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Edexcel Maths FP2


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2009-2013

Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryFriday 19 June 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. 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.

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 8 questions in this question paper. The total mark for this question 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.

Examiner’s use only

Team Leader’s use only

Question Leave Number Blank

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Centre No.

*M35144A0128*Turn over

Candidate No.

Paper Reference

6 6 6 8 0 1

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

Printer’s Log. No.

M35144AW850/R6668/57570 3/5/3/


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*M35144A0228*

1. (a) Express 12r r( )+

in partial fractions.

(1)

(b) Hence show that 42

3 51 21 r r

n nn nr

n

( )( )

( )( )+= +

+ +=∑ .

(5)

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PhysicsAndMathsTutor.com June 2009

4

*M35144A0428*

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2. Solve the equation

z3 = 4√2 – 4√2i ,

giving your answers in the form r(cos θ + i sin θ ), where – < θ .(6)

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*M35144A0628*

3. Find the general solution of the differential equation

sin cos sin sinxyx

y x x xdd

− = 2 ,

giving your answer in the form y = f(x).(8)

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4.

Figure 1

Figure 1 shows a sketch of the curve with polar equation

r = a + 3cos θ, a > 0, 0 θ < 2

The area enclosed by the curve is 107

2π.

Find the value of a.(8)

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*M35144A01228*

5.y = sec2 x

(a) Show that dd

2

24 26 4y

xx x= −sec sec .

(4)

(b) Find a Taylor series expansion of sec2 x in ascending powers of 4

πx −

, up to and

including the term in 3

4

πx −

.(6)

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*M35144A01628*

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

wz

zz=

+≠ −

ii,

The circle with equation |z | = 3 is mapped by T onto the curve C.

(a) Show that C is a circle and find its centre and radius.(8)

The region |z | < 3 in the z-plane is mapped by T onto the region R in the w-plane.

(b) Shade the region R on an Argand diagram.(2)

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*M35144A02028*

7. (a) Sketch the graph of y = |x2 – a2|, where a > 1, showing the coordinates of the points where the graph meets the axes.

(2)

(b) Solve |x2 – a2| = a2 – x, a > 1.(6)

(c) Find the set of values of x for which |x2 – a2| > a2 – x, a > 1.(4)

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*M35144A02428*

8.dd

dd

e2

2 5 6 2xt

xt

x t+ + = −

Given that x = 0 and ddxt= 2 at t = 0,

(a) find x in terms of t.(8)

The solution to part (a) is used to represent the motion of a particle P on the x-axis. At time t seconds, where t > 0, P is x metres from the origin O.

(b) Show that the maximum distance between O and P is 2 39√ m and justify that this

distance is a maximum.(7)

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*M35144A02828*


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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 15 marks)





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Paper Reference

6 6 6 8 0 1Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryThursday 24 June 2010 – MorningTime: 1 hour 30 minutes

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

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 8 questions in this question paper. The total mark for this paper is 75.There are 24 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.




1

2

3

4

5

6

7

8

Total

Surname Initial(s)

Signature

Centre No.

*N35388A0124*Turn over

Candidate No.

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


N35388AW850/R6668/57570 4/5/5



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2

*N35388A0224*

1. (a) Express 3(3 1)(3 2)r r− +


(2)

(b) Using your answer to part (a) and the method of differences, show that

1

3(3 1)(3 2)

n

r r r= − +∑ = 32(3 2)

nn + (3)

(c) Evaluate 1000

100

3(3 1)(3 2)r r r= − +∑ , giving your answer to 3 significant figures.

(2)

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*N35388A0424*

2. The displacement x metres of a particle at time t seconds is given by the differential equation

2

2

d cos 0d

xx x

t+ + =

When 0=t , 0=x and d 1d 2xt

= .

Find a Taylor series solution for x in ascending powers of t, up to and including the term in 3t .

(5)

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*N35388A0624*

3. (a) Find the set of values of x for which

243

xx

+ >+ (6)

(b) Deduce, or otherwise find, the values of x for which

243

xx

+ >+ (1)

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*N35388A0824*

4. z = − +8 8 3( √ )i

(a) Find the modulus of z and the argument of z.(3)

Using de Moivre’s theorem,

(b) find 3z ,(2)

(c) find the values of w such that 4w z= , giving your answers in the form a + ib, where ,a b∈ .

(5)

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*N35388A01024*

5.

Figure 1

Figure 1 shows the curves given by the polar equations

r = 2, 0 θ ,

and r = 1.5 + sin 3θ, 0 θ .

(a) Find the coordinates of the points where the curves intersect. (3)

The region S, between the curves, for which r >2 and for which r < (1.5 + sin 3θ), is shown shaded in Figure 1.

(b) Find, by integration, the area of the shaded region S, giving your answer in the form aπ + b√3, where a and b are simplified fractions.

