FP1 Edexcel

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Edexcel Maths FP1


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Past Paper Pack


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2009-2014

Examiner’s use only

Team Leader’s use only

Surname Initial(s)

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Candidate No.

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Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryFriday 30 January 2009 – AfternoonTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Orange) Nil

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions. Write your answers in the spaces provided in this question paper.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 10 questions in this question paper. The total mark for this paper is 75.There are 28 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

Paper Reference

6 6 6 7 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2009 Edexcel Limited.

Printer’s Log. No.

N34694AW850/R6667/57570 3/4/5/3

*N34694A0128*

physicsandmathstutor.com

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*N34694A0228*

1. f(x) = 2x3 – 8x2 + 7x – 3

Given that x = 3 is a solution of the equation f(x) = 0, solve f(x) = 0 completely.(5)

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January 2009physicsandmathstutor.com

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*N34694A0428*

2. (a) Show, using the formulae for r∑ and r∑ 2 , that

( ) ( )( )6 4 1 2 2 12

1r r n n n

r

n

=∑ + − = + +

(5)

(b) Hence, or otherwise, find the value of ( )6 4 12

11

20

r rr=∑ + − .

(2)

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*N34694A0628*

3. The rectangular hyperbola, H, has parametric equations x = 5t, y = 5t

, t ≠ 0.

(a) Write the cartesian equation of H in the form xy = c2.(1)

Points A and B on the hyperbola have parameters t = 1 and t = 5 respectively.

(b) Find the coordinates of the mid-point of AB.(3)

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*N34694A0828*

4. Prove by induction that, for n∈ +Z ,

11 11 r r

nnr

n

( )+=

+=∑

(5)

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*N34694A01028*

5. f(x) = 3 x + – 20

(a) Show that the equation f(x) = 0 has a root α in the interval [1.1, 1.2].(2)

(b) Find f ′(x).(3)

(c) Using x0 = 1.1 as a first approximation to α , apply the Newton-Raphson procedure once to f(x) to find a second approximation to α , giving your answer to 3 significant

figures.(4)

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√ 18√x


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*N34694A01228*

6. A series of positive integers u1, u2, u3, ... is defined by

u1 = 6 and un+1 = 6un – 5, for n . 1.

Prove by induction that un = 5 × 6 n – 1 + 1, for n . 1.(5)

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*N34694A01428*

7. Given that X =− −⎛

⎝⎜

⎞

⎠⎟

21 1

a , where a is a constant, and a ≠ 2,

(a) find X–1 in terms of a.(3)

Given that X + X–1 = I, where I is the 2×2 identity matrix,

(b) find the value of a.(3)

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*N34694A01628*

8. A parabola has equation y2 = 4ax, a > 0. The point Q (aq2, 2aq) lies on the parabola.

(a) Show that an equation of the tangent to the parabola at Q is

yq = x + aq2.(4)

This tangent meets the y-axis at the point R.

(b) Find an equation of the line l which passes through R and is perpendicular to the tangent at Q.

(3)

(c) Show that l passes through the focus of the parabola.(1)

(d) Find the coordinates of the point where l meets the directrix of the parabola.(2)

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*N34694A02028*

9. Given that z1 = 3 + 2i and z2 = 12 5− i

1z,

(a) find z2 in the form a + ib, where a and b are real.(2)

(b) Show on an Argand diagram the point P representing z1 and the point Q representing z2.

(2)

(c) Given that O is the origin, show that ∠ POQ = π2

.(2)

The circle passing through the points O, P and Q has centre C. Find

(d) the complex number represented by C,(2)

(e) the exact value of the radius of the circle.(2)

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*N34694A02428*

10. A = 3 2 0 0 3 2√

√

⎛

⎝⎜⎜

⎞

⎠⎟⎟ , B =

0 11 0

⎛

⎝⎜

⎞

⎠⎟ , C =

12

12

12

12

√ √

√ √

−⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

(a) Describe fully the transformations described by each of the matrices A, B and C.(4)

It is given that the matrix D = CA, and that the matrix E = DB.

(b) Find D.(2)

(c) Show that E =−⎛

⎝⎜

⎞

⎠⎟

3 33 3

.

(1)

The triangle ORS has vertices at the points with coordinates (0, 0), (–15, 15) and (4, 21). This triangle is transformed onto the triangle OR′S′ by the transformation described by E.

(d) Find the coordinates of the vertices of triangle OR′S′.(4)

(e) Find the area of triangle OR′S′ and deduce the area of triangle ORS.(3)

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√√ √

√

√

√




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryWednesday 17 June 2009 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Orange) Nil


Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions. You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.



Paper Reference

6 6 6 7 0 1



M35146AW850/R6667/57570 3/5/5/3

*M35146A0124*


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*M35146A0224*

1. The complex numbers z1and z2 are given by

z1 = 2 – i and z2 = –8 + 9i

(a) Show z1 and z2 on a single Argand diagram.(1)

Find, showing your working,

(b) the value of z1, (2)

(c) the value of arg z1, giving your answer in radians to 2 decimal places,(2)

(d) in the form a + bi, where a and b are real. (3)

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zz

2

1

June 2009physicsandmathstutor.com

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*M35146A0424*

2. (a) Using the formulae for , and , show that

(r + 1)(r + 3) = n(n + 1)(n + 2)(3n + k),

where k is a constant to be found.(7)

(b) Hence evaluate (r + 1)(r + 3).(2)

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rr

n

=∑

1r

r

n2

1=∑ r

r

n3

1=∑

rr

n

=∑

1

112

rr=∑

21

40


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*M35146A0824*

3. f (x) = (x2 + 4)(x2 + 8x + 25)

(a) Find the four roots of f (x) = 0. (5)

(b) Find the sum of these four roots.(2)

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*M35146A01024*

4. Given that α is the only real root of the equation

x3 – x2 – 6 = 0

(a) show that 2.2 < α < 2.3(2)

(b) Taking 2.2 as a first approximation to α , apply the Newton-Raphson procedure once to f (x) = x3 – x2 – 6 to obtain a second approximation to α , giving your answer to 3 decimal places.

(5)

(c) Use linear interpolation once on the interval [2.2, 2.3] to find another approximation to α , giving your answer to 3 decimal places.

(3)

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*M35146A01424*

5. R = , where a and b are constants and a > 0.

(a) Find R2 in terms of a and b. (3)

Given that R2 represents an enlargement with centre (0, 0) and scale factor 15,

(b) find the value of a and the value of b. (5)

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aa

⎛⎝⎜

2b

⎞⎠⎟


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*M35146A01624*

6. The parabola C has equation y2 = 16x.

(a) Verify that the point P(4t2, 8t) is a general point on C. (1)

(b) Write down the coordinates of the focus S of C. (1)

(c) Show that the normal to C at P has equation

y + tx = 8t + 4t3

(5)

The normal to C at P meets the x-axis at the point N.

