Paper Reference(s)

6674/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryWednesday 30 January 2008 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Green) 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, initial(s) and signature.Check that you have the correct question paper.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.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.Marks for individual questions and 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

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Centre No.

*N26319A0128*Turn over

Candidate No.

Paper Reference

6 6 7 4 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2008 Edexcel Limited.

Printer’s Log. No.

N26319AW850/R6674/57570 3/3/3/3/6300

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*N26319A0228*

1. Solve the differential equation

ddyx

y x− =3

to obtain y as a function of x.

(5)

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_________________________________________________________________________________________________________________________________ Q1

(Total 5 marks)

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*N26319A0428*

2. Given that x = − 12 is the real solution of the equation

2 11 14 10 03 2x x x− + + = ,

find the two complex solutions of this equation.

(6)

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Question 2 continued

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_________________________________________________________________________________________________________________________________Q2

(Total 6 marks)

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*N26319A0628*

3. (a) Simplify the expression ( )( ) ( ),x xx

x+ +

−− −

3 91

3 5 giving your answer in the form

a x b x cx

( )( ) ,+ +−1

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

(b) Hence, or otherwise, solve the inequality

( )( ) .x xx

x+ +

−> −

3 91

3 5

(4)

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Question 3 continued

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Q3

(Total 8 marks)

Question 3 continued

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*N26319A01028*

4.f ( ) tan , .x x x

xx= + −

− − < <3

222 π π

The equation f(x) = 0 has a root α in the interval [0.7, 0.8].

(a) Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to α. Give your answer to 3 decimal places.

(4)

(b) Taking 0.75 as a first approximation to α, apply the Newton–Raphson procedure once to f(x) to obtain a second approximation to α. Give your answer to 3 decimal places.

(4)

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_________________________________________________________________________________________________________________________________ Q4

(Total 8 marks)

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*N26319A01228*

5. (a) Express 5 4

1 2r

r r r+

+ +( )( ) in partial fractions.(4)

(b) Hence, or otherwise, show that

5 4

1 27 11

2 1 21

2rr r r

n nn nr

n ++ +

=+

+ +=∑ ( )( ) ( )( )

.

(5)

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*N26319A01428*

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Question 5 continued

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_________________________________________________________________________________________________________________________________ Q5

(Total 9 marks)

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*N26319A01628*

6. z = − +2 i.

(a) Express in the form a + ib

(i) 1z

,

(ii) z2.(4)

(b) Show that z z2 5− = √ 2.(2)

(c) Find arg (z2 – z ).(2)

(d) Display z and z2 – z on a single Argand diagram.(2)

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Question 6 continued

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*N26319A01828*

Question 6 continued

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Question 6 continued

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_________________________________________________________________________________________________________________________________ Q6

(Total 10 marks)

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*N26319A02028*

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

3 22

22d

ddd

yx

yx

y x− − = .(8)

(b) Find the particular solution for which, at and dd

x yyx

= = =0 2 3, .

(6)

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Question 7 continued

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*N26319A02228*

Question 7 continued

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Question 7 continued

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_________________________________________________________________________________________________________________________________ Q7

(Total 14 marks)

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*N26319A02428*

8. Figure 1

Figure 1 shows the curve C1 which has polar equation r = a (3 + 2cosθ ), 0 θ < 2π, and the circle C2 with equation r = 4a, 0 θ < 2π, where a is a positive constant.

(a) Find, in terms of a, the polar coordinates of the points where the curve C1 meets the circle C2.

(4)

The regions enclosed by the curves C1 and C2 overlap and this common region R is shaded in the figure.

(b) Find, in terms of a, an exact expression for the area of the shaded region R.(8)

(c) In a single diagram, copy the two curves in Figure 1 and also sketch the curve C3 with polar equation r = 2a cos θ, 0 θ < 2π. Show clearly the coordinates of the points of intersection of C1, C2 and C3 with the initial line, θ = 0.

(3)

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θ = 0

C2 C1

O

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Question 8 continued

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*N26319A02628*

Question 8 continued

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*N26319A02728*

Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 15 marks)

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*N26319A02828*

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