)

B1: CAO

B1: Correct statements regarding row and column reduction

M1: Simplifying the initial matrix by reducing rows and then columns

A1: CAO

B1: Correct statements regarding both max. number of lines to cover zeros and augmentation

M1: Develop an improved solution – need to see one double covered +e; one uncovered –e;

and one single covered unchanged. 3 lines needed to 4 lines needed (so getting to the

optimal table)

A1: CSO on final table (so must have scored all previous marks in this part ) + deduction of

the correct allocation

(c)

B1: possible values of ijx defined

B1: definine the set of values for i and j

B1: Correct objective function and either ‘minimise’ or ‘maximise’ (dependent on if problem

is defined in terms of original values or modified values)

M1: at least four equations, unit coefficient and equal to 1

A1: CAO (all eight equations)


355


4(a)

M1

A1

M1

A1

M1

A1

3.3

1.1b

3.4

1.1b

3.4

1.1b

(6)

(b) EMV is 1.48 (tokens) per game (correct to 3 sf)

Analysis: Play the game and if the player doesn’t pick a 2 – 6 on

the first go then they should pick again

B1

B1

3.4

3.2a

(2)

(8 marks)

Notes:

(a)

M1: Tree diagram with at least three end pay-offs, two decision nodes and two chance nodes

A1: Correct structure of tree diagram with each arc labelled correctly (including probabilities)

M1: At least three end-pay offs consistent with their stated probabilities; all five attempted

A1: CAO for end-pay offs

M1: End chance node follow through their end pay-offs and other chance/decision nodes

completed

A1: CAO for decision and chance nodes including double lines through inferior options

(b)

B1: Correct EMV

B1: Correct analysis


356


5(a) Column 2 dominates column 4 B1 2.5

Because 2 2,0 1 and 3 2 B1 2.4

(2)

(b) Row minima: -2, -1, -1 max is -1

Column maxima: 4, 2, 3 min is 2

M1

A1

1.1b

1.1b

Play safe is A plays 2 or 3 and B plays 2 A1 1.1b

Row maximin (-1) Column minimax (2) so not stable A1 2.4

(4)

(c) e.g.

4 2 3 6 0 5

3 1 2 5 1 4

1 2 0 1 4 2

1 2 3

2 3

1 2 3

Subject to 6 5 0

4 0

5 4 2 0

V p p p

V p p

V p p p

B1

B1

B1

1.1b

3.3

3.3

(3)

(d) Substitute p values to obtain 19 19 20 19

, ,7 7 7 7

V V M1 3.4

Value of the game to player A 19 5

27 7

M1

A1

1.1b

1.1b

(3)

(12 marks)

Notes:

(a)

B1: Correct statement – must include the word ‘dominate’

B1: Correct inequalities – must be clear that all three inequalities must hold

(b)

M1: Attempt at row minima and column maxima – condone one error

A1: Correct max(row min) and min(col max)

A1: Correct play safe for both players

A1: Correct reasoning that the game is not stable (accept 1 2 + statement)

(c)

B1: Correct augmentation to make all entries non-negative

B1: At least one (of the three) equations or inequalities correct in V, 1 2 3, ,p p p (with all ip terms

in the constraint equations having correct signs)

B1: CAO - all three constraints correct involving V and ip expressed as inequalities

(d)

M1: Substitute p values to obtain three values for V

M1: Their least value of V minus their augmented value

A1: CAO for the value of the game to player A


357


6(a)

1 3 3 3 5 7 21C

2 8 5 3 6 3 19C

B1

B1

1.1b

1.1b

(2)

(b) e.g. the minimum flow out of the source S is at least 5 + 3 + 4 =

12 and the maximum flow into the sink T is 6 + 10 = 16

B1

2.4

(1)

(c) The minimum flow into G is 1 + 1 + 1 + 3 = 6 but the maximum

flow out of G is 6 therefore the arcs into G must be at their lower

capacities

B1

2.4

(1)

(d)

M1

A1

3.1a

1.1b

(2)

(e)

Maximum flow is 15 B1 1.1b

The minimum flow out of the source is 12 but the flow out of C

is at least 3 + 4 = 7 B1 2.4

Therefore the minimum flow through the network is 5 + 3 + 3 +

4 = 15 which is equal to the maximum flow B1 2.2a

(3)

(f) Increase the upper capacity of arc BF to at least 9 and therefore

increase the flow in this arc to 9 B1 2.1

Therefore increase the flow in FH and HT to 10 B1 2.4

The flow in GT decreases to 5 and all other arcs are unchanged B1 2.2a

(3)

(12 marks)


358

Notes:

(a)

B1: correct capacity for 1C

B1: correct capacity for 2C

(b)

B1: correct statement regarding the min. flow out of the sink and max. flow into the sink

(c)

B1: correct statement regarding the flow into node G

(d)

M1: consistent flow pattern ( 12) throughout the network - so the flow into each node must

equal the flow out of each node (and this flow must be greater than or equal to 12 but not

necessarily the maximum flow of 15) - one number only on each arc

A1: CAO

(e)

B1: CAO (for max. flow)

B1: Consideration of both the min. flow from the source and the flow through node C

B1: Completely correct argument that the max. flow = min. flow

(f) B1: Correct argument regarding increasing the upper capacity of arc BF and hence the flow in

that arc

B1: Correct reasoning regarding increasing the flow in arcs FH and HT

B1: Correct deduction that the flow in GT decreases to 5 and conclude that all other arcs are

unchanged


359


7

Stage Stat

e

Actio

n

Dest Value

May 2 2 0 700 + 1500 = 2200*

(4) 1 3 0 350 + 1500 = 1850*

0 4 0 1500 + 800 = 2300*

April 2 1 0 700 + 1500 + 2300 = 4500

(3) 2 1 700 + 1500 + 1850 = 4050*

3 2 700 + 1500 + 2200 = 4400

1 2 0 350 + 1500 + 2300 = 4150

3 1 350 + 1500 + 1850 = 3700*

4 2 350 + 1500 + 800 +2200 = 4850

0 3 0 1500 + 2300 = 3800*

4 1 1500 + 800 + 1850 = 4150

5 2 1500 + 800 + 2200 = 4500

March 2 4 0 700 + 1500 +800 + 3800 = 6800

(6) 5 1 700 + 1500 + 800 + 3700 = 6700*

1 5 0 350 + 1500 + 800 + 3800 = 6450*

Feb 2 1 1 700 + 1500 + 6450 = 8650*

(2) 2 2 700 + 1500 + 6700 = 8900

1 2 1 350 + 1500 + 6450 = 8300*

3 2 350 + 1500 + 6700 = 8550

0 3 1 1500 + 6450 = 7950*

4 2 1500 + 800 + 6700 = 9000

Jan 0 3 0 1500 + 7950 = 9450*

(3) 4 1 1500 + 800 + 8300 = 10600

5 2 1500 + 800 + 8650 = 10950

M1

A1

M1

A1

M1

A1ft

M1

A1ft

M1

A1

3.1b

1.1b

3.1b

1.1b

1.1b

1.1b

1.1b

1.1b

1.1b

1.1b


Number

made 3 3 5 3 4

B1 1.1b

Minimum production cost: £9450 B1 1.1b

(12 marks)

Notes:

All M marks – must bring optimal result from previous stage into calculations so for the

second stage (April) if none of their 2200, 1850 or 2300 (the optimal results from May) are

used then M0. Ignore extra rows. Condone and credit rows that have been crossed out if

they can still be read. Must have right ‘ingredients’ (storage costs, additional space costs,

overhead cost) at least once per stage. Must have values in two of the three colums (State,

Action, Dest). If no working seen then the number stated in the Value column must be

correct to imply the correct method has been used


360

1M1: First stage (May) completed. At least 3 rows, ‘something’ in each cell (but see M mark

guidance above) including the correct structure (e.g. no value greater than 5 in the action

column) in each of the first four columns

1A1: CAO for first stage.