(d) Find the area of triangle PSN in terms of t, giving your answer in its simplest form. (4)

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*M35146A02024*

7. A = , where a is a constant.

(a) Find the value of a for which the matrix A is singular. (2)

B =

(b) Find B–1. (3)

The transformation represented by B maps the point P onto the point Q.

Given that Q has coordinates (k – 6, 3k + 12) , where k is a constant,

(c) show that P lies on the line with equation y = x + 3. (3)

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a−

⎛⎝⎜ 1

− ⎞⎠⎟

24

31−

⎛⎝⎜

− ⎞⎠⎟

24


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*M35146A02224*

8. Prove by induction that, for n +] ,

(a) f(n) = 5n + 8n + 3 is divisible by 4,(7)

(b) =(7)

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32

⎛⎝⎜

−−

⎞⎠⎟

21

n 2 12nn

+⎛⎝⎜

−−

⎞⎠⎟

21 2

nn




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

9

Total

Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 1 February 2010 – AfternoonTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) Nil





Paper Reference

6 6 6 7 0 1



N35143AW850/R6667/57570 3/5/5/3

*N35143A0124*


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*N35143A0224*

1. The complex numbers z1 and z2 are given by

z1= 2 + 8i and z2 = 1 – i


(a) zz

1

2 in the form a + bi, where a and b are real,

(3)

(b) the value of zz

1

2,

(2)

(c) the value of arg

zz

1

2, giving your answer in radians to 2 decimal places.

(2)

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*N35143A0424*

2. f ( )x xx

= −3 1122

(a) Write down, to 3 decimal places, the value of f(1.3) and the value of f(1.4).(1)

The equation f(x) = 0 has a root α between 1.3 and 1.4

(b) Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains α.

(3)

(c) Taking 1.4 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α, giving your answer to 3 decimal places.

(5)

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*N35143A0624*

3. A sequence of numbers is defined by

1

1

2,5 4, 1.n n

uu u n+

== − !

Prove by induction that, for n∈ +Z , un = 5n–1 + 1.(4)

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*N35143A0824*

4.

B P

A S x

yy2 =12x

Figure 1

Figure 1 shows a sketch of part of the parabola with equation y2 =12x .

The point P on the parabola has x-coordinate 13

.

The point S is the focus of the parabola.

(a) Write down the coordinates of S.(1)

The points A and B lie on the directrix of the parabola. The point A is on the x-axis and the y-coordinate of B is positive.

Given that ABPS is a trapezium,

(b) calculate the perimeter of ABPS.(5)

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*N35143A01024*

5. A =−+

⎛

⎝⎜

⎞

⎠⎟

aa

52 4 , where a is real.

(a) Find det A in terms of a. (2)

(b) Show that the matrix A is non-singular for all values of a.(3)

Given that a = 0,

(c) find A–1. (3)

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*N35143A01224*

6. Given that 2 and 5 + 2i are roots of the equation

x x cx d3 212 0− + + = , c, d ∈ \,

(a) write down the other complex root of the equation.(1)

(b) Find the value of c and the value of d.(5)

(c) Show the three roots of this equation on a single Argand diagram.(2)

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*N35143A01424*

7. The rectangular hyperbola H has equation xy c= 2 , where c is a constant.

The point P ct ct

, ⎛⎝⎜

⎞⎠⎟ is a general point on H.

(a) Show that the tangent to H at P has equation

t y x ct2 2+ =(4)

The tangents to H at the points A and B meet at the point (15c, – c).

(b) Find, in terms of c, the coordinates of A and B.(5)

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*N35143A01824*

8. (a) Prove by induction that, for any positive integer n,

r n nr

n3

1

2 214

1=∑ = +( )

(5)

(b) Using the formulae for

rr

n

=∑

1 and r

r

n3

1=∑ , show that

( ) ( )( )r r n n nr

n3

1

23 2 14

2 7+ + = + +=∑

(5)

(c) Hence evaluate ( )r rr

3

15

25

3 2+ +=∑

(2)

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*N35143A02224*

9. M = √−√

√ √

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

12

12

12

12

(a) Describe fully the geometrical transformation represented by the matrix M. (2)

The transformation represented by M maps the point A with coordinates (p, q) onto the point B with coordinates (3 2, 4 2).

(b) Find the value of p and the value of q. (4)

(c) Find, in its simplest surd form, the length OA, where O is the origin.(2)

(d) Find M2.(2)

The point B is mapped onto the point C by the transformation represented by M2.

(e) Find the coordinates of C.(2)

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryTuesday 22 June 2010 – AfternoonTime: 1 hour 30 minutes



Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.


Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner.Answers without working may not gain full credit.

Paper Reference

6 6 6 7 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy.©2010 Edexcel Limited.


N35387AW850/R6667/57570 4/3

*N35387A0128*


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*N35387A0228*

1. z = 2 – 3i

(a) Show that 2 5 12i.z = − −(2)


(b) the value of 2 ,z(2)

(c) the value of 2arg( ),z giving your answer in radians to 2 decimal places.(2)

(d) Show z and 2z on a single Argand diagram.(1)

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2. , where a is a real constant.

(a) Given that a 2, find M–1.(3)

(b) Find the values of a for which M is singular.(2)

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⎟⎟⎠⎞

⎜⎜⎝⎛=

aa6

32M


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*N35387A0628*

3.

(a) Show that 0)(f =x has a root between 1.4 and 1.5(2)

(b) Starting with the interval [1.4,1.5], use interval bisection twice to find an interval of width 0.025 that contains .

(3)

(c) Taking 1.45 as a first approximation to , apply the Newton-Raphson procedure once

to 27)(f 3 +−=x

xx to obtain a second approximation to , giving your answer to

3 decimal places.(5)

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3 7f ( ) 2, 0x x xx

= − +


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*N35387A01028*

4.

Given that 2f ( ) ( 3)( ),x x x ax b= + + + where a and b are real constants,

(a) find the value of a and the value of b.(2)

(b) Find the three roots of f ( ) 0.x =(4)

(c) Find the sum of the three roots of f ( ) 0.x =(1)

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3 2f ( ) 44 150x x x x= + + +


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*N35387A01228*

5. The parabola C has equation 2 20 .y x=

(a) Verify that the point 2(5 ,10 )P t t is a general point on C.(1)

The point A on C has parameter 4.t = The line l passes through A and also passes through the focus of C.

