PhysicsAndMathsTutor.com

Physics & Maths TutorTypewritten TextEdexcel Maths FP2

Physics & Maths TutorTypewritten TextPast Paper Pack

Physics & Maths TutorTypewritten Text

Physics & Maths TutorTypewritten Text2009-2013

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.

Examiners use only

Team Leaders use only

Question Leave Number Blank

1

2

3

4

5

6

7

8

Total

Surname Initial(s)

Signature

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.

Printers Log. No.

M35144AW850/R6668/57570 3/5/3/

PhysicsAndMathsTutor.com

Leave blank

2

*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)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

4*M35144A0428*

Leave blank

2. Solve the equation

z3 = 42 42i ,

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

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

6

*M35144A0628*

3. Find the general solution of the differential equation

sin cos sin sinx yx

y x x xdd

= 2 ,

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

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

8

*M35144A0828*

4.

Figure 1

Figure 1 shows a sketch of the curve with polar equation

r = a + 3cos , a > 0, 0 - < 2

The area enclosed by the curve is 1072.

Find the value of a.(8)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

O Initial line

PhysicsAndMathsTutor.com June 2009

Leave blank

9

Turn over*M35144A0928*

Question 4 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

12

*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

x

, up to and

including the term in 3

4

x

.(6)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

13

Turn over*M35144A01328*

Question 5 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

16

*M35144A01628*

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

w zz

z=+

i

i,

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)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

17

Turn over*M35144A01728*

Question 6 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

20

*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)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

21

Turn over*M35144A02128*

Question 7 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

24

*M35144A02428*

8.dd

dd

e2

2 5 6 2x

txt

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)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

PhysicsAndMathsTutor.com June 2009

Leave blank

28

*M35144A02828*

Question 8 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 15 marks)

PhysicsAndMathsTutor.com June 2009

Examiners use only

Team Leaders use only

Question Leave Number Blank

1

2

3

4

5

6

7

8

9

10

Total

Surname Initial(s)

Signature

Turn over

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.

Examiners use only

Team Leaders use only

Question Leave Number Blank

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.

Printers Log. No.

N35388AW850/R6668/57570 4/5/5

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com

Leave blank

2

*N35388A0224*

1. (a) Express 3(3 1)(3 2)r r +

in partial fractions.

(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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

4

*N35388A0424*

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

2

2

d cos 0d

x x xt

+ + =

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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

6

*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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

8

*N35388A0824*

4. z = +8 8 3( )i (a) Find the modulus of z and the argument of z.

(3)

Using de Moivres 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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

10

*N35388A01024*

5.

Figure 1

Figure 1 shows the curves given by the polar equations

r = 2, 0 - - ,

and r = 1.5 + sin 3, 0 - - . (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 3), is shown shaded in Figure 1.

(b) Find, by integration, the area of the shaded region S, giving your answer in the form a + b3, where a and b are simplified fractions.

(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

S

r = 2

r = 1.5 + sin 3

= 0O

= 2

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

11

Turn over*N35388A01124*

Question 5 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

14

*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 30wz

= .

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

equation of this circle.(5)

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

15

Turn over*N35388A01524*

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

18

*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

z z xx

= (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)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

19

Turn over*N35388A01924*

Question 7 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

22

*N35388A02224*

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

2

2

d 25 3cos5d

y y xx

+ =(4)

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

2

2

d 25 3cos5d

y y xx

+ =(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 - . (2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Leave blank

24

*N35388A02424*

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 14 marks)

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com June 2010

Examiners use only

Team Leaders use only

Surname Initial(s)

Signature

Centre No.

Turn over

Candidate No.

Question Leave Number Blank

1

2

3

4

5

6

7

8

Total

Paper Reference(s)

6668/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced Sub