2M1: Second stage (April) completed. At least 9 rows, something in each cell (see M mark

guidance above) including the correct structure for the fifth (Value) column (e.g. bringing

forward values from the previous stage)

2A1: CAO for second stage. No extra rows

3M1: Third stage (March) completed. At least 3 rows, something in each cell (see M mark

guidance above)

3A1ft: CAO on the ft for third stage. No extra rows

4M1: Fourth stage (February) completed. At least 6 rows, something in each cell (see M mark

guidance above)

4A1ft: CAO on the ft for fourth stage. No extra rows

5M1: Fifth stage (January) completed. At least 3 rows, something in each cell (see M mark

guidance above)

5A1: CAO for the fifth stage. No extra rows

1B1: CAO – but must have scored all previous M marks

2B1: CAO – condone lack of units - but must have scored all previous M marks


361


362

Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE

Mathematics and

Further Mathematics

Mathematical formulae and statistical tables

For first certification from June 2018 for:

Advanced Subsidiary GCE in Mathematics (8MA0)

Advanced GCE in Mathematics (9MA0)

Advanced Subsidiary GCE in Further Mathematics (8FM0)

For first certification from June 2019 for:

Advanced GCE in Further Mathematics (9FM0)

This copy is the property of Pearson. It is not to be removed from the

examination room or marked in any way.

Contents

1 Introduction 1

2 AS Level in Mathematics 2

Pure Mathematics 2

Statistics 2

Mechanics 3

3 A Level in Mathematics 4

Pure Mathematics 4

Statistics 6

Mechanics 7

4 AS Level in Further Mathematics 8

Pure Mathematics 8

Statistics 12

Mechanics 14

5 A Level in Further Mathematics 15

Pure Mathematics 15

Statistics 21

Mechanics 25

6 Statistical Tables 26

Binomial Cumulative Distribution Function 26

Percentage Points Of The Normal Distribution 31

Poisson Cumulative Distribution Function 32

Percentage Points of the 2 Distribution 33

Critical Values for Correlation Coefficients 34

Random Numbers 35

Percentage Points of Student’s t Distribution 36

Percentage Points of the F Distribution 37

Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics 1

Mathematical Formulae and Statistical Tables – Issue 1 – Accredited pre-publication - April 2017


1 Introduction The formulae in this booklet have been arranged by qualification. Students sitting AS or A Level

Further Mathematics papers may be required to use the formulae that were introduced in AS or

A Level Mathematics papers.

It may also be the case that students sitting Mechanics and Statistics papers will need to use

formulae introduced in the appropriate Pure Mathematics papers for the qualification they are

sitting.

2 Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics



2 AS Level in Mathematics

Pure Mathematics

Mensuration

Surface area of sphere = 4 r 2

Area of curved surface of cone = r slant height

Binomial series

1 2 2

2( )

1 n n n r r nn n

n na b a b a b b

r

na b a

(n ℕ)

where C( )

n

r

n n!

r r ! n r !

Logarithms and exponentials

loglog

log

ba

b

xx

a

lnex a xa

Differentiation

First Principles

0

f ( ) f ( )f ( ) = lim

h

x h xx

h

Statistics

Probability

P(A) = 1 – P(A)

Standard deviation

Standard deviation = (Variance)

Interquartile range = IQR = Q3 – Q1

For a set of n values 1 2, ,... ,...i nx x x x

22 2 ( )

S ( ) ixx i i

xx x x

n

Standard deviation =

2

2or

S xx

xx

n n




Statistical tables

The following statistical tables are required for A Level Mathematics:

Binomial Cumulative Distribution Function (see page 25)

Random Numbers (see page 34)

Mechanics

Kinematics

For motion in a straight line with constant acceleration:

v = u + at

s = ut + ½ at2

s = vt - ½ at2

v2 = u2 + 2as

s = ½ (u + v)t




3 A Level in Mathematics

Pure Mathematics

Mensuration

Surface area of sphere = 4 r 2

Area of curved surface of cone = r slant height

Arithmetic series

Sn = 1

2n(a + l) =

1

2n[2a + (n 1)d]

Binomial series

1 2 2

21 n n n r r nn n

n na b a b a b b

r

n( a b ) a

(n ℕ)

where C( )

n

r

n n!

r r ! n r !

2( 1) ( 1) ( 1)(1 ) 1 ( 1 )

1 2 1 2

n rn n n n n rx nx x x x , n

r

Logarithms and exponentials

loglog

log

ba

b

xx

a

lnx a xae

Geometric series

Sn = (1 )

1

na r

r

S =1

a

r for r < 1

Numerical integration

The trapezium rule:

b

a

xy d 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where

b ah

n




Trigonometric identities

sin ( ) sin cos cos sinA B A B A B

cos( ) cos cos sin sinA B A B A B

1

2

tan tantan ( ) ( ( ) )

1 tan tan

A BA B A B k

A B

sin sin 2sin cos2 2

A B A BA B

sin sin 2cos sin2 2

A B A BA B

cos cos 2cos cos2 2

A B A BA B

cos cos 2sin sin2 2

A B A BA B

Differentiation

First Principles

0

f ( ) f ( )f ( ) = lim

h

x h xx

h

f(x) f (x)

tan kx k sec2 kx

seckx kseckx tankx

cotkx – kcosec2kx

cosec kx – kcosec kx cot kx

f( )

g( )

x

x

2

f ( ) g( ) f( ) g ( )

(g( ))

x x x x

x




Integration (+ constant)

f(x) f( ) d

x x

sec2 kx k

1 tan kx

tankx k

1ln seckx

cot kx k

1ln sinkx

coseckx 12

1 1ln cosec cot , ln tan( )kx kx kx

k k

seckx 1 12 4

1 1ln sec tan , ln tan( )kx kx kx

k k

xx

uvuvx

x

vu d

d

dd

d

d

Numerical solution of equations

The Newton-Raphson iteration for solving 0)f( x : )(f

)f(1

n

n

nnx

xxx

Statistics

Probability

P(A) = 1 – P(A)

P( ) P( ) P( ) P( )A B A B A B

P( ) P( )P( )A B A B | A

)P()|P()P()|P(

)P()|P()|P(

AABAAB

AABBA

For independent events A and B,

P(BA) = P(B), P(AB) = P(A),

P(A B) = P(A) P(B)




Standard deviation

Standard deviation = (Variance)

Interquartile range = IQR = Q3 – Q1

For a set of n values 1 2, ,... ,...i nx x x x

n

xxxxS i

iixx

2

22)(

)(

Standard deviation =

2

2xxxS

xn n

or Discrete distributions

Distribution of X P(X = x) Mean Variance

Binomial ),B( pn (1 )x n xnp p

x

np )1( pnp

Sampling distributions

For a random sample of n observations from 2N( , )

~ N(0, 1)/

X

n

Statistical tables

The following statistical tables are required for A Level Mathematics:


Percentage Points Of The Normal Distribution (30)

Critical Values for Correlation Coefficients: Product Moment Coefficient (see page 33)


Mechanics

Kinematics

For motion in a straight line with constant acceleration:

v = u + at

s = ut + ½ at2

s = vt - ½ at2

v2 = u2 + 2as

s = ½ (u + v)t




4 AS Level in Further Mathematics

Students sitting a AS Level Further Mathematics paper may also require those formulae listed for

A Level Mathematics in Section 3.

Pure Mathematics

Summations

)12)(1(6

1

1

2

nnnrn

r

22

4

1

1

3 )1(

nnrn

r

Matrix transformations

Anticlockwise rotation through about O:

cos sin

sincos

Reflection in the line xy )(tan :

2cos2sin

2sin 2cos

Area of a sector

A = d

2

1 2r (polar coordinates)

Complex numbers

{ (cos i sin )} (cos i sin )n n

r r n n

The roots of 1nz are given by

2 i

ek

nz

, for 1 , ,2 ,1 ,0 nk




Maclaurin’s and Taylor’s Series

)0(f!

)0(f!2

)0(f)0f()f( )(2

rr

r

xxxx

xr

xxxx

rx allfor

!