(b) Find the gradient of l.(4)

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*N35387A01428*

6. Write down the 2 × 2 matrix that represents

(a) an enlargement with centre (0, 0) and scale factor 8,(1)

(b) a reflection in the x-axis.(1)

Hence, or otherwise,

(c) find the matrix T that represents an enlargement with centre (0, 0) and scale factor 8, followed by a reflection in the x-axis.

(2)

6 14 2

⎛ ⎞= ⎜ ⎟⎝ ⎠

A and 1

,6

kc

⎛ ⎞= ⎜ ⎟−⎝ ⎠B where k and c are constants.

(d) Find AB.(3)

Given that AB represents the same transformation as T,

(e) find the value of k and the value of c.(2)

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*N35387A01828*

7.

(a) Show that )2(4)(f6)1(f kkk −=+ .(3)

(b) Hence, or otherwise, prove by induction that, for ,n +∈ )(f n is divisible by 8.(4)

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nnn 62)(f +=


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*N35387A02028*

8. The rectangular hyperbola H has equation 2 ,xy c= where c is a positive constant.

The point A on H has x-coordinate 3c.

(a) Write down the y-coordinate of A.(1)

(b) Show that an equation of the normal to H at A is

3 27 80y x c= −(5)

The normal to H at A meets H again at the point B.

(c) Find, in terms of c, the coordinates of B.(5)

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*N35387A02428*

9. (a) Prove by induction that

2

1

1 ( 1)(2 1)6

n

rr n n n

=

= + +∑(6)

Using the standard results for 1

n

rr

=∑ and 2

1,

n

rr

=∑

(b) show that2

1

1( 2)( 3) ( ),3

n

rr r n n an b

=

+ + = + +∑

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

(c) Hence show that2

2

1

1( 2)( 3) (7 27 26)3

n

r nr r n n n

= +

+ + = + +∑(3)

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 31 January 2011 – AfternoonTime: 1 hour 30 minutes






Paper Reference

6 6 6 7 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy.©2011 Edexcel Limited.


N35406AW850/R6667/57570 5/5/5/3

*N35406A0132*


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*N35406A0232*

1.

Express in the form a + bi, where a and b are real constants,

(a) z 2 ,(2)

(b) zw

.(3)

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Q1

(Total 5 marks)

z = 5 – 3i, w = 2 + 2i


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*N35406A0332* Turn over

2.A B= ⎛

⎝⎜

⎞⎠⎟ =

− −⎛⎝⎜

⎞⎠⎟

2 05 3

3 15 2

,

(a) Find AB.(3)

Given that

C =−⎛

⎝⎜

⎞⎠⎟

1 00 1

(b) describe fully the geometrical transformation represented by C,(2)

(c) write down C100.(1)

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(Total 6 marks)


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*N35406A0432*

3.

The root of the equation f (x) = 0 lies in the interval 1 6 1 8. , .[ ]. (a) Use linear interpolation once on the interval 1 6 1 8. , .[ ] to find an approximation to .

Give your answer to 3 decimal places.(4)

(b) Differentiate f (x) to find ′f ( ).x(2)

(c) Taking 1.7 as a first approximation to , apply the Newton-Raphson process once to f (x) to obtain a second approximation to . Give your answer to 3 decimal places.

(4)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

f ( ) ,x x x x= − −5 4 6 02 32 !


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*N35406A0832*

4. Given that 2 – 4i is a root of the equation

z p z q2 0+ + = ,

where p and q are real constants,

(a) write down the other root of the equation,(1)

(b) find the value of p and the value of q.(3)

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*N35406A01032*

5. (a) Use the results for r rr

n

r

n

= =∑ ∑

1

2

1, and r

r

n3

1=∑ , to prove that

r r r n n n nr

n

( )( ) ( )( )( )+ + = + + +=

∑ 1 5 14

1 2 71

for all positive integers n.(5)

(b) Hence, or otherwise, find the value of

r r rr

( )( )+ +=∑ 1 5

20

50

(2)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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6. y

x

P

S

C

O

Q

Figure 1

Figure 1 shows a sketch of the parabola C with equation y 2 = 36x. The point S is the focus of C.

(a) Find the coordinates of S.(1)

(b) Write down the equation of the directrix of C.(1)

Figure 1 shows the point P which lies on C, where y > 0, and the point Q which lies on the directrix of C. The line segment QP is parallel to the x-axis.

Given that the distance PS is 25,

(c) write down the distance QP,(1)

(d) find the coordinates of P,(3)

(e) find the area of the trapezium OSPQ.(2)

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7. z = – 24 – 7i

(a) Show z on an Argand diagram.(1)

(b) Calculate arg z, giving your answer in radians to 2 decimal places.(2)

It is given that

w = a + bi, a , b

Given also that w = 4 and arg w = 56π ,

(c) find the values of a and b,(3)

(d) find the value of zw .(3)


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*N35406A02232*

8.A =

−−

⎛⎝⎜

⎞⎠⎟

2 21 3

(a) Find det A.(1)

(b) Find A–1.(2)

The triangle R is transformed to the triangle S by the matrix A. Given that the area of triangle S is 72 square units,

(c) find the area of triangle R.(2)

The triangle S has vertices at the points (0, 4), (8, 16) and (12, 4).

(d) Find the coordinates of the vertices of R.(4)

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*N35406A02632*

9. A sequence of numbers u1 , u2 , u3 , u4 , ... is defined by

un+ =1 u un + =14 2 2,

Prove by induction that, for n +∈! ,

unn= −( )2

34 1

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10. The point P tt

t6 6 0, , ,⎛⎝⎜

⎞⎠⎟

≠ lies on the rectangular hyperbola H with equation xy = 36.

(a) Show that an equation for the tangent to H at P is

yt

xt

= − +1 122

(5)

The tangent to H at the point A and the tangent to H at the point B meet at the point −( )9 12, .

(b) Find the coordinates of A and B.(7)

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryWednesday 22 June 2011 – MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them.


Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75. There are 32 pages in this question paper. Any blank pages are indicated.


Paper Reference

6 6 6 7 0 1



P38168AW850/R6667/57570 5/5/5/3

*P38168A0132*


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2

*P38168A0232*

1.

(a) Show that the equation f ( ) 0x = has a root between 1x = and 2.x =(2)

(b) Starting with the interval [ ]1, 2 , use interval bisection twice to find an interval of width 0.25 which contains .

(3)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

f ( ) 3 3 7xx x= + −


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4

*P38168A0432*

2.

