edexcel gce january 2010 fp3 (old syllabus) question paper
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FP3 Old Syllabus Jan 2010 Paper. Enjoy GuysTRANSCRIPT
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Paper Reference(s)
6676/01Edexcel GCEFurther Pure Mathematics FP3Advanced/Advanced SubsidiaryWednesday 3 February 2010 – 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, 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 7 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 6 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2010 Edexcel Limited.
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1. 2d 2sindy x yx= +
It is given that 1=y at 0=x .
Use the approximation 1 0
0
dd
y y yh x− ⎛ ⎞= ⎜ ⎟
⎝ ⎠ with a step length of 1.0=h to find estimates of
y at 1.0=x and at 2.0=x , giving your answers to 4 decimal places.(5)
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___________________________________________________________________________ Q1
(Total 5 marks)
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2. Given that 3 lny x x= ,
(a) find ddyx
, 2
2
dd
yx
and 3
3
dd
yx
. (5)
(b) Find the Taylor series expansion of 3 lnx x in ascending powers of )1( −x up to and including the term in 3( 1)x − .
(3)
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(Total 8 marks)
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3. (a) Use De Moivre’s theorem to show that
cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ(5)
(b) Hence find the two positive solutions of
5 332 40 10 1 0x x x− + + = ,
giving your answers to 3 decimal places. (6)
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(Total 11 marks)
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4. For n∈ +, show, using mathematical induction, that
(i)
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
12)2(01001
123011001
nnnn
n
,
(5)
(ii) )52( 13 ++n is divisible by 7. (5)
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(Total 10 marks)
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5. Given that
a = i + 7j + 9k and b = − i + 3j + k,
(a) show that )2( kjiba −+=× c , and state the value of the constant c. (2)
The plane Π1 passes through the point (3, 1, 3) and the vector a × b is perpendicular to Π1.
(b) Find a cartesian equation for the plane Π1.(2)
The line 1l has equation r = i – 2k + λa.
(c) Show that the line 1l lies in the plane Π1. (2)
The line 2l has equation r = i + j + k + μb. The line 2l lies in a plane Π2, which is parallel to the plane Π1.
(d) Find a cartesian equation of the plane Π2. (2)
(e) Find the distance between the planes Π1 and Π2. (3)
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(Total 11 marks)
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6. M =
11 5√3
5√3 1
−
−
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Given that λ1 and λ2 are the eigenvalues of M and λ1 > λ2 ,
(a) show that λ1 = 16 and find the value of λ2 .(4)
(b) Find eigenvectors corresponding to the eigenvalues λ1 and λ2 .(4)
Given that there is an orthogonal matrix P such that P–1MP is the diagonal matrix D,
where D = λ
λ1 0
0 2
⎛
⎝⎜
⎞
⎠⎟ ,
(c) find the matrix P, (2)
(d) verify that P–1MP = D.(4)
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(Total 14 marks)
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7. The point P represents the complex number z in an Argand diagram. Point P moves on the curve C given by the equation
| z – 4 + 4i| = 2| z – 1 + i|
(a) Show that C is a circle whose equation may be written z k= , giving the exact value of k.
(5)
(b) Draw an Argand diagram showing the circle C and the points representing the complex numbers 1 – i and 4 – 4i.
(3)
(c) For the points on the circle C, find the maximum and minimum values of 4 4iz − + .
(3)
The transformation T from the z-plane to the w-plane is given by 8w zz
= + .
(d) Show that T maps the curve C onto a line segment in the w-plane and define this line segment by giving its equation and the coordinates of its end points.
(5)
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TOTAL FOR PAPER: 75 MARKS
END
Q7
(Total 16 marks)
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