(7)

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S

r = 2

r = 1.5 + sin 3θ

θ = 0O

θ =π2



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*N35388A01424*

6. A complex number z is represented by the point P in the Argand diagram.

(a) Given that 6z z− = , sketch the locus of P.(2)

(b) Find the complex numbers z which satisfy both 6z z− = and 3 4i 5z − − = .(3)

The transformation T from the z-plane to the w-plane is given by 30w

z= .

(c) Show that T maps 6z z− = onto a circle in the w-plane and give the cartesian

equation of this circle.(5)



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*N35388A01824*

7. (a) Show that the transformation 12z y= transforms the differential equation

12

d 4 tan 2dy

y x yx

− = (I)

into the differential equation

d 2 tan 1d

zz x

x− = (II)

(5)

(b) Solve the differential equation (II) to find z as a function of x. (6)

(c) Hence obtain the general solution of the differential equation (I). (1)

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*N35388A02224*

8. (a) Find the value of λ for which y = λx sin 5x is a particular integral of the differential equation

2

2

d 25 3cos5d

yy x

x+ =

(4)

(b) Using your answer to part (a), find the general solution of the differential equation

2

2

d 25 3cos5d

yy x

x+ =

(3)

Given that at 0=x , 0=y and d 5dyx

= ,

(c) find the particular solution of this differential equation, giving your solution in the form =y f(x).

(5)

(d) Sketch the curve with equation =y f(x) for 0 x π. (2)

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24

*N35388A02424*


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END

Q8

(Total 14 marks)





Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


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Total

Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryThursday 23 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.

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 8 0 1

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


P35413AW850/R6668/57570 5/5/3/3

*P35413A0128*


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*P35413A0228*

1. Find the set of values of x for which

(7)

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4

*P35413A0428*

2.

(a) Show that

,

where k is a constant to be found.(3)

Given that, at and ,

(b) find a series solution for y in ascending powers of x, up to and including the term in .

(4)

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6

*P35413A0628*


giving your answer in the form .(8)

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*P35413A0828*

4. Given that

,

(a) find the values of the constants A, B and C.(2)

(b) Show that

(2)

(c) Using the result in part (b) and the method of differences, show that

(5)

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*P35413A0928* Turn over


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*P35413A01228*

5. The point P represents the complex number z on an Argand diagram, where

The locus of P as z varies is the curve C.

(a) Find a cartesian equation of C.(2)

(b) Sketch the curve C. (2)

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

The point Q is mapped by T onto the point R. Given that R lies on the real axis,

(c) show that Q lies on C.(5)

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*P35413A01628*

6.

A

C

NO

R

0Figure 1

The curve C shown in Figure 1 has polar equation

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

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

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

Find the exact area of the shaded region R.(9)

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20

*P35413A02028*

7. (a) Use de Moivre’s theorem to show that

(5)

Hence, given also that

(b) find all the solutions of

in the interval Give your answers to 3 decimal places.(6)

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*P35413A02428*

8. The differential equation

describes the motion of a particle along the x-axis.

(a) Find the general solution of this differential equation. (8)

(b) Find the particular solution of this differential equation for which, at

and .

(5)

On the graph of the particular solution defined in part (b), the first turning point for is the point A.

(c) Find approximate values for the coordinates of A.(2)

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27

*P35413A02728*


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END

Q8

(Total 15 marks)




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryFriday 22 June 2012 – 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.



Paper Reference

6 6 6 8 0 1

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


P40104AW850/R6668/57570 5/5

*P40104A0128*


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*P40104A0228*

1. Find the set of values of x for which

(5)

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*P40104A0428*

2. The curve C has polar equation

r = +1 2 0 2cos ,θ θ� � π

At the point P on C, the tangent to C is parallel to the initial line.

Given that O is the pole, find the exact length of the line OP. (7)

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*P40104A0628*

3. (a) Express the complex number in the form . (3)

(b) Solve the equation

giving the roots in the form . (5)

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*P40104A0828*


(9)

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*P40104A01228*

5.

(a) Show that

(2)

Given that y = 1 at x = 1,

(b) find a series solution for y in ascending powers of , up to and including the term in .

(8)

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*P40104A01628*

6. (a) Express


(2)

(b) Hence prove, by the method of differences, that

where a and b are constants to be found. (6)

(c) Hence show that

(3)

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*P40104A02028*

7. (a) Show that the substitution transforms the differential equation

(I)

into the differential equation

(II) (3)

(b) By solving differential equation (II), find a general solution of differential equation (I) in the form y x= ( )f .

(6)

Given that y = 2 at x = 1 ,

(c) find the value of d

d

y

x at x = 1

(2)

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*P40104A02428*

8. The point P represents a complex number z on an Argand diagram such that

z z− = −6 2 3i

(a) Show that, as z varies, the locus of P is a circle, stating the radius and the coordinates of the centre of this circle.