(a) Find the modulus of 1.z(1)

(b) Find, in radians, the argument of 1,z giving your answer to 2 decimal places.(2)

The solutions to the quadratic equation

2 10 28 0z z− + =

are 2z and 3.z

(c) Find 2z and 3,z giving your answers in the form i p q± , where p and q are integers.(3)

(d) Show, on an Argand diagram, the points representing your complex numbers 1 2,z z and 3 .z

(2)__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

z1 2= − + i


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8

*P38168A0832*

3. (a) Given that1 22 1

⎛ ⎞=⎜ ⎟−⎝ ⎠

A

(i) find 2 ,A

(ii) describe fully the geometrical transformation represented by 2.A(4)

(b) Given that

B =−

−⎛⎝⎜

⎞⎠⎟

0 101

describe fully the geometrical transformation represented by B.(2)

(c) Given that1 12

9k

k+⎛ ⎞= ⎜ ⎟

⎝ ⎠C

where k is a constant, find the value of k for which the matrix C is singular.(3)

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*P38168A01232*

4.

(a) Use differentiation to find f ( ).x′(2)

The root of the equation f ( ) 0x = lies in the interval [ ]0.7 , 0.9 .

(b) Taking 0.8 as a first approximation to , apply the Newton-Raphson process once to f ( )x to obtain a second approximation to . Give your answer to 3 decimal places.

(4)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

f ( ) ,x xx

x x= + − − ≠2 52

3 1 0


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*P38168A01432*

5. where a and b are constants.

Given that the matrix A maps the point with coordinates (4, 6) onto the point with coordinates (2, 8) ,−

(a) find the value of a and the value of b.(4)

A quadrilateral R has area 30 square units. It is transformed into another quadrilateral S by the matrix A. Using your values of a and b,

(b) find the area of quadrilateral S.(4)

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______________________________________________________________________________________________________________________________________________________

A =−

−⎛⎝⎜

⎞⎠⎟

42

ab

,


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*P38168A01832*

6. Given that find the value of x and the value of y such that

z z+ = − +∗3 1 13 i i

where z∗ is the complex conjugate of z.(7)

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i ,z x y= +


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*P38168A02032*

7. (a) Use the results for and to show that

2

1

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

n

r

r n n n=

− = + −∑ for all positive integers n.

(6)

(b) Hence show that

( )3

2 2

1

2(2 1)3

n

r n

r n an b= +

− = +∑ where a and b are integers to be found.

(4)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

1

n

rr

=∑ 2

1

n

rr

=∑


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*P38168A02432*

8. The parabola C has equation

The point P t t12 242,( ) is a general point on C.

(a) Find the equation of the directrix of C.(2)

(b) Show that the equation of the tangent to C at P t t12 242,( ) is

2 0− +y t 1x t =2

(4)

The tangent to C at the point ( )3, 12 meets the directrix of C at the point X.

(c) Find the coordinates of X.(4)

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2 48 .y x=


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*P38168A02832*

9. Prove by induction, that for

(a) 3 06 1

3 03 3 1 1

⎛⎝⎜

⎞⎠⎟

=−

⎛⎝⎜

⎞⎠⎟

n n

n( ),

(6)

(b) 2 1f ( ) 7 5nn −= + is divisible by 12.(6)

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,n +∈




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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 30 January 2012 – MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.



Paper Reference

6 6 6 7 0 1

This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. ©2012 Pearson Education Ltd.


P40086AW850/R6667/57570 5/4/5

*P40086A0124*


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*P40086A0224*

1. Given that z1 1= − i ,

(a) find arg( z1 ).(2)

Given also that z2 3 4= + i, find, in the form a b+ i , a b, ,∈

(b) z z1 2 ,(2)

(c) zz

2

1

.(3)

In part (b) and part (c) you must show all your working clearly.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40086A0424*

2. (a) Show that f ( )x x x= + −4 1 has a real root in the interval [ . , . ]0 5 1 0 .(2)

(b) Starting with the interval [ . , . ],0 5 1 0 use interval bisection twice to find an interval of width 0.125 which contains .

(3)

(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice tof ( )x to obtain an approximate value of Give your answer to 3 decimal places.

(5)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40086A0624*

3. A parabola C has cartesian equation C y x2 16= . The point P t4 2,( t8 ) is a general point on C.

(a) Write down the coordinates of the focus F and the equation of the directrix of F C.(3)

(b) Show that the equation of the normal to C at C P isP y tx t t+ = +8 4 3.(5)

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8

*P40086A0824*

4. A right angled triangle T has verticesT A ( , ),1 1 B ( , )2 1 and C ( , ).2 4 When T is transformed T

by the matrix P = ⎛⎝⎜

⎞⎠⎟

01

1 0

, the image is ′T .

(a) Find the coordinates of the vertices of ′T .(2)

(b) Describe fully the transformation represented by P. (2)

The matrices Q = ⎛⎝⎜

−−

⎞⎠⎟

43

21

and R = ⎛⎝⎜

⎞⎠⎟

13

2 4

represent two transformations. When T isT

transformed by the matrix QR, the image is ′′T .

(c) Find QR.(2)

(d) Find the determinant of QR.(2)

(e) Using your answer to part (d), find the area of ′′T .(3)

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*P40086A01224*

5. The roots of the equation

z z z3 28 22 20 0− + − =

are z z z1 2 3, and .

(a) Given that z1 3= + i, find z2 and z3.(4)

(b) Show, on a single Argand diagram, the points representing z z z1 2 3, . and (2)

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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*P40086A01424*

6. (a) Prove by induction

r n nr

n3

1

2 214

1= +=

∑ ( )(5)

(b) Using the result in part (a), show that

( ) ( )r n n n nr

n3

1

3 22 14

2 8=

∑ − = + + −(3)

(c) Calculate the exact value of ( )rr

3

20

50

2=∑ − .

(3)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40086A01824*

7. A sequence can be described by the recurrence formula

u u n un n+ = + =1 12 1 1 1, ,

(a) Find u2 and u3 .(2)

(b) Prove by induction that unn= −2 1

(5)

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*P40086A02024*

8. A = ⎛⎝⎜

⎞⎠⎟

0 12 3

(a) Show that A is non-singular.(2)

(b) Find B such that BA2 = A.(4)

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*P40086A02224*

9. The rectangular hyperbola H has cartesian equationH xy = 9

The points P pp

Q qq

3 3 3 3, ,⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

and lie on H, where HH p q≠ ± .

(a) Show that the equation of the tangent at P isP x p y p+ =2 6 .(4)

(b) Write down the equation of the tangent at Q.(1)

The tangent at the point Pt and the tangent at the point P Q intersect at R.t

(c) Find, as single fractions in their simplest form, the coordinates of R in terms of p and q.

