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This publication may only be reproduced in accordance with London Qualifications Limited copyright policy. Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited Paper Reference(s) 6667 Edexcel GCE Further Pure Mathematics FP1 Advanced Level Specimen Paper Time: 1 hour 30 minutes Materials required for examination Items included with question papers Answer Book (AB16) Nil Graph Paper (ASG2) Mathematical Formulae (Lilac) Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI-89, TI-92, Casio CFX-9970G, Hewlett Packard HP 48G. Instructions to Candidates In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper reference (6667), your surname, initials and signature. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. Full marks may be obtained for answers to ALL questions. This paper has eight questions. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.

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6674/01This publication may only be reproduced in accordance with
London Qualifications Limited copyright policy.

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI-89, TI-92, Casio CFX-9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper

reference (6667), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

2

1 1

xx

(a) Show that the root of the equation f(x) = 0 lies in the interval 43 .

(2)

(b) Taking 3.6 as your starting value, apply the Newton-Raphson procedure once to f(x) to

obtain a second approximation to . Give your answer to 4 decimal places.

(5)

3. Find the set of values of x for which

. 2

1

3

4. ,552f 23 ppxxxx .

The equation f (x) = 0 has (1 – 2i) as a root.

Solve the equation and determine the value of p.

(7)

5. (a) Obtain the general solution of the differential equation

.1.0 d

d tS

t

S

(6)

(b) The differential equation in part (a) is used to model the assets, £S million, of a bank

t years after it was set up. Given that the initial assets of the bank were £200 million, use

your answer to part (a) to estimate, to the nearest £ million, the assets of the bank

10 years after it was set up.

(4)

44 ,2cos22

(2)

(b) Find the polar coordinates of the points where tangents to C are parallel to the initial line.

(6)

(c) Find the area of the region bounded by C.

(4)

7. Given that z = 3 + 4i and zw = 14 + 2i, find

(a) w in the form p + iq where p and q are real,

(4)

(b) the modulus of z and the argument of z in radians to 2 decimal places

(4)

(c) the values of the real constants m and n such that

i2010 nzwmz .

(i) t

(5)

(b) Use you answers to part (a) to show that the substitution tx e transforms the

differential equation

.2 d

d 2

(6)

END

This publication may only be reproduced in accordance with London Qualifications Limited copyright policy.

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI 89, TI 92, Casio CFX-9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper

reference (6668), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

S

1. The displacement x of a particle from a fixed point O at time t is given by .sinh tx

At time T the displacement . 3

4 x

(3)

(a)

, 1

1

d

d

2xx

y

3.

Figure 1 shows the curve C with equation .cosh xy The tangent at P makes an angle

with the x-axis and the arc length from A(0, 1) to P(x, y) is s.

(a) Show that .sinh xs

(3)

(a) By considering the gradient of the tangent at P show that the intrinsic equation of C is

s = tan .

(2)

.

2

1

(b) Hence obtain 3I , giving your answers in terms of .

(4)

(7)

The curve C has equation .422 xy

(b) Use your answer to part (a) to find the area of the finite region bounded by C, the

positive x-axis, the positive y-axis and the line x = 2, giving your answer in the

form qp ln where p and q are constants to be found.

(4)

6.

The parametric equations of the curve C shown in Fig. 2 are

.20,cos1,sin ttayttax

(6)

(5)

y

xO

U

7. (a) Using the definitions of xsinh and xcosh in terms of exponential functions, express

xtanh in terms of xe and .e x

(1)

(2)

(d) Hence obtain xd

d (artanh x) and use integration by parts to show that

artanh x dx = x artanh x + 21ln 2

1 x + constant.

2

2

2

b

y

a

x

(a) Show that an equation of the normal to C at tan,sec baP is

.tansin 22 baaxby

(6)

The normal at P cuts the coordinate axes at A and B. The mid-point of AB is M.

(b) Find, in cartesian form, an equation of the locus of M as varies.

(7)

END

This publication may only be reproduced in accordance with London Qualifications Limited copyright policy.

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP3), the paper

reference (6669), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

NM

with a step length of 0.1 to estimate the values of y

at 1.0x and ,2.0x giving your answers to 2 significant figures.

(6)

1

i

z w

maps the circle 1z in the z-plane to the line i1 ww in the w-plane.

(4)

The region 1z in the z-plane is mapped to the region R in the w-plane.

(b) Shade the region R on an Argand diagram.

(2)

x

y

Find a series solution of the differential equation in ascending powers of (x – 1) up to and

including the term in (x 1) 3 .

(7)

5.

(4)

(6)

11

6. The points A, B, C, and D have position vectors

kjdkjicjibkia 4,23,3,2

respectively.

(a) Find ACAB and hence find the area of triangle ABC.

(7)

(2)

(c) Find the perpendicular distance of D from the plane containing A, B and C.