!21)exp(e

2

2 31ln (1 ) ( 1) ( 1 1)

2 3

rrx x x

x x xr

xr

xxxxx

rr allfor

)!12()1(

!5!3sin

1253

xr

xxxx

rr allfor

)!2()1(

!4!21cos

242

3 5 2 1

arctan ( 1) ( 1 1)3 5 2 1

rrx x x

x x xr

Vectors

Vector product:

1221

3113

2332

321

321ˆ sin

baba

baba

baba

bbb

aaa

kji

nbaba

)()()(

321

321

321

bac.acb.cba.

ccc

bbb

aaa

If A is the point with position vector kjia 321 aaa and the direction vector b is given by

kjib 321 bbb , then the straight line through A with direction vector b has cartesian equation

)( 3

3

2

2

1

1

b

az

b

ay

b

ax

The plane through A with normal vector kjin 321 nnn has cartesian equation

1 2 3 0 where n x n y n z d d a.n

The plane through non-collinear points A, B and C has vector equation

cbaacabar )1()()(

The plane through the point with position vector a and parallel to b and c has equation

cbar ts

The perpendicular distance of ) , ,( from 1 2 3 0n x n y n z d is 1 2 3

2 2 2

1 2 3

n n n d

n n n




Hyperbolic functions

1sinhcosh 22 xx

xxx coshsinh22sinh

xxx 22 sinhcosh2cosh

)1( 1lnarcosh }{ 2 xxxx

}{ 1lnarsinh 2 xxx

12

1artanh ln ( 1)

1

xx x

x

Differentiation

f(x) f(x)

xarcsin 21

1

x

xarccos 21

1

x

xarctan 21

1

x

xsinh xcosh

xcosh xsinh

xtanh x2sech

xarsinh 21

1

x

xarcosh 1

1

2 x

artanh x 21

1

x




Integration (+ constant; 0a where relevant)

f(x)

xx d)f(

xsinh xcosh

xcosh xsinh

xtanh xcoshln

22

1

xa

)( arcsin axa

x

22

1

xa

a

x

aarctan

1

22

1

ax

)( ln,arcosh }{ 22 axaxxa

x

22

1

xa

}{ 22ln,arsinh axxa

x

22

1

xa )( artanh

1ln

2

1ax

a

x

axa

xa

a

22

1

ax

ax

ax

a

ln

2

1




Statistics

Discrete distributions

For a discrete random variable X taking values ix with probabilities P(X = xi)

Expectation (mean): E(X) = = ix P(X = ix )

Variance: Var(X) = 2 = ( ix – )2 P(X = ix ) = 2

ix P(X = ix ) – 2


Standard discrete distributions:

Distribution of X P(X = x) Mean Variance

Binomial B(n, p) 1n xx

np p

x

np np(1 – p)

Poisson Po ( ) e!

x

x

Continuous distributions

For a continuous random variable X having probability density function f

Expectation (mean): xxxX d)f()E(

Variance: 2222 d)f(d)f()()Var( xxxxxxX

For a function )g( X : xxxX d)f()g())E(g(

Cumulative distribution function:

0

0 0F( ) P( ) f ( ) d

x

x X x t t

Standard continuous distribution:

Distribution of X P.D.F. Mean Variance

Normal ) ,N( 2

2

21

e2

1

x

2

Uniform (Rectangular) on [a, b] ab

1

1

2( )a b

2

121 )( ab




Correlation and regression

For a set of n pairs of values ) ,(ii

yx

22 2 ( )

S ( ) ixx i i

xx x x

n

22 2 ( )

S ( ) iyy i i

yy y y

n

( )( )S ( )( ) i i

xy i i i i

x yx x y y x y

n

The product moment correlation coefficient is:

2 2 2 22 2

( )( )S ( )( )

S S ( ) ( ) ( ) ( )

{ }{ }

i ii i

xy i i

xx yy i i i ii i

x yx y

x x y y nrx x y y x y

x yn n

The regression coefficient of y on x is 2

S ( )( )

S ( )

xy i i

xx i

x x y yb

x x

Least squares regression line of y on x is bxay where xbya

Residual Sum of Squares (RSS) =

2

2S

S S 1S

xy

yy yy

xx

r

Spearman’s rank correlation coefficient is

2

2

61

( 1)s

d

n nr

Non-parametric tests

Goodness-of-fit test and contingency tables: 2

2

~)(

i

ii

E

EO

Statistical tables

The following statistical tables are required for AS Level Further Mathematics:


Poisson Cumulative Distribution Function (see page 31)

Percentage Points of the 2 Distribution (see page 32)

Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman’s Coefficient

(see page 33)





Mechanics

Centres of mass

For uniform bodies:

Triangular lamina: 23

along median from vertex

Circular arc, radius r, angle at centre 2 :

sinr from centre

Sector of circle, radius r, angle at centre 2 : 2 sin

3

r

from centre




5 A Level in Further Mathematics

Students sitting a A Level Further Mathematics paper may also require those formulae listed for

A Level Mathematics in Section 3.

Pure Mathematics

Summations

)12)(1(6

1

1

2

nnnrn

r

22

4

1

1

3 )1(

nnrn

r

Matrix transformations

Anticlockwise rotation through about O:

cos sin

sincos

Reflection in the line xy )(tan :

2cos2sin

2sin 2cos

Area of a sector

A = d

2

1 2r (polar coordinates)

Complex numbers

{ (cos i sin )} (cos i sin )n n

r r n n

The roots of 1nz are given by

2 i

ek

nz

, for 1 , ,2 ,1 ,0 nk




Maclaurin’s and Taylor’s Series

)0(f!

)0(f!2

)0(f)0f()f( )(2

rr

r

xxxx

xr

xxxx

rx allfor

!

!21)exp(e

2

2 31ln (1 ) ( 1) ( 1 1)

2 3

rrx x x

x x xr

xr

xxxxx

rr allfor

)!12()1(

!5!3sin

1253

xr

xxxx

rr allfor

)!2()1(

!4!21cos

242

3 5 2 1

arctan ( 1) ( 1 1)3 5 2 1

rrx x x

x x xr

Vectors

Vector product:

1221

3113

2332

321

321ˆ sin

baba

baba

baba

bbb

aaa

kji

nbaba

)()()(

321

321

321

bac.acb.cba.

ccc

bbb

aaa

If A is the point with position vector kjia 321 aaa and the direction vector b is given by

kjib 321 bbb , then the straight line through A with direction vector b has cartesian equation

)( 3

3

2

2

1

1

b

az

b

ay

b

ax

The plane through A with normal vector kjin 321 nnn has cartesian equation

1 2 3 0 where n x n y n z d d a.n

The plane through non-collinear points A, B and C has vector equation

cbaacabar )1()()(

The plane through the point with position vector a and parallel to b and c has equation

cbar ts The perpendicular distance of ) , ,( from 1 2 3 0n x n y n z d is

1 2 3

2 2 2

1 2 3

n n n d

n n n




Hyperbolic functions

1sinhcosh 22 xx

xxx coshsinh22sinh

xxx 22 sinhcosh2cosh

)1( 1lnarcosh }{ 2 xxxx

}{ 1lnarsinh 2 xxx

12

1artanh ln ( 1)

1

xx x

x

Conics

Ellipse Parabola Hyperbola

Rectangular

Hyperbola

Standard

Form 1

2

2

2

2

b

y

a

x axy 42 1

2

2

2

2

b

y

a

x

2cxy

Parametric

Form )sin ,cos( ba )2 ,( 2 atat

(a sec , b tan )

(a cosh , b sinh )

t

cct,

Eccentricity 1e

)1( 222 eab 1e

1e

)1( 222 eab e = 2

Foci )0 ,( ae )0 ,(a )0 ,( ae ( 2c , 2c )