(4)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

___________________________________________________________________________




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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryFriday 1 June 2012 – MorningTime: 1 hour 30 minutes






Paper Reference

6 6 6 7 0 1



P40688AW850/R6667/57570 5/5/4

*P40688A0132*


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2

*P40688A0232*

1. f ( )x x x x= − − −2 6 7 43 2

(a) Show that f ( 4 ) = 0 (1)

(b) Use algebra to solve f ( x ) = 0 completely.(4)

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*P40688A0432*

2. (a) Given that

A B= ⎛⎝⎜

⎞⎠⎟

=−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

3 1 34 5 5

1 11 20 1

and

find AB.(2)

(b) Given that

C D= ⎛⎝⎜

⎞⎠⎟

= ⎛⎝⎜

⎞⎠⎟

3 28 6

5 24

, ,kk

where k is a constant

and

E = C + D

find the value of k for which E has no inverse.(4)

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6

*P40688A0632*

3.

f ( ) ,x xx

x x= + − −2 34

3 7 0√

>

A root ! of the equation f ( x ) = 0 lies in the interval 3 5, .[ ] Taking 4 as a first approximation to !", apply the Newton-Raphson process once to f ( x )

to obtain a second approximation to ! . Give your answer to 2 decimal places.(6)

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*P40688A0832*

4. (a) Use the standard results for rr

n3

1=∑

and r

r

n

=∑

1 to show that

3 2= + 13( )+ −( )r r n n 2nr

n2

1

6 3 14

+=

∑


(b) Hence find the exact value of

r rr

3

16

30

6 3+ −( )=∑

(2)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40688A01032*

5.

Q

S

CP

O

y

x

Figure 1

Figure 1 shows a sketch of the parabola #"with equation y2 = 8x . The point $"lies on C, where y > 0, and the point Q lies on C, where y < 0 The line segment PQ is parallel to the y-axis.

Given that the distance PQ is 12,

(a) write down the y-coordinate of P,(1)

(b) find the x-coordinate of P.(2)

Figure 1 shows the point S which is the focus of C. The line l passes through the point P and the point S.

(c) Find an equation for l in the form ax + %&"+ c = 0, where a, b and c are integers.(4)

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14

*P40688A01432*

6. < <f ( ) tan ,x x x x= ⎛⎝⎜

⎞⎠⎟ + − −

23 6 ϖ ϖ

(a) Show that the equation f ( x ) = 0 has a root ! in the interval 1 2, .[ ] (2)

(b) Use linear interpolation once on the interval 1 2,[ ] to find an approximation to !"( Give your answer to 2 decimal places.

(3)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40688A01632*

7. z = −2 3i√

(a) Calculate arg )", giving your answer in radians to 2 decimal places.(2)

Use algebra to express

(b) z + z"2 in the form a b+ i√ 3, where a and b are integers,(3)

(c) zz

+−

71

in the form c d+ i√ 3, where c and d are integers.(4)

Given that

w = *"– 3i

where * is a real constant, and arg( ) ,4 5 32

− + = −i w ϖ

(d) find the value of *.(2)

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*P40688A02032*

8. The rectangular hyperbola H has equation xy = c2 , where c is a positive constant.

The point P ct ct

t, , ,⎛⎝⎜

⎞⎠⎟ ≠ 0 is a general point on H.

(a) Show that an equation for the tangent to H at P is

x + t2 y = 2ct (4)

The tangent to H at the point P meets the x-axis at the point +"and the y-axis at the point B.

Given that the area of the triangle OAB, where O is the origin, is 36,

(b) find the exact value of c, expressing your answer in the form k √ 2, where k is an integer.

(4)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P40688A02432*

9. M =−

⎛⎝⎜

⎞⎠⎟

3 42 5

(a) Find det M.(1)

The transformation represented by M maps the point ,"(2a – 7, a – 1), where a is a constant, onto the point ,-"(25, –14)."

(b) Find the value of a.(3)

The point ."has coordinates (6, 0).

Given that O is the origin,

(c) find the area of triangle ORS. (2)

Triangle ORS is mapped onto triangle /.-,-"by the transformation represented by M.

(d) Find the area of triangle OR'S' .(2)

Given that

A =−⎛

⎝⎜⎞⎠⎟

0 11 0

(e) describe fully the single geometrical transformation represented by A. (2)

The transformation represented by A followed by the transformation represented by B is equivalent to the transformation represented by M.

(f) Find B.(4)

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*P40688A03032*

10. Prove by induction that, for n∈ +! ,

f ( n ) = 22n – 1 + 32n – 1 is divisible by 5. (6)

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 28 January 2013 – MorningTime: 1 hour 30 minutes




Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75. There are 28 pages in this question paper. Any blank pages are indicated.


Paper Reference

6 6 6 7 0 1



P41485AW850/R6667/57570 5/5/5/5/6/

*P41485A0128*


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2

*P41485A0228*

1. Show, using the formulae for rr

n

=∑

1and r

r

n2

1=∑ , that

3 2 1 2 1 2 12

1r n n n

r

n

−( ) = + −=

∑ ( )( ), for all positive integers n.(5)

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*P41485A0428*

2.z =

+50

3 4i

Find, in the form a b+ i where a b, ∈! ,

(a) z,(2)

(b) z2 .(2)

Find

(c) z ,(2)

(d) arg z2 , giving your answer in degrees to 1 decimal place.(2)

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*P41485A0628*

3.f ( )x x x= + −−2 5

12

12 , x! 0

(a) Find ′f ( )x .(2)

The equation f ( )x = 0 has a root ! in the interval [4.5, 5.5].

(b) Using x0 5= as a first approximation to !, apply the Newton-Raphson procedure once to f ( )x to find a second approximation to !, giving your answer to 3 significant figures.

(4)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P41485A0828*

4. The transformation U, represented by the 2 2× matrix P, is a rotation through 90° anticlockwise about the origin.

(a) Write down the matrix P.(1)

The transformation V, represented by the 2 2× matrix Q, is a reflection in the line y x= − .

(b) Write down the matrix Q.(1)

Given that U followed by V is transformation T, which is represented by the matrix R,

(c) express R in terms of P and Q,(1)

(d) find the matrix R,(2)

(e) give a full geometrical description of T as a single transformation.(2)

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*P41485A01228*

5. f ( ) ( )( )x x x x= + − +4 9 6 342 2

(a) Find the four roots of f ( )x = 0

Give your answers in the form x p q= + i , where p and q are real.(5)

(b) Show these four roots on a single Argand diagram.(2)

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*P41485A01428*

6. X = ⎛⎝⎜

⎞⎠⎟

13 2

a , where a is a constant.

(a) Find the value of a for which the matrix X is singular.(2)

Y =−⎛

⎝⎜⎞⎠⎟

1 13 2

(b) Find Y−1 .(2)

The transformation represented by Y maps the point A onto the point B.

