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Paper Reference(s)
6674/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryFriday 30 January 2009 – AfternoonTime: 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, 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.
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 28 pages in this question paper. Any blank pages are indicated.
Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.
Paper Reference
6 6 7 4 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2009 Edexcel Limited.
Printer’s Log. No.
N30024AW850/R6674/57570 3/3/3/3/
*N30024A0128*
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*N30024A0228*
1. (a) Show that .(4)
(b) Hence, or otherwise, find the value of .(2)
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( ) ( )r r n nr
n2
1
211
34− − = −
=∑
( )r rr
2
10
20
1− −=∑
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(Total 6 marks)
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2. f(x) = 2x4 – 14x3 + 33x2 – 26x + 10.
Given that x = 3 + i is a solution of the equation f(x) = 0, solve f(x) = 0 completely.(6)
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(Total 6 marks)
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3. Find the set of values of x for which
(6)
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x x
x
3 5 12
34
+ −−
> .
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(Total 6 marks)
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*N30024A0828*
4.
Figure 1
Figure 1 shows part of the curve with equation y = f(x), where
The point A, with x-coordinate p, is a stationary point on the curve.
The equation f(x) = 0 has a root α in the interval 0.6 < α < 0.7.
(a) Explain why is not suitable to use as a first approximation to α when applying the Newton-Raphson procedure to f(x).
(1)
(b) Using as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 decimal places.
(5)
(c) By considering the change of sign of f(x) over an appropriate interval, show that your answer to part (b) is accurate to 3 decimal places.
(2)
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y
xO
A
f ( ) sin ( ).x x x= − −1 2
x p0 =
x0 0 6= .
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(Total 8 marks)
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5. Given that and ,
(a) find in the form a + ib, where a and b are real.(2)
(b) Show, on an Argand diagram, the point P representing and the point Q representing .
(2)
(c) Given that O is the origin, show that .(2)
The circle passing through the points O, P and Q has centre C. Find
(d) the complex number represented by C,(2)
(e) the exact value of the radius of the circle.(2)
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z1 3 2= + i zz2
1
12 5= − i
z2
z1
∠ =POQ π2
z2
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Question 5 continued
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(Total 10 marks)
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*N30024A01628*
6. The curve has polar equation r = 6 sin θ.
(a) Find a cartesian equation of . (2)
The curve has polar equation r = (9√6) cos 2θ , 0 θ .
(b) Express the polar equation of in terms of sin θ only.(1)
The tangent to at the point P is parallel to the initial line.
(c) Use differentiation to show that, at P, sin θ .(4)
(d) Find the value of r at P.(2)
(e) Hence, or otherwise, show that and have a common tangent parallel to theinitial line.
(3)
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C1
C1
C2
π4
C2
C2
C2C1
= 1
6√
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Question 6 continued
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7.
(a) Show that xex is a particular integral of the differential equation, where λ is a constant to be found.
(4)
(b) Find the general solution of the differential equation.(4)
(c) Find the particular solution for which (5)
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d
d
d
de
2
24 5 4
y
x
y
xy x+ − = .
yy
xx= − = − =2
3
4
30and
d
dat .
λ
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(Total 13 marks)
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8. (a) Show that the substitution transforms the differential equation
(I)
into the differential equation
(II)(4)
(b) Solve the differential equation (II).(5)
(c) Hence show that
where c is an arbitrary constant, is a general solution of the differential equation (I). (2)
Given that
(d) find the value of y at (3)
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yt
= 1
yx c x
=+1
cos sin,
√
sin cos ,xy
xy x y x
d
d+ = < <2 0 π
d
dcosec
t
xt x x x− = − < <cot , .0 π.
y x= =2
3 4at ,
π
x =2
.π
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 14 marks)