Directrices e

ax ax

e

ax x + y = 2c

Asymptotes none none b

y

a

x 0 ,0 yx




Differentiation

f(x) f(x)

xarcsin 21

1

x

xarccos 21

1

x

xarctan 21

1

x

xsinh xcosh

xcosh xsinh

xtanh x2sech

xarsinh 21

1

x

xarcosh 1

1

2 x

artanh x 21

1

x




Integration (+ constant; 0a where relevant)

f(x)

xx d)f(

xsinh xcosh

xcosh xsinh

xtanh xcoshln

22

1

xa

)( arcsin axa

x

22

1

xa

a

x

aarctan

1

22

1

ax

)( ln,arcosh }{ 22 axaxxa

x

22

1

xa

}{ 22ln,arsinh axxa

x

22

1

xa )( artanh

1ln

2

1ax

a

x

axa

xa

a

22

1

ax

ax

ax

a

ln

2

1

Arc length

xx

ys d

d

d1

2

(cartesian coordinates)

tt

y

t

xs d

d

d

d

d22

(parametric form)

2

2 drd

ds r

(polar form)




Surface area of revolution

Sx =

2d

2 1 dd

yy x

x

(cartesian coordinates)

Sx =

2 2d d

2 dd d

x yy t

t t

(parametric form)

Sx =

2

2 dr2 sin d

dr r

(polar form)




Statistics


For a discrete random variable X taking values ix with probabilities P(X = xi)

Expectation (mean): E(X) = = xi P(X = xi)

Variance: Var(X) = 2 = (xi – )2 P(X = xi) = 2

ix P(X = xi) – 2

For a function )g( X : E(g(X)) = g(xi) P(X = xi)

The probability generating function of X is G ( ) EX

Xt t and

2

E( ) G (1) and Var( ) G (1) G (1) G (1)X X X X

X X

For Z = X + Y, where X and Y are independent: G ( ) G ( ) G ( )Z X Y

t t t


Standard discrete distributions:

Distribution of X P(X = x) Mean Variance P.G.F.

Binomial B(n, p) 1n xx

np p

x

np np(1 – p) 1

np pt

Poisson Po ( ) e!

x

x

1

et

Geometric Geo(p)

on 1, 2, …

11

xp p

1

p

2

1 p

p

1 (1 )

pt

p t

Negative binomial

on r, r + 1, …

1(1 )

1

r x rx

p pr

r

p

2

(1 )r p

p

1 (1 )

r

pt

p t

Continuous distributions

For a continuous random variable X having probability density function f

Expectation (mean): xxxX d)f()E(

Variance: 2222 d)f(d)f()()Var( xxxxxxX

For a function )g( X : xxxX d)f()g())E(g(

Cumulative distribution function:

0

0 0F( ) P( ) f ( ) d

x

x X x t t




Standard continuous distribution:

Distribution of X P.D.F. Mean Variance

Normal ) ,N( 2

2

21

e2

1

x

2

Uniform (Rectangular) on [a, b] ab

1

1

2( )a b

2

121 )( ab

Correlation and regression

For a set of n pairs of values ) ,(ii

yx

22 2 ( )

S ( ) ixx i i

xx x x

n

22 2 ( )

S ( ) iyy i i

yy y y

n

( )( )S ( )( ) i i

xy i i i i

x yx x y y x y

n

The product moment correlation coefficient is

2 2 2 22 2

( )( )S ( )( )

S S ( ) ( ) ( ) ( )

{ }{ }

i ii i

xy i i

xx yy i i i ii i

x yx y

x x y y nrx x y y x y

x yn n

The regression coefficient of y on x is 2

S ( )( )

S ( )

xy i i

xx i

x x y yb

x x

Least squares regression line of y on x is bxay where xbya

Residual Sum of Squares (RSS) =

2

2S

S S 1S

xy

yy yy

xx

r

Spearman’s rank correlation coefficient is

2

2

61

( 1)s

d

n nr

Expectation algebra

For independent random variables X and Y

E( ) E( ) E( )XY X Y , 2 2Var( ) Var( ) Var( )aX bY a X b Y




Sampling distributions

(i) Tests for mean when is known

For a random sample n

XXX , , ,21 of n independent observations from a distribution having

mean and variance 2

:

X is an unbiased estimator of , with n

X2

)Var(

2S is an unbiased estimator of 2 , where

1

)( 2

2

n

XXS i

For a random sample of n observations from 2

N( , ) , ~ N(0, 1)/

X

n

For a random sample of xn observations from ) ,N( 2

xx and, independently, a random

sample of yn observations from ) ,N( 2

yy , )1 ,0N(~

)()(

22

y

y

x

x

yx

nn

YX

(ii) Tests for variance and mean when is not known

For a random sample of n observations from ) ,N( 2 :

2

12

2

~)1(

n

Sn

1~

/

n

tnS

X (also valid in matched-pairs situations)

For a random sample of xn observations from ) ,N( 2

xx and, independently, a random

sample of y

n observations from ) ,N( 2

yy

1 ,122

22

~/

/ ynxn

yy

xx FS

S

If 222

yx (unknown) then

2

2

( ) ( )~

1 1

x y

n nx y

p

x y

X Yt

Sn n

where

2 2

2( 1) ( 1)

2

x x y y

p

x y

n S n SS

n n




Non-parametric tests

Goodness-of-fit test and contingency tables: 2

2

~)(

i

ii

E

EOStatistical tables

The following statistical tables are required for A Level Further Mathematics:


Percentage Points Of The Normal Distribution (see page 30)

Poisson Cumulative Distribution Function (see page 31)

Percentage Points of the 2 Distribution (see page 32)

Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman’s Coefficient

(see page 33)


Percentage Points of Student’s t Distribution (see page 35)

Percentage Points of the F Distribution (see page 36




Mechanics

Centres of mass

For uniform bodies:

Triangular lamina: 32

along median from vertex

Circular arc, radius r, angle at centre 2 :

sinr from centre

Sector of circle, radius r, angle at centre 2 : 2 sin

3

r

from centre

Solid hemisphere, radius r: r8

3 from centre

Hemispherical shell, radius r: r21

from centre

Solid cone or pyramid of height h: 14

h above the base on the line from centre of base to vertex

Conical shell of height h: 13

h above the base on the line from centre of base to vertex

Motion in a circle

Transverse velocity: rv

Transverse acceleration: rv

Radial acceleration: r

vr

22




6 Statistical Tables

Binomial Cumulative Distribution Function

The tabulated value is P(X x), where X has a binomial distribution with index n and parameter p.

p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n = 5, x = 0 0.7738 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0312

1 0.9774 0.9185 0.8352 0.7373 0.6328 0.5282 0.4284 0.3370 0.2562 0.1875

2 0.9988 0.9914 0.9734 0.9421 0.8965 0.8369 0.7648 0.6826 0.5931 0.5000

3 1.0000 0.9995 0.9978 0.9933 0.9844 0.9692 0.9460 0.9130 0.8688 0.8125

4 1.0000 1.0000 0.9999 0.9997 0.9990 0.9976 0.9947 0.9898 0.9815 0.9688

n = 6, x = 0 0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156

1 0.9672 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.1094

2 0.9978 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443 0.4415 0.3438

3 0.9999 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.6563

4 1.0000 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906

5 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844

n = 7, x = 0 0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078 1 0.9556 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.0625

2 0.9962 0.9743 0.9262 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.2266

3 0.9998 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.5000

4 1.0000 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0.8471 0.7734

5 1.0000 1.0000 0.9999 0.9996 0.9987 0.9962 0.9910 0.9812 0.9643 0.9375

6 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9984 0.9963 0.9922

n = 8, x = 0 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039 1 0.9428 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.0632 0.0352

2 0.9942 0.9619 0.8948 0.7969 0.6785 0.5518 0.4278 0.3154 0.2201 0.1445

3 0.9996 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.5941 0.4770 0.3633

4 1.0000 0.9996 0.9971 0.9896 0.9727 0.9420 0.8939 0.8263 0.7396 0.6367

5 1.0000 1.0000 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.8555

6 1.0000 1.0000 1.0000 0.9999 0.9996 0.9987 0.9964 0.9915 0.9819 0.9648

7 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9983 0.9961

n = 9, x = 0 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020 1 0.9288 0.7748 0.5995 0.4362 0.3003 0.1960 0.1211 0.0705 0.0385 0.0195