Given that B has coordinates (1 – ", 7" – 2), where " is a constant,

(c) find, in terms of ", the coordinates of point A.(4)

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*P41485A01828*

7. The rectangular hyperbola, H, has cartesian equation xy = 25

The point P 5 5p p,⎛⎝⎜

⎞⎠⎟ , and the point Q 5 5q q,⎛

⎝⎜⎞⎠⎟ , where p, q!"!#$!p!"!q, are points on

the rectangular hyperbola H.

(a) Show that the equation of the tangent at point P is

p y x p2 10+ =(4)

(b) Write down the equation of the tangent at point Q.(1)

The tangents at P and Q meet at the point N.

Given p q+ ≠ 0 ,

(c) show that point N has coordinates 10 10pqp q p q+ +

⎛⎝⎜

⎞⎠⎟

, .(4)

The line joining N to the origin is perpendicular to the line PQ.

(d) Find the value of p q2 2 .(5)

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*P41485A02228*

8. (a) Prove by induction that, for n ∈ +! ,

r r n n nr

n

( ) ( )( )+ = + +=

∑ 3 1 51

13

(6)

(b) A sequence of positive integers is defined byu

u u n n nn n

1

1

13 1

== + + ∈+

+

,( ),

!

Prove by induction that

u n nn = − +2 1 1( ) , n ∈ +!(5)

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P41485A02628*

9.

S

P

y y2 = 36x

xO N

Figure 1

Figure 1 shows a sketch of part of the parabola with equation y x2 36= .

The point P (4, 12) lies on the parabola.

(a) Find an equation for the normal to the parabola at P.(5)

This normal meets the x-axis at the point N and S is the focus of the parabola, as shown in Figure 1.

(b) Find the area of triangle PSN.(4)

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________




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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryMonday 10 June 2013 – MorningTime: 1 hour 30 minutes








P43138AW850/R6667/57570 5/5/5/5/5

*P43138A0132*

Paper Reference

6 6 6 7 0 1


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2

*P43138A0232*

1.

M =−

− −⎛⎝⎜

⎞⎠⎟

x xx x

23 6 4 11

Given that the matrix M is singular, find the possible values of x.(4)

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*P43138A0432*

2. f(x) = cos(x2) – x + 3, 0 < x < !

(a) Show that the equation f(x) = 0 has a root " in the interval [2.5, 3].(2)

(b) Use linear interpolation once on the interval [2.5, 3] to find an approximation for ", giving your answer to 2 decimal places.

(3)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P43138A0632*

3. Given that x = 12

is a root of the equation

2x3 – 9x2 + kx – 13 = 0, k ! " find

(a) the value of k,(3)

(b) the other 2 roots of the equation.(4)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P43138A0832*

4. The rectangular hyperbola H has Cartesian equation xy = 4

The point P tt

2 2,⎛⎝⎜

⎞⎠⎟ lies on H, where t ! 0

(a) Show that an equation of the normal to H at the point P is

ty – t3x = 2 – 2t4

(5)

The normal to H at the point where t = – 12

meets H again at the point Q.

(b) Find the coordinates of the point Q.(4)

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12

*P43138A01232*

5. (a) Use the standard results for rr

n

=∑

1 and r

r

n2

1=∑ to show that

( )( ) ( )r r n n nr

n

=∑ + + = + +

1

22 3 13

9 26


(b) Hence show that

( )( ) ( )r r n an bn cr n

n

= +∑ + + = + +

1

322 3 2

3

where a, b and c are integers to be found.(4)

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P43138A01632*

6. A parabola C has equation y2 = 4ax, a > 0

The points P(ap2, 2ap) and Q(aq2, 2aq) lie on C, where p ! 0, q ! 0, p ! q.

(a) Show that an equation of the tangent to the parabola at P is

py – x = ap2

(4)

(b) Write down the equation of the tangent at Q.(1)

The tangent at P meets the tangent at Q at the point R.

(c) Find, in terms of p and q, the coordinates of R, giving your answers in their simplest form.

(4)

Given that R lies on the directrix of C,

(d) find the value of pq.(2)

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*P43138A02032*

7. z1 = 2 + 3i, z2 = 3 + 2i, z3 = a + bi, a, b ! "

(a) Find the exact value of |z1 + z2|.(2)

Given that w z zz

= 1 3

2,

(b) find w in terms of a and b, giving your answer in the form x + iy, x, y ! "(4)

Given also that w = 1713

– 713

i,

(c) find the value of a and the value of b,(3)

(d) find arg w, giving your answer in radians to 3 decimal places.(2)

_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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*P43138A02432*

8.

A =−

−⎛⎝⎜

⎞⎠⎟

6 24 1

and I is the 2 × 2 identity matrix.

(a) Prove that

A2 = 7A + 2I(2)

(b) Hence show that

A–1 = 12 (A – 7I)

(2)

The transformation represented by A maps the point P onto the point Q.

Given that Q has coordinates (2k + 8, – 2k – 5), where k is a constant,

(c) find, in terms of k, the coordinates of P.(4)

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*P43138A02832*

9. (a) A sequence of numbers is defined by

u1 = 8

un + 1 = 4un – 9n, n #$1

Prove by induction that, for n !$%+,

un = 4n + 3n +$1(5)

(b) Prove by induction that, for m !$%+,

3 41 1

2 1 41 2

−−

⎛⎝⎜

⎞⎠⎟

=+ −

−⎛⎝⎜

⎞⎠⎟

m m mm m

(5)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


June 2013 Rphysicsandmathstutor.com

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Paper Reference

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Pearson Edexcel InternationalAdvanced Level

You must have:

Mathematical Formulae and Statistical Tables (Pink)

6667A/01Wednesday 29 January 2014 – MorningTime: 1 hour 30 minutes

Candidates may use any calculator allowed by the regulations of the

Joint Council for Qualifications. Calculators must not have the facility

for symbolic algebra 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).

Coloured pencils and highlighter pens must not be used. 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. When a calculator is used, the answer should be given to an appropriate

degree of accuracy.

Information 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.

*P43019A0132*P43019A©2014 Pearson Education Ltd.

5/5/5/5/

Further Pure Mathematics FP1Advanced/Advanced Subsidiary

PhysicsAndMathsTutor.com January 2014 (IAL)

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*P43019A0232*

1. f(x) = 2x – 5 cos x, where x is in radians.

(a) Show that the equation f(x) = 0 has a root in the interval [1, 1.4].(2)

(b) Starting with the interval [1, 1.4], use interval bisection twice to find an interval of width 0.1 which contains

(3)

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Question 1 continued

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(Total 5 marks)


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*P43019A0432*

2.