(6)

The point Q represents a complex number z on an Argand diagram such that

arg ( )z − = −6 34π

(b) Sketch, on the same Argand diagram, the locus of P and the locus of Q as z varies. (4)

(c) Find the complex number for which both z z− = −6 2 3i and arg ( )z − = −6 34π

(4)

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END

28

*P40104A02828*

Q8

(Total 14 marks)

Leave blank




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6668/01REdexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryFriday 21 June 2013 – MorningTime: 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.

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 8 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.




P42955AW850/R6668/57570 5/5/

*P42955A0132*

Paper Reference

6 6 6 8 0 1 R


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2

*P42955A0232*

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

w zz

= + 2ii

z��

The transformation maps points on the real axis in the z-plane onto a line in the w-plane.

Find an equation of this line.(4)

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PhysicsAndMathsTutor.com June 2013 (R)

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*P42955A0432*

2. Use algebra to find the set of values of x for which

63

11

xx x− +

>

(7)

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*P42955A0832*

3. (a) Express

21 3( )( )r r+ +


(2)

(b) Hence show that

21 3

5 136 2 31 ( )( )

( )( )( )r rn nn nr

n

+ += +

+ +=∑

(4)

(c) Evaluate 2

1 310

100

( )( )r rr + +=∑ , giving your answer to 3 significant figures.

(2)

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*P42955A01232*

4. Given that

y yx

yx

ydd

dd

2

2

2

5 0+ ⎛⎝⎜

⎞⎠⎟

+ =

(a) find dd

3

3y

x in terms of d

d

2

2y

x, d

dyx

and y.(4)

Given that y = 2 and ddyx

= 2 at x = 0

(b) find a series solution for y in ascending powers of x, up to and including the term in x3.(5)

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*P42955A01632*

5. (a) Find, in the form y = f(x), the general solution of the equation

ddyx

+ 2y tan x = sin 2x, 0 < x < 2π

(6)

Given that y = 2 at x = 3π

(b) find the value of y at x =

6π

, giving your answer in the form a + k ln b,

where a and b are integers and k is rational.(4)

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*P42955A02032*

6. The complex number z = ei�, where � is real.

(a) Use de Moivre’s theorem to show that

1 2cosnnz nθ

z+ =

where n is a positive integer.(2)

(b) Show that

cos5 �� 116

(cos5� + 5cos3� + 10cos� )(5)

(c) Hence find all the solutions of

cos5� + 5cos3� + 12cos� = 0

in the interval 0 �� < 2�(4)

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*P42955A02432*

7. (a) Find the value of � for which �2e3 is a particular integral of the differential equation

dd

2

2y

t – 6 d

dyt

+ 9y = 6e3� � � � � � � 0(5)

(b) Hence find the general solution of this differential equation.(3)

Given that when = 0, y = 5 and ddyt

= 4

(c) find the particular solution of this differential equation, giving your solution in the form y = f().

(5)

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*P42955A02832*

8.

Figure 1

Figure 1 shows a closed curve C with equation

r = 3(cos2�)12 , where –

4π

< ��4π

, 34π

< � ��54π

The lines PQ, SR, PS and QR are tangents to C, where PQ and SR are parallel to the initial line and PS and QR are perpendicular to the initial line. The point O is the pole.

(a) Find the total area enclosed by the curve C, shown unshaded inside the rectangle in Figure 1.

(4)

(b) Find the total area of the region bounded by the curve C and the four tangents, shown shaded in Figure 1.

(9)

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P

O

Q

Initial line

S R


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*P42955A03232*


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END

Q8

(Total 13 marks)




Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.


1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryFriday 21 June 2013 – MorningTime: 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 for 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 8 0 1



P43149AW850/R6668/57570 5/5/5

*P43149A0128*


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*P43149A0228*

1. (a) Express 2

2 1 2 3( )( )r r ++ in partial fractions.

(2)

(b) Using your answer to (a), find, in terms of n,

r

n

1=∑ 3

2 1 2 3( )( )r r+ +

Give your answer as a single fraction in its simplest form.(3)

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*P43149A0428*

2. z = 5�3 – 5i

Find

(a) |z|,(1)

(b) arg(z), in terms of �.(2)

2 cos isin4 4π πw ⎛ ⎞= +

⎝ ⎠ Find

(c) wz

,(1)

(d) arg ,wz

⎛⎝⎜

⎞⎠⎟

in terms of �.(2)

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*P43149A0628*

3. dd

2

2 4 0yx

y x+ − =sin

Given that y yx

x= = =12

18

0 and dd

at ,

find a series expansion for y in terms of x, up to and including the term in x3.(5)

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*P43149A0828*

4. (a) Given thatz = r (cos � + i sin �), r � �

prove, by induction, that zn = rn (cos �� + i sin ��), n � �+

(5)3 33 cos isin4 4

w π π⎛ ⎞= +⎜ ⎟⎝ ⎠

(b) Find the exact value of w 5, giving your answer in the form a + ib, where a, b ��.(2)

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*P43149A01228*

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

x yx

y xdd

+ =2 4 2

(5)

(b) Find the particular solution for which y = 5 at x = 1, giving your answer in the form y = f(x).