(3)

(8)

(4)

NO

p z

z p

(2)

find the values of the constants A, B and C.

(7)

The region R bounded by the curve with equation , 22

,cos 2

rotated through 2 about the x-axis.

(c) Find the volume of the solid generated.

(6)

END

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

13

Question

1

4f and 3f are of opposite sign and so xf has root in (3, 4) M1 A1 (2)

(b) 6.30 x

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

NQ

Question

den:2x = veveve

den:32 x = veveve

den:3 x = veveve M1 A1

Set of values 3: 2: 3 and 2 xxxxxx A1 (7)

(7 marks)

4. If 1 – 2i is a root, then so is 1 + 2i B1

i21i21 xx are factors of f(x) M1

so 522 xx is a factor of f (x) A1

1252f 2 xxxx M1 A1 ft

Third root is 2 1 and p = 12 A1 A1 (7)

(7 marks)

(b) 0at 200 tS

300 i.e. 100200 CC

At St ,10 100100e300

(10 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

15

Question

(b) Tangent parallel to initial line when sinry is stationary

Consider therefore

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

NS

Question

w i43i43

arg z = – 3

(c) Equating real and imaginary parts M1

,10143 nm A1

2024 nm A1

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

17

Question

021 mm

Particular integral

t t

2 12 eee

(14 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

NU

Question

(5 marks)

(7 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

19

Question

(b) Gradient of tangent

(c)

2sec

d

d

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OM

Question

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

21

Question

ttx cosh24sinh44 2

tt d 12cosh2

(7)

2

0

2

0

2

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OO

Question

(6)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

23

Question

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OQ

Question

a

a by

(b) M: A normal cuts

tanat 0 22

tan sin

(15 marks)

qìêå=lîÉêOR

1

A1 (4)

B1 (line)

B1 (shading)

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

OS

Question

1

Assume true for n = k. Then

(7 marks)

1 d

32 1

(7 marks)

27

Question

(b) from obtainedr Eigenvecto 2

e.g.

(10 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

OU

Question

Number Scheme Marks

6. (a) 1 3, ,1 AB ; 1 3, ,1AC . M1 A1

ACAB

131

131

Area of ABC 2

B, and C is

(12 marks)

29

Question

24 ,13 ,17

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

PM

Question

p z

z p

(c) V

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI-89, TI-92, Casio CFX-9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper

reference (6667), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

2

1 1

xx

(a) Show that the root of the equation f(x) = 0 lies in the interval 43 .

(2)

(b) Taking 3.6 as your starting value, apply the Newton-Raphson procedure once to f(x) to

obtain a second approximation to . Give your answer to 4 decimal places.

(5)

3. Find the set of values of x for which

. 2

1

3

4. ,552f 23 ppxxxx .

The equation f (x) = 0 has (1 – 2i) as a root.

Solve the equation and determine the value of p.

(7)

5. (a) Obtain the general solution of the differential equation

.1.0 d

d tS

t

S

(6)

(b) The differential equation in part (a) is used to model the assets, £S million, of a bank

t years after it was set up. Given that the initial assets of the bank were £200 million, use

your answer to part (a) to estimate, to the nearest £ million, the assets of the bank

10 years after it was set up.

(4)

44 ,2cos22

(2)

(b) Find the polar coordinates of the points where tangents to C are parallel to the initial line.

(6)

(c) Find the area of the region bounded by C.

(4)

7. Given that z = 3 + 4i and zw = 14 + 2i, find

(a) w in the form p + iq where p and q are real,

(4)

(b) the modulus of z and the argument of z in radians to 2 decimal places

(4)

(c) the values of the real constants m and n such that

i2010 nzwmz .

(i) t

(5)

(b) Use you answers to part (a) to show that the substitution tx e transforms the

differential equation

.2 d

d 2

(6)

END

This publication may only be reproduced in accordance with London Qualifications Limited copyright policy.

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI 89, TI 92, Casio CFX-9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper

reference (6668), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

S

1. The displacement x of a particle from a fixed point O at time t is given by .sinh tx

At time T the displacement . 3

4 x

(3)

(a)

, 1

1

d

d

2xx

y

3.

Figure 1 shows the curve C with equation .cosh xy The tangent at P makes an angle

with the x-axis and the arc length from A(0, 1) to P(x, y) is s.

(a) Show that .sinh xs

(3)

(a) By considering the gradient of the tangent at P show that the intrinsic equation of C is

s = tan .

(2)

.

2

1

(b) Hence obtain 3I , giving your answers in terms of .

(4)

(7)

The curve C has equation .422 xy

(b) Use your answer to part (a) to find the area of the finite region bounded by C, the

positive x-axis, the positive y-axis and the line x = 2, giving your answer in the

form qp ln where p and q are constants to be found.

(4)

6.