(i) A =−−

⎛⎝⎜

⎞⎠⎟

4 103 k

k, .where is a constant

The triangle T is transformed to the triangle T by the transformation represented by A.

Given that the area of triangle T is twice the area of triangle T, find the possible values of k.

(4)

(ii) Given that

B C=−

−⎛⎝⎜

⎞⎠⎟

=−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

1 22 5

31

201

822

,

find BC.(3)

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(Total 7 marks)


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*P43019A0632*

3. A rectangular hyperbola has parametric equations

x t yt

t= = ≠2 2 0, ,

Points P and Q on this hyperbola have parameters t t= =12

4 and respectively.

The line L, which passes through the origin O, is perpendicular to the chord PQ.

Find an equation for L.(4)

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*P43019A0832*

4. f ( ) ,x xx

x= − − >2 6 3 012

2

A root of the equation f (x) = 0 lies in the interval [3, 4].

Taking 3.5 as a first approximation to , apply the Newton-Raphson process once to f (x) to obtain a second approximation to Give your answer to 3 decimal places.

(5)

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*P43019A01032*

5. z = 5 i 3, w = 3 – i

(a) Find the value of |w |.(1)

Find in the form a + ib, where a and b are real constants,

(b) zw, showing clearly how you obtained your answer,(2)

(c) zw

, showing clearly how you obtained your answer.

(3)

Given that

arg( ) = , where is a real constant,3πz λ λ+

(d) find the value of .(2)

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*P43019A01432*

6. (a) Use the standard results for r rr

n

r

n3

1 1= =∑ ∑ and to show that for all positive integers n,

r r r n n n n ar

n

( )( ) ( )( )( )+ − = + − +=

∑ 1 1 14

1 11

where a is an integer to be determined.(4)

(b) Hence find the value of n, where n > 1, that satisfies

r r r rr

n

r

n

( )( )+ − == =

∑ ∑1 1 101

2

1

(5)

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(Total 9 marks)


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*P43019A01832*

7. P M=−

−⎛⎝⎜

⎞⎠⎟

=−

−⎛⎝⎜

⎞⎠⎟

3 22

6 72

a ab b

a ab b

, s

where a and b are non-zero constants.

(a) Find P−1, leaving your answer in terms of a and b.(3)

Given that

M = PQ

(b) find the matrix Q, giving your answer in its simplest form.(3)

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*P43019A02032*

8. The parabola C has equation y2 = 4ax, where a is a positive constant.

The point P(ap2, 2ap) lies on the parabola C.

(a) Show that an equation of the normal to C at P is

y + px = ap3 + 2ap(5)

The normal to C at the point P meets the x-axis at the point (6a, 0) and meets the directrix of C at the point D. Given that p > 0,

(b) find, in terms of a, the coordinates of D.(4)

Given also that the directrix of C cuts the x-axis at the point X,

(c) find, in terms of a, the area of the triangle XPD, giving your answer in its simplest form.

(3)

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*P43019A02432*

9. Given that z = x + iy, where x , y , find the value of x and the value of y such that

(3 – i)z* + 2iz = 9 – i

where z* is the complex conjugate of z.(8)

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*P43019A02832*

10. (i) A sequence of numbers u1, u2, u3,…, is defined by

un + 1 = 5un + 3, u1 = 3

Prove by induction that, for n +,

unn= −3

45 1( )

(5)

(ii) Prove by induction that, for n +,

f(n) = 5(5n) – 4n – 5 is divisible by 16.(6)

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TOTAL FOR PAPER: 75 MARKS

END

Q10

(Total 11 marks)




Surname Initial(s)

Signature

Centre No.

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Candidate No.


1

2

3

4

5

6

7

8

9

Total

Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryTuesday 10 June 2014 – MorningTime: 1 hour 30 minutes






Paper Reference

6 6 6 7 0 1



P43153AW850/R6667/57570 5/5/5/1/

*P43153A0128*

PhysicsAndMathsTutor.com June 2014

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*P43153A0228*

1. The complex numbers z1 and z2 are given by

z1 = p + 2i and z2 = 1 – 2i

where p is an integer.

(a) Find zz

1

2 in the form a + bi where a and b are real. Give your answer in its simplest

form in terms of p.(4)

Given that zz

1

213= ,

(b) find the possible values of p.(4)

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*P43153A0428*

2.

f(x) = x3 – 5

232x

+ 2x – 3, x > 0

(a) Show that the equation f(x) = 0 has a root in the interval [1.1, 1.5].(2)

(b) Find f x).(2)

(c) Using x0 = 1.1 as a first approximation to , apply the Newton-Raphson procedure once to f(x) to find a second approximation to , giving your answer to 3 decimal places.

(3)

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*P43153A0628*

3. Given that 2 and 1 – 5i are roots of the equation

x3 + px2 + 30x + q = 0, p, q

(a) write down the third root of the equation.(1)

(b) Find the value of p and the value of q.(5)

(c) Show the three roots of this equation on a single Argand diagram.(2)

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*P43153A0828*

4. (i) Given that

A B= −⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

=−⎛

⎝⎜⎞⎠⎟

134

215

21

13

41

and ,

(a) find AB.

(b) Explain why AB BA.(4)

(ii) Given that

C =−⎛

⎝⎜⎞⎠⎟

23

2kk

k, where is a real number

find C–1, giving your answer in terms of k.(3)

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*P43153A01028*


n

r

n

= =∑ ∑

1

2

1 and to show that

( ) ( )2 1 13

4 12 2

1r n n

r

n

− = −=

∑ (6)

(b) Hence show that

( ) ( )2 1 12 2

2 1

4

r an bnr n

n

− = −= +∑

where a and b are constants to be found.(3)

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*P43153A01428*

6. The rectangular hyperbola H has cartesian equation xy = c2.

The point P ct ct

, ,⎛⎝⎜

⎞⎠⎟ t > 0, is a general point on H.

(a) Show that an equation of the tangent to H at the point P is

t2 y + x = 2ct(4)

An equation of the normal to H at the point P is t3x – ty = ct4 – c

Given that the normal to H at P meets the x-axis at the point A and the tangent to H at P meets the x-axis at the point B,

(b) find, in terms of c and t, the coordinates of A and the coordinates of B.(2)

Given that c = 4,

(c) find, in terms of t, the area of the triangle APB. Give your answer in its simplest form.(3)

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*P43153A01828*

7. (i) In each of the following cases, find a 2 × 2 matrix that represents

(a) a reflection in the line y = –x,

(b) a rotation of 135° anticlockwise about (0, 0),

(c) a reflection in the line y = –x followed by a rotation of 135° anticlockwise about (0, 0).

(4)

(ii) The triangle T has vertices at the points (1, k), (3, 0) and (11, 0), where k is a constant.