2 0.9916 0.9470 0.8591 0.7382 0.6007 0.4628 0.3373 0.2318 0.1495 0.0898

3 0.9994 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.2539

4 1.0000 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000

5 1.0000 0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0.9006 0.8342 0.7461

6 1.0000 1.0000 1.0000 0.9997 0.9987 0.9957 0.9888 0.9750 0.9502 0.9102

7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9986 0.9962 0.9909 0.9805

8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992 0.9980

n = 10, x = 0 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010 1 0.9139 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0.0464 0.0233 0.0107

2 0.9885 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.0547

3 0.9990 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.1719

4 0.9999 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770

5 1.0000 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.6230

6 1.0000 1.0000 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.8281

7 1.0000 1.0000 1.0000 0.9999 0.9996 0.9984 0.9952 0.9877 0.9726 0.9453

8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983 0.9955 0.9893

9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990




p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n = 12, x = 0 0.5404 0.2824 0.1422 0.0687 0.0317 0.0138 0.0057 0.0022 0.0008 0.0002 1 0.8816 0.6590 0.4435 0.2749 0.1584 0.0850 0.0424 0.0196 0.0083 0.0032

2 0.9804 0.8891 0.7358 0.5583 0.3907 0.2528 0.1513 0.0834 0.0421 0.0193

3 0.9978 0.9744 0.9078 0.7946 0.6488 0.4925 0.3467 0.2253 0.1345 0.0730

4 0.9998 0.9957 0.9761 0.9274 0.8424 0.7237 0.5833 0.4382 0.3044 0.1938

5 1.0000 0.9995 0.9954 0.9806 0.9456 0.8822 0.7873 0.6652 0.5269 0.3872

6 1.0000 0.9999 0.9993 0.9961 0.9857 0.9614 0.9154 0.8418 0.7393 0.6128

7 1.0000 1.0000 0.9999 0.9994 0.9972 0.9905 0.9745 0.9427 0.8883 0.8062

8 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983 0.9944 0.9847 0.9644 0.9270

9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9992 0.9972 0.9921 0.9807

10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9968

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998

n = 15, x = 0 0.4633 0.2059 0.0874 0.0352 0.0134 0.0047 0.0016 0.0005 0.0001 0.0000 1 0.8290 0.5490 0.3186 0.1671 0.0802 0.0353 0.0142 0.0052 0.0017 0.0005

2 0.9638 0.8159 0.6042 0.3980 0.2361 0.1268 0.0617 0.0271 0.0107 0.0037

3 0.9945 0.9444 0.8227 0.6482 0.4613 0.2969 0.1727 0.0905 0.0424 0.0176

4 0.9994 0.9873 0.9383 0.8358 0.6865 0.5155 0.3519 0.2173 0.1204 0.0592

5 0.9999 0.9978 0.9832 0.9389 0.8516 0.7216 0.5643 0.4032 0.2608 0.1509

6 1.0000 0.9997 0.9964 0.9819 0.9434 0.8689 0.7548 0.6098 0.4522 0.3036

7 1.0000 1.0000 0.9994 0.9958 0.9827 0.9500 0.8868 0.7869 0.6535 0.5000

8 1.0000 1.0000 0.9999 0.9992 0.9958 0.9848 0.9578 0.9050 0.8182 0.6964

9 1.0000 1.0000 1.0000 0.9999 0.9992 0.9963 0.9876 0.9662 0.9231 0.8491

10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9972 0.9907 0.9745 0.9408

11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9937 0.9824

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9963

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

n = 20, x = 0 0.3585 0.1216 0.0388 0.0115 0.0032 0.0008 0.0002 0.0000 0.0000 0.0000 1 0.7358 0.3917 0.1756 0.0692 0.0243 0.0076 0.0021 0.0005 0.0001 0.0000

2 0.9245 0.6769 0.4049 0.2061 0.0913 0.0355 0.0121 0.0036 0.0009 0.0002

3 0.9841 0.8670 0.6477 0.4114 0.2252 0.1071 0.0444 0.0160 0.0049 0.0013

4 0.9974 0.9568 0.8298 0.6296 0.4148 0.2375 0.1182 0.0510 0.0189 0.0059

5 0.9997 0.9887 0.9327 0.8042 0.6172 0.4164 0.2454 0.1256 0.0553 0.0207

6 1.0000 0.9976 0.9781 0.9133 0.7858 0.6080 0.4166 0.2500 0.1299 0.0577

7 1.0000 0.9996 0.9941 0.9679 0.8982 0.7723 0.6010 0.4159 0.2520 0.1316

8 1.0000 0.9999 0.9987 0.9900 0.9591 0.8867 0.7624 0.5956 0.4143 0.2517

9 1.0000 1.0000 0.9998 0.9974 0.9861 0.9520 0.8782 0.7553 0.5914 0.4119

10 1.0000 1.0000 1.0000 0.9994 0.9961 0.9829 0.9468 0.8725 0.7507 0.5881

11 1.0000 1.0000 1.0000 0.9999 0.9991 0.9949 0.9804 0.9435 0.8692 0.7483

12 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 0.9940 0.9790 0.9420 0.8684

13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9935 0.9786 0.9423

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9936 0.9793

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9941

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000




p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n = 25, x = 0 0.2774 0.0718 0.0172 0.0038 0.0008 0.0001 0.0000 0.0000 0.0000 0.0000 1 0.6424 0.2712 0.0931 0.0274 0.0070 0.0016 0.0003 0.0001 0.0000 0.0000

2 0.8729 0.5371 0.2537 0.0982 0.0321 0.0090 0.0021 0.0004 0.0001 0.0000

3 0.9659 0.7636 0.4711 0.2340 0.0962 0.0332 0.0097 0.0024 0.0005 0.0001

4 0.9928 0.9020 0.6821 0.4207 0.2137 0.0905 0.0320 0.0095 0.0023 0.0005

5 0.9988 0.9666 0.8385 0.6167 0.3783 0.1935 0.0826 0.0294 0.0086 0.0020

6 0.9998 0.9905 0.9305 0.7800 0.5611 0.3407 0.1734 0.0736 0.0258 0.0073

7 1.0000 0.9977 0.9745 0.8909 0.7265 0.5118 0.3061 0.1536 0.0639 0.0216

8 1.0000 0.9995 0.9920 0.9532 0.8506 0.6769 0.4668 0.2735 0.1340 0.0539

9 1.0000 0.9999 0.9979 0.9827 0.9287 0.8106 0.6303 0.4246 0.2424 0.1148

10 1.0000 1.0000 0.9995 0.9944 0.9703 0.9022 0.7712 0.5858 0.3843 0.2122

11 1.0000 1.0000 0.9999 0.9985 0.9893 0.9558 0.8746 0.7323 0.5426 0.3450

12 1.0000 1.0000 1.0000 0.9996 0.9966 0.9825 0.9396 0.8462 0.6937 0.5000

13 1.0000 1.0000 1.0000 0.9999 0.9991 0.9940 0.9745 0.9222 0.8173 0.6550

14 1.0000 1.0000 1.0000 1.0000 0.9998 0.9982 0.9907 0.9656 0.9040 0.7878

15 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9971 0.9868 0.9560 0.8852

16 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992 0.9957 0.9826 0.9461

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9988 0.9942 0.9784

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9927

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9980

20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995

21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

n = 30, x = 0 0.2146 0.0424 0.0076 0.0012 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.5535 0.1837 0.0480 0.0105 0.0020 0.0003 0.0000 0.0000 0.0000 0.0000