The parametric equations of the curve C shown in Fig. 2 are

.20,cos1,sin ttayttax

(6)

(5)

y

xO

U

7. (a) Using the definitions of xsinh and xcosh in terms of exponential functions, express

xtanh in terms of xe and .e x

(1)

(2)

(d) Hence obtain xd

d (artanh x) and use integration by parts to show that

artanh x dx = x artanh x + 21ln 2

1 x + constant.

2

2

2

b

y

a

x

(a) Show that an equation of the normal to C at tan,sec baP is

.tansin 22 baaxby

(6)

The normal at P cuts the coordinate axes at A and B. The mid-point of AB is M.

(b) Find, in cartesian form, an equation of the locus of M as varies.

(7)

END

This publication may only be reproduced in accordance with London Qualifications Limited copyright policy.

Edexcel Foundation is a registered charity. ©2003 London Qualifications Limited

Paper Reference(s)

Materials required for examination Items included with question papers

Answer Book (AB16) Nil

Graph Paper (ASG2) Mathematical Formulae (Lilac)

Candidates may use any calculator EXCEPT those with the facility for symbolic

algebra, differentiation and/or integration. Thus candidates may NOT use

calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G,

Hewlett Packard HP 48G.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your

centre number, candidate number, the unit title (Further Pure Mathematics FP3), the paper

reference (6669), your surname, initials and signature.

When a calculator is used, the answer should be given to an appropriate degree of

accuracy.

Full marks may be obtained for answers to ALL questions.

This paper has eight questions.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

NM

with a step length of 0.1 to estimate the values of y

at 1.0x and ,2.0x giving your answers to 2 significant figures.

(6)

1

i

z w

maps the circle 1z in the z-plane to the line i1 ww in the w-plane.

(4)

The region 1z in the z-plane is mapped to the region R in the w-plane.

(b) Shade the region R on an Argand diagram.

(2)

x

y

Find a series solution of the differential equation in ascending powers of (x – 1) up to and

including the term in (x 1) 3 .

(7)

5.

(4)

(6)

11

6. The points A, B, C, and D have position vectors

kjdkjicjibkia 4,23,3,2

respectively.

(a) Find ACAB and hence find the area of triangle ABC.

(7)

(2)

(c) Find the perpendicular distance of D from the plane containing A, B and C.

(3)

(8)

(4)

NO

p z

z p

(2)

find the values of the constants A, B and C.

(7)

The region R bounded by the curve with equation , 22

,cos 2

rotated through 2 about the x-axis.

(c) Find the volume of the solid generated.

(6)

END

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

13

Question

1

4f and 3f are of opposite sign and so xf has root in (3, 4) M1 A1 (2)

(b) 6.30 x

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

NQ

Question

den:2x = veveve

den:32 x = veveve

den:3 x = veveve M1 A1

Set of values 3: 2: 3 and 2 xxxxxx A1 (7)

(7 marks)

4. If 1 – 2i is a root, then so is 1 + 2i B1

i21i21 xx are factors of f(x) M1

so 522 xx is a factor of f (x) A1

1252f 2 xxxx M1 A1 ft

Third root is 2 1 and p = 12 A1 A1 (7)

(7 marks)

(b) 0at 200 tS

300 i.e. 100200 CC

At St ,10 100100e300

(10 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

15

Question

(b) Tangent parallel to initial line when sinry is stationary

Consider therefore

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

NS

Question

w i43i43

arg z = – 3

(c) Equating real and imaginary parts M1

,10143 nm A1

2024 nm A1

EDEXCEL FURTHER PURE MATHEMATICS FP1 (6667) SPECIMEN PAPER MARK SCHEME

17

Question

021 mm

Particular integral

t t

2 12 eee

(14 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

NU

Question

(5 marks)

(7 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

19

Question

(b) Gradient of tangent

(c)

2sec

d

d

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OM

Question

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

21

Question

ttx cosh24sinh44 2

tt d 12cosh2

(7)

2

0

2

0

2

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OO

Question

(6)

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

23

Question

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) SPECIMEN PAPER MARK SCHEME

OQ

Question

a

a by

(b) M: A normal cuts

tanat 0 22

tan sin

(15 marks)

qìêå=lîÉêOR

1

A1 (4)

B1 (line)

B1 (shading)

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

OS

Question

1

Assume true for n = k. Then

(7 marks)

1 d

32 1

(7 marks)

27

Question

(b) from obtainedr Eigenvecto 2

e.g.

(10 marks)

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

OU

Question

Number Scheme Marks

6. (a) 1 3, ,1 AB ; 1 3, ,1AC . M1 A1

ACAB

131

131

Area of ABC 2

B, and C is

(12 marks)

29

Question

24 ,13 ,17

EDEXCEL FURTHER PURE MATHEMATICS FP3 (6669) SPECIMEN PAPER MARK SCHEME

PM

Question

p z

z p

(c) V