Triangle T is transformed onto the triangle T by the matrix

6 21 2

−⎛⎝⎜

⎞⎠⎟

Given that the area of triangle T is 364 square units, find the value of k.(6)

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*P43153A02228*

8. The points P(4k2, 8k) and Q(k2, 4k), where k is a constant, lie on the parabola C with equation y2 = 16x.

The straight line l1 passes through the points P and Q.

(a) Show that an equation of the line l1 is given by

3ky – 4x = 8k2

(4)

The line l2 is perpendicular to the line l1 and passes through the focus of the parabola C. The line l2 meets the directrix of C at the point R.

(b) Find, in terms of k, the y coordinate of the point R.(7)

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*P43153A02628*

9. Prove by induction that, for n +,

f(n) = 8n – 2n

is divisible by 6(6)

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*P43153A02828*


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END

Q9

(Total 6 marks)




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

9

Total

Paper Reference(s)

6667/01R

Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryTuesday 10 June 2014 – MorningTime: 1 hour 30 minutes








P43152AW850/R6667/57570 5/5/5/1/1/1/1/

*P43152A0128*

Paper Reference

6 6 6 7 0 1 R

PhysicsAndMathsTutor.com June 2014 (R)

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2

*P43152A0228*

1. The roots of the equation

2z3 – 3z2 + 8z + 5 = 0

are z1, z2 and z3

Given that z1 = 1 + 2i, find z2 and z3(5)

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*P43152A0428*

2.f(x) = 3cos2x + x – 2, – x

(a) Show that the equation f(x) = 0 has a root in the interval [2, 3].(2)

(b) Use linear interpolation once on the interval [2, 3] to find an approximation to .

Give your answer to 3 decimal places.(3)

(c) The equation f(x) = 0 has another root in the interval [–1, 0]. Starting with this interval, use interval bisection to find an interval of width 0.25 which contains

(4)

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*P43152A0628*

3. (i)

A =

−⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

12

12

12

12

(a) Describe fully the single transformation represented by the matrix A.(2)

The matrix B represents an enlargement, scale factor –2, with centre the origin.

(b) Write down the matrix B.(1)

(ii)

M =−

⎛⎝⎜

⎞⎠⎟

32 3

kk, where is a positive constant.

Triangle T has an area of 16 square units.

Triangle T is transformed onto the triangle T by the transformation represented by the matrix M.

Given that the area of the triangle T is 224 square units, find the value of k.(3)

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*P43152A0828*

4. The complex number z is given by

z pp

= ++

23

ii

where p is an integer.

(a) Express z in the form a + bi where a and b are real. Give your answer in its simplest form in terms of p.

(4)

(b) Given that arg(z) = , where tan = 1 find the possible values of p.(5)

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*P43152A01228*


n

r

n

= =∑ ∑

1

3

1 and to show that

r r n n n nr

n

=∑ − = + + −

1

2 3 14

1 3 2( ) ( )( )( )

(5)

(b) Calculate the value of r rr =∑ −

10

502 3( )

(3)

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*P43152A01428*

6. A B=−

⎛⎝⎜

⎞⎠⎟

=−⎛

⎝⎜⎞⎠⎟

2 11 0

1 10 1

and

Given that M = (A + B)(2A – B),

(a) calculate the matrix M,(6)

(b) find the matrix C such that MC = A.(4)

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*P43152A01628*

7. The parabola C has cartesian equation y2 = 4ax, a 0

The points P(ap2, 2ap) and P (ap2, –2ap) lie on C.

(a) Show that an equation of the normal to C at the point P is

y + px = 2ap + ap3

(5)

(b) Write down an equation of the normal to C at the point P .(1)

The normal to C at P meets the normal to C at P at the point Q.

(c) Find, in terms of a and p, the coordinates of Q.(2)

Given that S is the focus of the parabola,

(d) find the area of the quadrilateral SPQP .(3)

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*P43152A02028*

8. The rectangular hyperbola H has equation xy = c2, where c is a positive constant.

The point P ct ct

t, , ,⎛⎝⎜

⎞⎠⎟ ≠ 0 is a general point on H.

An equation for the tangent to H at P is given by

ytx c

t= − +1 2

2

The points A and B lie on H.

The tangent to H at A and the tangent to H at B meet at the point −⎛⎝⎜

⎞⎠⎟

67

127

c c, .

Find, in terms of c, the coordinates of A and the coordinates of B.(5)

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*P43152A02428*

9. (a) Prove by induction that, for n +,

( )r nr n

r

n

+ =−

=∑ 1 2 21

1

(5)

(b) A sequence of numbers is defined by

u1 = 0, u2 = 32,

un+2 = 6un+1 – 8un n 1

Prove by induction that, for n +,

un = 4n+1 – 2n+3

(7)

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END

Q9

(Total 12 marks)


8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009

Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.

Summations

)12)(1(61

1

2 ++=∑=

nnnrn

r

2241

1

3 )1( +=∑=

nnrn

r

Numerical solution of equations

The Newton-Raphson iteration for solving 0)f( =x : )(f)f(

1n

nnn x

xxx

′−=+

Conics

Parabola Rectangular

Hyperbola

Standard Form axy 42 = xy = c2

Parametric Form (at2, 2at) ⎟

⎠⎞

⎜⎝⎛

tcct,

Foci )0 ,(a Not required

Directrices ax −= Not required

Matrix transformations

Anticlockwise rotation through θ about O: ⎟⎟⎠

⎞⎜⎜⎝

⎛ −θθθθ

cos sinsincos

Reflection in the line xy )(tanθ= : ⎟⎟⎠

⎞⎜⎜⎝

⎛− θθ

θθ2cos2sin2sin 2cos

In FP1, θ will be a multiple of 45°.

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4π r 2

Area of curved surface of cone = π r × slant height

Arithmetic series

un = a + (n – 1)d

Sn = 21 n(a + l) =

21 n[2a + (n − 1)d]

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5

Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2 = b2 + c2 – 2bc cos A

Binomial series

2

1

)( 221 nrrnnnnn bbarn

ban

ban

aba ++⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+=+ −−− KK (n ∈ ℕ)

where )!(!

!C rnr

nrn

rn

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

∈<+×××

+−−++×−++=+ nxx

rrnnnxnnnxx rn ,1(

21)1()1(

21)1(1)1( 2 K

K

KK ℝ)

Logarithms and exponentials

ax

xb

ba log

loglog =

Geometric series un = arn − 1

Sn = r ra n

−−

1)1(

S∞ = r

a−1

for ⏐r⏐ < 1

Numerical integration

The trapezium rule: ⎮⌡⌠

b

a

xy d ≈ 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where

nabh −=

FP1 Edexcel

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