2 0.8122 0.4114 0.1514 0.0442 0.0106 0.0021 0.0003 0.0000 0.0000 0.0000

3 0.9392 0.6474 0.3217 0.1227 0.0374 0.0093 0.0019 0.0003 0.0000 0.0000

4 0.9844 0.8245 0.5245 0.2552 0.0979 0.0302 0.0075 0.0015 0.0002 0.0000

5 0.9967 0.9268 0.7106 0.4275 0.2026 0.0766 0.0233 0.0057 0.0011 0.0002

6 0.9994 0.9742 0.8474 0.6070 0.3481 0.1595 0.0586 0.0172 0.0040 0.0007

7 0.9999 0.9922 0.9302 0.7608 0.5143 0.2814 0.1238 0.0435 0.0121 0.0026

8 1.0000 0.9980 0.9722 0.8713 0.6736 0.4315 0.2247 0.0940 0.0312 0.0081

9 1.0000 0.9995 0.9903 0.9389 0.8034 0.5888 0.3575 0.1763 0.0694 0.0214

10 1.0000 0.9999 0.9971 0.9744 0.8943 0.7304 0.5078 0.2915 0.1350 0.0494

11 1.0000 1.0000 0.9992 0.9905 0.9493 0.8407 0.6548 0.4311 0.2327 0.1002

12 1.0000 1.0000 0.9998 0.9969 0.9784 0.9155 0.7802 0.5785 0.3592 0.1808

13 1.0000 1.0000 1.0000 0.9991 0.9918 0.9599 0.8737 0.7145 0.5025 0.2923

14 1.0000 1.0000 1.0000 0.9998 0.9973 0.9831 0.9348 0.8246 0.6448 0.4278

15 1.0000 1.0000 1.0000 0.9999 0.9992 0.9936 0.9699 0.9029 0.7691 0.5722

16 1.0000 1.0000 1.0000 1.0000 0.9998 0.9979 0.9876 0.9519 0.8644 0.7077

17 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9955 0.9788 0.9286 0.8192

18 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9986 0.9917 0.9666 0.8998

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9971 0.9862 0.9506

20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9950 0.9786

21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9984 0.9919

22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9974

23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993

24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998

25 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000




p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n = 40, x = 0 0.1285 0.0148 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1 0.3991 0.0805 0.0121 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000

2 0.6767 0.2228 0.0486 0.0079 0.0010 0.0001 0.0000 0.0000 0.0000 0.0000

3 0.8619 0.4231 0.1302 0.0285 0.0047 0.0006 0.0001 0.0000 0.0000 0.0000

4 0.9520 0.6290 0.2633 0.0759 0.0160 0.0026 0.0003 0.0000 0.0000 0.0000

5 0.9861 0.7937 0.4325 0.1613 0.0433 0.0086 0.0013 0.0001 0.0000 0.0000

6 0.9966 0.9005 0.6067 0.2859 0.0962 0.0238 0.0044 0.0006 0.0001 0.0000

7 0.9993 0.9581 0.7559 0.4371 0.1820 0.0553 0.0124 0.0021 0.0002 0.0000

8 0.9999 0.9845 0.8646 0.5931 0.2998 0.1110 0.0303 0.0061 0.0009 0.0001

9 1.0000 0.9949 0.9328 0.7318 0.4395 0.1959 0.0644 0.0156 0.0027 0.0003

10 1.0000 0.9985 0.9701 0.8392 0.5839 0.3087 0.1215 0.0352 0.0074 0.0011

11 1.0000 0.9996 0.9880 0.9125 0.7151 0.4406 0.2053 0.0709 0.0179 0.0032

12 1.0000 0.9999 0.9957 0.9568 0.8209 0.5772 0.3143 0.1285 0.0386 0.0083

13 1.0000 1.0000 0.9986 0.9806 0.8968 0.7032 0.4408 0.2112 0.0751 0.0192

14 1.0000 1.0000 0.9996 0.9921 0.9456 0.8074 0.5721 0.3174 0.1326 0.0403

15 1.0000 1.0000 0.9999 0.9971 0.9738 0.8849 0.6946 0.4402 0.2142 0.0769

16 1.0000 1.0000 1.0000 0.9990 0.9884 0.9367 0.7978 0.5681 0.3185 0.1341

17 1.0000 1.0000 1.0000 0.9997 0.9953 0.9680 0.8761 0.6885 0.4391 0.2148

18 1.0000 1.0000 1.0000 0.9999 0.9983 0.9852 0.9301 0.7911 0.5651 0.3179

19 1.0000 1.0000 1.0000 1.0000 0.9994 0.9937 0.9637 0.8702 0.6844 0.4373

20 1.0000 1.0000 1.0000 1.0000 0.9998 0.9976 0.9827 0.9256 0.7870 0.5627

21 1.0000 1.0000 1.0000 1.0000 1.0000 0.9991 0.9925 0.9608 0.8669 0.6821

22 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9970 0.9811 0.9233 0.7852

23 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 0.9917 0.9595 0.8659

24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9966 0.9804 0.9231

25 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9988 0.9914 0.9597

26 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9966 0.9808

27 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9988 0.9917

28 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9968

29 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989

30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997

31 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000




p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

n = 50, x = 0 0.0769 0.0052 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1 0.2794 0.0338 0.0029 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2 0.5405 0.1117 0.0142 0.0013 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000

3 0.7604 0.2503 0.0460 0.0057 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000

4 0.8964 0.4312 0.1121 0.0185 0.0021 0.0002 0.0000 0.0000 0.0000 0.0000

5 0.9622 0.6161 0.2194 0.0480 0.0070 0.0007 0.0001 0.0000 0.0000 0.0000

6 0.9882 0.7702 0.3613 0.1034 0.0194 0.0025 0.0002 0.0000 0.0000 0.0000

7 0.9968 0.8779 0.5188 0.1904 0.0453 0.0073 0.0008 0.0001 0.0000 0.0000

8 0.9992 0.9421 0.6681 0.3073 0.0916 0.0183 0.0025 0.0002 0.0000 0.0000

9 0.9998 0.9755 0.7911 0.4437 0.1637 0.0402 0.0067 0.0008 0.0001 0.0000

10 1.0000 0.9906 0.8801 0.5836 0.2622 0.0789 0.0160 0.0022 0.0002 0.0000

11 1.0000 0.9968 0.9372 0.7107 0.3816 0.1390 0.0342 0.0057 0.0006 0.0000

12 1.0000 0.9990 0.9699 0.8139 0.5110 0.2229 0.0661 0.0133 0.0018 0.0002

13 1.0000 0.9997 0.9868 0.8894 0.6370 0.3279 0.1163 0.0280 0.0045 0.0005

14 1.0000 0.9999 0.9947 0.9393 0.7481 0.4468 0.1878 0.0540 0.0104 0.0013

15 1.0000 1.0000 0.9981 0.9692 0.8369 0.5692 0.2801 0.0955 0.0220 0.0033

16 1.0000 1.0000 0.9993 0.9856 0.9017 0.6839 0.3889 0.1561 0.0427 0.0077

17 1.0000 1.0000 0.9998 0.9937 0.9449 0.7822 0.5060 0.2369 0.0765 0.0164

18 1.0000 1.0000 0.9999 0.9975 0.9713 0.8594 0.6216 0.3356 0.1273 0.0325

19 1.0000 1.0000 1.0000 0.9991 0.9861 0.9152 0.7264 0.4465 0.1974 0.0595

20 1.0000 1.0000 1.0000 0.9997 0.9937 0.9522 0.8139 0.5610 0.2862 0.1013

21 1.0000 1.0000 1.0000 0.9999 0.9974 0.9749 0.8813 0.6701 0.3900 0.1611

22 1.0000 1.0000 1.0000 1.0000 0.9990 0.9877 0.9290 0.7660 0.5019 0.2399

23 1.0000 1.0000 1.0000 1.0000 0.9996 0.9944 0.9604 0.8438 0.6134 0.3359

24 1.0000 1.0000 1.0000 1.0000 0.9999 0.9976 0.9793 0.9022 0.7160 0.4439

25 1.0000 1.0000 1.0000 1.0000 1.0000 0.9991 0.9900 0.9427 0.8034 0.5561

26 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9955 0.9686 0.8721 0.6641

27 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9981 0.9840 0.9220 0.7601

28 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9993 0.9924 0.9556 0.8389

29 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9966 0.9765 0.8987

30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9986 0.9884 0.9405

31 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9947 0.9675

32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9978 0.9836

33 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9923

34 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9967

35 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9987

36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995

37 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998

38 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000




Percentage Points Of The Normal Distribution

The values z in the table are those which a random variable Z N(0, 1) exceeds with probability p;

that is, P(Z > z) = 1 (z) = p.

p z p z

0.5000 0.0000 0.0500 1.6449

0.4000 0.2533 0.0250 1.9600

0.3000 0.5244 0.0100 2.3263

0.2000 0.8416 0.0050 2.5758

0.1500 1.0364 0.0010 3.0902

0.1000 1.2816 0.0005 3.2905




Poisson Cumulative Distribution Function

The tabulated value is P(X x), where X has a Poisson distribution with parameter .