(2)

(c) (i) Find the exact values of the coordinates of the turning points of the curve with equation y = f(x), making your method clear.

(ii) Sketch the curve with equation y = f(x), showing the coordinates of the turning points.

(5)

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*P43149A01628*

6. (a) Use algebra to find the exact solutions of the equation

|2x2 + 6x – 5| = 5 – 2x(6)

(b) On the same diagram, sketch the curve with equation y = |2x2 + 6x – 5| and the line with equation y = 5 – 2x, showing the x-coordinates of the points where the line crosses the curve.

(3)

(c) Find the set of values of x for which

|2x2 + 6x – 5| > 5 – 2x(3)

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*P43149A02028*

7. (a) Show that the transformation y = xv transforms the equation

4 8 8 422

22 4x y

xx y

xx y xd

ddd

− + + =( ) (I)

into the equation

4 42

2dd

vx

v x+ = (II)(6)

(b) Solve the differential equation (II) to find v as a function of x.(6)

(c) Hence state the general solution of the differential equation (I).(1)

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*P43149A02428*

8.

Figure 1

Figure 1 shows a curve C with polar equation sin 2 , 0 ,2πr a θ θ= � � and a half-line l.

The half-line l meets C at the pole O and at the point P. The tangent to C at P is parallel to the initial line. The polar coordinates of P are (R, ��.

(a) Show that 1cos =φ√3 (6)

(b) Find the exact value of R.(2)

The region S, shown shaded in Figure 1, is bounded by C and l.

(c) Use calculus to show that the exact area of S is

136

9 12a arccos⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠⎟

√2√3

(7)

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P

O

C

S

��0

l


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*P43149A02828*


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END

Q8

(Total 15 marks)


Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 – Issue 1 – September 2009 9

Further Pure Mathematics FP2 Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1–C4. Area of a sector

A = ⎮⌡⌠ θd

21 2r (polar coordinates)

Complex numbers

θθθ sinicosei += )sini(cos)}sini(cos{ θθθθ nnrr nn +=+

The roots of 1=nz are given by nk

zi2

eπ

= , for 1 , ,2 ,1 ,0 −= nk K

Maclaurin’s and Taylor’s Series

KK )0(f!

)0(f!2

)0(f)0f()f( )(2

+++′′+′+= rr

rxxxx

KK )(f!

)( )(f!2

)()(f)()f()f( )(2

+−

++′′−+′−+= a

raxaaxaaxax r

r

KK )(f!

)(f!2

)(f)f()f( )(2

+++′′+′+=+ arxaxaxaxa r

r

xrxxxx

rx allfor

!

!21)exp(e

2

KK +++++==

)11( )1( 32

)1(ln 132

≤<−+−+−+−=+ + xrxxxxx

rr KK

xr

xxxxxr

r allfor )!12(

)1( !5!3

sin1253

KK ++

−+−+−=+

xr

xxxxr

r allfor )!2(

)1( !4!2

1cos242

KK +−+−+−=

)11( 12

)1( 53

arctan1253

≤≤−++

−+−+−=+

xr

xxxxxr

r KK

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°.

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2009 7

Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.

Integration (+ constant)

f(x) ⎮⌡⌠ xx d)f(

sec2 kx k1 tan kx

xtan xsecln

xcot xsinln

xcosec )tan(ln,cotcosecln 21 xxx +−

xsec )tan(ln,tansecln 41

21 π++ xxx

⎮⌡⌠ ⎮⌡

⌠−= xxuvuvx

xvu d

ddd

dd

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009

Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.

Logarithms and exponentials

xax a=lne

Trigonometric identities

BABABA sincoscossin)(sin ±=± BABABA sinsincoscos)(cos m=±

))(( tantan1tantan)(tan 2

1 π+≠±±=± kBABABABA

m

2cos

2sin2sinsin BABABA −+=+

2sin

2cos2sinsin BABABA −+=−

2cos

2cos2coscos BABABA −+=+

2sin

2sin2coscos BABABA −+−=−

Differentiation

f(x) f ′(x)

tan kx k sec2 kx

sec x sec x tan x

cot x –cosec2 x

cosec x –cosec x cot x

)g()f(

xx

))(g(

)(g)f( )g()(f2x

xxxx ′−′

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 −=

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]

paper reference(s) · 2016. 8. 9. · paper reference(s) 6668/01. edexcel gce. further pure...

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