= 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

x = 0 0.6065 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067

1 0.9098 0.7358 0.5578 0.4060 0.2873 0.1991 0.1359 0.0916 0.0611 0.0404

2 0.9856 0.9197 0.8088 0.6767 0.5438 0.4232 0.3208 0.2381 0.1736 0.1247

3 0.9982 0.9810 0.9344 0.8571 0.7576 0.6472 0.5366 0.4335 0.3423 0.2650

4 0.9998 0.9963 0.9814 0.9473 0.8912 0.8153 0.7254 0.6288 0.5321 0.4405

5 1.0000 0.9994 0.9955 0.9834 0.9580 0.9161 0.8576 0.7851 0.7029 0.6160

6 1.0000 0.9999 0.9991 0.9955 0.9858 0.9665 0.9347 0.8893 0.8311 0.7622

7 1.0000 1.0000 0.9998 0.9989 0.9958 0.9881 0.9733 0.9489 0.9134 0.8666

8 1.0000 1.0000 1.0000 0.9998 0.9989 0.9962 0.9901 0.9786 0.9597 0.9319

9 1.0000 1.0000 1.0000 1.0000 0.9997 0.9989 0.9967 0.9919 0.9829 0.9682

10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 0.9972 0.9933 0.9863

11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9976 0.9945

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992 0.9980

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

= 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

x = 0 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001 0.0000

1 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008 0.0005

2 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042 0.0028

3 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149 0.0103

4 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403 0.0293

5 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885 0.0671

6 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.1649 0.1301

7 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.2687 0.2202

8 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918 0.3328

9 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218 0.4579

10 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453 0.5830

11 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520 0.6968

12 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364 0.7916

13 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981 0.8645

14 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400 0.9165

15 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.9665 0.9513

16 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.9823 0.9730

17 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.9911 0.9857

18 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957 0.9928

19 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.9980 0.9965

20 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9991 0.9984

21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9993

22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997




Percentage Points of the 2 Distribution

The values in the table are those which a random variable with the 2 distribution on degrees of

freedom exceeds with the probability shown.

0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005

1 0.000 0.000 0.001 0.004 0.016 2.705 3.841 5.024 6.635 7.879

2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597

3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838

4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860

5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.086 16.750

6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548

7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278

8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955

9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589

10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188

11 2.603 3.053 3.816 4.575 5.580 17.275 19.675 21.920 24.725 26.757

12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300

13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819

14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319

15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801

16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267

17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718

18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156

19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582

20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997

21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401

22 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796

23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181

24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.558

25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928

26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290

27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645

28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993

29 13.121 14.256 16.047 17.708 19.768 39.088 42.557 45.722 49.588 52.336

30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672




Critical Values for Correlation Coefficients

These tables concern tests of the hypothesis that a population correlation coefficient is 0. The values

in the tables are the minimum values which need to be reached by a sample correlation coefficient in order to be significant at the level shown, on a one-tailed test.

Product Moment Coefficient Sample

Size, n

Spearman’s Coefficient

0.10 0.05 Level

0.025 0.01 0.005 0.05

Level

0.025 0.01

0.8000 0.9000 0.9500 0.9800 0.9900 4 1.0000 - -

0.6870 0.8054 0.8783 0.9343 0.9587 5

0.9000 1.0000 1.0000

0.6084 0.7293 0.8114 0.8822 0.9172 6 0.8286 0.8857 0.9429

0.5509 0.6694 0.7545 0.8329 0.8745 7 0.7143 0.7857 0.8929

0.5067 0.6215 0.7067 0.7887 0.8343 8 0.6429 0.7381 0.8333

0.4716 0.5822 0.6664 0.7498 0.7977 9 0.6000 0.7000 0.7833

0.4428 0.5494 0.6319 0.7155 0.7646 10

0.5636 0.6485 0.7455

0.4187 0.5214 0.6021 0.6851 0.7348 11 0.5364 0.6182 0.7091

0.3981 0.4973 0.5760 0.6581 0.7079 12 0.5035 0.5874 0.6783

0.3802 0.4762 0.5529 0.6339 0.6835 13 0.4835 0.5604 0.6484

0.3646 0.4575 0.5324 0.6120 0.6614 14 0.4637 0.5385 0.6264

0.3507 0.4409 0.5140 0.5923 0.6411 15

0.4464 0.5214 0.6036

0.3383 0.4259 0.4973 0.5742 0.6226 16 0.4294 0.5029 0.5824

0.3271 0.4124 0.4821 0.5577 0.6055 17 0.4142 0.4877 0.5662

0.3170 0.4000 0.4683 0.5425 0.5897 18 0.4014 0.4716 0.5501

0.3077 0.3887 0.4555 0.5285 0.5751 19 0.3912 0.4596 0.5351

0.2992 0.3783 0.4438 0.5155 0.5614 20

0.3805 0.4466 0.5218

0.2914 0.3687 0.4329 0.5034 0.5487 21 0.3701 0.4364 0.5091

0.2841 0.3598 0.4227 0.4921 0.5368 22 0.3608 0.4252 0.4975

0.2774 0.3515 0.4133 0.4815 0.5256 23 0.3528 0.4160 0.4862

0.2711 0.3438 0.4044 0.4716 0.5151 24 0.3443 0.4070 0.4757

0.2653 0.3365 0.3961 0.4622 0.5052 25

0.3369 0.3977 0.4662

0.2598 0.3297 0.3882 0.4534 0.4958 26 0.3306 0.3901 0.4571

0.2546 0.3233 0.3809 0.4451 0.4869 27 0.3242 0.3828 0.4487

0.2497 0.3172 0.3739 0.4372 0.4785 28 0.3180 0.3755 0.4401

0.2451 0.3115 0.3673 0.4297 0.4705 29 0.3118 0.3685 0.4325

0.2407 0.3061 0.3610 0.4226 0.4629 30

0.3063 0.3624 0.4251

0.2070 0.2638 0.3120 0.3665 0.4026 40 0.2640 0.3128 0.3681

0.1843 0.2353 0.2787 0.3281 0.3610 50 0.2353 0.2791 0.3293

0.1678 0.2144 0.2542 0.2997 0.3301 60 0.2144 0.2545 0.3005

0.1550 0.1982 0.2352 0.2776 0.3060 70 0.1982 0.2354 0.2782

0.1448 0.1852 0.2199 0.2597 0.2864 80

0.1852 0.2201 0.2602

0.1364 0.1745 0.2072 0.2449 0.2702 90 0.1745 0.2074 0.2453

0.1292 0.1654 0.1966 0.2324 0.2565 100 0.1654 0.1967 0.2327




Random Numbers

86 13 84 10 07 30 39 05 97 96 88 07 37 26 04 89 13 48 19 20

60 78 48 12 99 47 09 46 91 33 17 21 03 94 79 00 08 50 40 16

78 48 06 37 82 26 01 06 64 65 94 41 17 26 74 66 61 93 24 97

80 56 90 79 66 94 18 40 97 79 93 20 41 51 25 04 20 71 76 04

99 09 39 25 66 31 70 56 30 15 52 17 87 55 31 11 10 68 98 23

56 32 32 72 91 65 97 36 56 61 12 79 95 17 57 16 53 58 96 36

66 02 49 93 97 44 99 15 56 86 80 57 11 78 40 23 58 40 86 14

31 77 53 94 05 93 56 14 71 23 60 46 05 33 23 72 93 10 81 23

98 79 72 43 14 76 54 77 66 29 84 09 88 56 75 86 41 67 04 42

50 97 92 15 10 01 57 01 87 33 73 17 70 18 40 21 24 20 66 62

90 51 94 50 12 48 88 95 09 34 09 30 22 27 25 56 40 76 01 59

31 99 52 24 13 43 27 88 11 39 41 65 00 84 13 06 31 79 74 97

22 96 23 34 46 12 67 11 48 06 99 24 14 83 78 37 65 73 39 47

06 84 55 41 27 06 74 59 14 29 20 14 45 75 31 16 05 41 22 96

08 64 89 30 25 25 71 35 33 31 04 56 12 67 03 74 07 16 49 32

86 87 62 43 15 11 76 49 79 13 78 80 93 89 09 57 07 14 40 74

94 44 97 13 77 04 35 02 12 76 60 91 93 40 81 06 85 85 72 84

63 25 55 14 66 47 99 90 02 90 83 43 16 01 19 69 11 78 87 16

11 22 83 98 15 21 18 57 53 42 91 91 26 52 89 13 86 00 47 61

01 70 10 83 94 71 13 67 11 12 36 54 53 32 90 43 79 01 95 15




Percentage Points of Student’s t Distribution

The values in the table are those which a random variable with Student’s t distribution on degrees

of freedom exceeds with the probability shown.

0.10 0.05 0.025 0.01 0.005

1 3.078 6.314 12.706 31.821 63.657

2 1.886 2.920 4.303 6.965 9.925

3 1.638 2.353 3.182 4.541 5.841

4 1.533 2.132 2.776 3.747 4.604

5 1.476 2.015 2.571 3.365 4.032

6 1.440 1.943 2.447 3.143 3.707

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

11 1.363 1.796 2.201 2.718 3.106

12 1.356 1.782 2.179 2.681 3.055

13 1.350 1.771 2.160 2.650 3.012

14 1.345 1.761 2.145 2.624 2.977

15 1.341 1.753 2.131 2.602 2.947

16 1.337 1.746 2.120 2.583 2.921

17 1.333 1.740 2.110 2.567 2.898

18 1.330 1.734 2.101 2.552 2.878

19 1.328 1.729 2.093 2.539 2.861

20 1.325 1.725 2.086 2.528 2.845

21 1.323 1.721 2.080 2.518 2.831

22 1.321 1.717 2.074 2.508 2.819

23 1.319 1.714 2.069 2.500 2.807

24 1.318 1.711 2.064 2.492 2.797

25 1.316 1.708 2.060 2.485 2.787

26 1.315 1.706 2.056 2.479 2.779

27 1.314 1.703 2.052 2.473 2.771

28 1.313 1.701 2.048 2.467 2.763

29 1.311 1.699 2.045 2.462 2.756

30 1.310 1.697 2.042 2.457 2.750

32 1.309 1.694 2.037 2.449 2.738

34 1.307 1.691 2.032 2.441 2.728

36 1.306 1.688 2.028 2.435 2.719

38 1.304 1.686 2.024 2.429 2.712

40 1.303 1.684 2.021 2.423 2.704

45 1.301 1.679 2.014 2.412 2.690

50 1.299 1.676 2.009 2.403 2.678

55 1.297 1.673 2.004 2.396 2.668

60 1.296 1.671 2.000 2.390 2.660

70 1.294 1.667 1.994 2.381 2.648

80 1.292 1.664 1.990 2.374 2.639

90 1.291 1.662 1.987 2.369 2.632

100 1.290 1.660 1.984 2.364 2.626

110 1.289 1.659 1.982 2.361 2.621

120 1.289 1.658 1.980 2.358 2.617




Percentage Points of the F Distribution

The values in the table are those which a random variable with the F distribution on 1 and 2

degrees of freedom exceeds with probability 0.05 or 0.01.

Probability

1

2

1 2 3 4 5 6 8 10 12 24

0.05

1 161.4 199.5 215.7 224.6 230.2 234.0 238.9 241.9 243.9 249.1 254.3

2 18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.40 19.41 19.46 19.50

3 10.13 9.55 9.28 9.12 9.01 8.94 8.85 8.79 8.74 8.64 8.53

4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.96 5.91 5.77 5.63

5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.74 4.68 4.53 4.37

6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.06 4.00 3.84 3.67

7 5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.64 3.57 3.41 3.23

8 5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.35 3.28 3.12 2.93

9 5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.14 3.07 2.90 2.71

10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.98 2.91 2.74 2.54

11 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.85 2.79 2.61 2.40

12 4.75 3.89 3.49 3.26 3.11 3.00 2.85 2.75 2.69 2.51 2.30

14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.60 2.53 2.35 2.13

16 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.49 2.42 2.24 2.01

18 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.41 2.34 2.15 1.92

20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.35 2.28 2.08 1.84

25 4.24 3.39 2.99 2.76 2.60 2.49 2.34 2.24 2.16 1.96 1.71

30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.16 2.09 1.89 1.62

40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.08 2.00 1.79 1.51

60 4.00 3.15 2.76 2.53 2.37 2.25 2.10 1.99 1.92 1.70 1.39

120 3.92 3.07 2.68 2.45 2.29 2.18 2.02 1.91 1.83 1.61 1.25

3.84 3.00 2.60 2.37 2.21 2.10 1.94 1.83 1.75 1.52 1.00

0.01

1 4052. 5000. 5403. 5625. 5764. 5859. 5982. 6056. 6106. 6235. 6366.

2 98.50 99.00 99.17 99.25 99.30 99.33 99.37 99.40 99.42 99.46 99.50

3 34.12 30.82 29.46 28.71 28.24 27.91 27.49 27.23 27.05 26.60 26.13

4 21.20 18.00 16.69 15.98 15.52 15.21 14.80 14.55 14.37 13.93 13.45

5 16.26 13.27 12.06 11.39 10.97 10.67 10.29 10.05 9.89 9.47 9.02

6 13.70 10.90 9.78 9.15 8.75 8.47 8.10 7.87 7.72 7.31 6.88

7 12.20 9.55 8.45 7.85 7.46 7.19 6.84 6.62 6.47 6.07 5.65

8 11.30 8.65 7.59 7.01 6.63 6.37 6.03 5.81 5.67 5.28 4.86

9 10.60 8.02 6.99 6.42 6.06 5.80 5.47 5.26 5.11 4.73 4.31

10 10.00 7.56 6.55 5.99 5.64 5.39 5.06 4.85 4.17 4.33 3.91

11 9.65 7.21 6.22 5.67 5.32 5.07 4.74 4.54 4.40 4.02 3.60

12 9.33 6.93 5.95 5.41 5.06 4.82 4.50 4.30 4.16 3.78 3.36

14 8.86 6.51 5.56 5.04 4.70 4.46 4.14 3.94 3.80 3.43 3.00

16 8.53 6.23 5.29 4.77 4.44 4.20 3.89 3.69 3.55 3.18 2.75

18 8.29 6.01 5.09 4.58 4.25 4.01 3.71 3.51 3.37 3.00 2.57

20 8.10 5.85 4.94 4.43 4.10 3.87 3.56 3.37 3.23 2.86 2.42

25 7.77 5.57 4.68 4.18 3.86 3.63 3.32 3.13 2.99 2.62 2.17

30 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.98 2.84 2.47 2.01

40 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.80 2.66 2.29 1.80

60 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.63 2.50 2.12 1.60

120 6.85 4.79 3.95 3.48 3.17 2.96 2.66 2.47 2.34 1.95 1.38

6.63 4.61 3.78 3.32 3.02 2.80 2.51 2.32 2.18 1.79 1.00

If an upper percentage point of the F distribution on 1 and 2 degrees of freedom is f , then the corresponding

lower percentage point of the F distribution on 2 and 1 degrees of freedom is 1/ f.

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