Please check the examination details below before entering your candidate information
Candidate surname
Pearson Edexcel
Other names
Centre Number
Candidate Number
International
Advanced Level
Sample Assessment Materials for first teaching September 2018
(Time: 1 hour 30 minutes)
Paper Reference WFM02/01
Mathematics
International Advanced Subsidiary/Advanced Level
Further Pure Mathematics FP2
You must have:
Mathematical Formulae and Statistical Tables, calculator
Total Marks
Candidates may use any calculator permitted by Pearson regulations.
Calculators must not have the facility for symbolic algebra manipulation,
differentiation and integration, or have retrievable mathematical
formulae stored in them.
Instructions
black ink or ball-point pen.
•• Use
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
in the boxes at the top of this page with your name, centre number and
• Fill
candidate number.
Answer all questions and ensure that your answers to parts of questions are
• clearly
labelled.
the questions in the spaces provided
• Answer
– there may be more space than you need.
should show sufficient working to make your methods clear.
• You
Answers without working may not gain full credit.
answers should be given to three significant figures unless
• Inexact
otherwise stated.
Information
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
•• AThere
are 8 questions in this question paper. The total mark for this paper is 75.
The
marks
each question are shown in brackets
• – use this asfora guide
as to how much time to spend on each question.
Advice
each question carefully before you start to answer it.
•• Read
Try to answer every question.
your answers if you have time at the end.
•• Check
If you change your mind about an answer, cross it out and put your new answer
and any working underneath.
Turn over
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©2018 Pearson Education Ltd.
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Answer ALL questions. Write your answers in the spaces provided.
Using algebra, find the set of values of x for which
x
2
<
x+2 x+5
(7)
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Q1
(Total for Question 1 is 7 marks)
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2.
(a) Express
1
in partial fractions.
(1)
(b) Hence show that
n
∑ ( r + 6 ) ( r + 8)
2
=
r =1
n ( an + b )
56 ( n + 7 ) ( n + 8)
where a and b are integers to be found.
(4)
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( r + 6 ) ( r + 8)
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Q2
(Total for Question 2 is 5 marks)
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(a) Show that the substitution z = y–2 transforms the differential equation
(I)
into the differential equation
2
dz
− 4 xz = −2 xe − x
dx
(II)
(b) Solve differential equation (II) to find z as a function of x.
(4)
(5)
(c) Hence find the general solution of differential equation (I), giving your answer in the
form y2 = f(x).
(1)
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2
dy
+ 2 xy = xe − x y 3
dx
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Q3
(Total for Question 3 is 10 marks)
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A transformation T from the z-plane to the w-plane is given by
z −1
, z ≠ −1
z +1
The line in the z-plane with equation y = 2x is mapped by T onto the curve C in the
w-plane.
(a) Show that C is a circle and find its centre and radius.
(7)
The region y < 2x in the z-plane is mapped by T onto the region R in the w-plane.
(b) Sketch circle C on an Argand diagram and shade and label region R.
(2)
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w=
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Q4
(Total for Question 4 is 9 marks)
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Given that y = cot x,
d2 y
= 2 cot x + 2 cot 3 x
d x2
(3)
(b) Hence show that
d3 y
= p cot 4 x + q cot 2 x + r
3
dx
where p, q and r are integers to be found.
(3)
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(a) show that
π

(c) Find the Taylor series expansion of cot x in ascending powers of  x −  up to and

3
3
π

including the term in  x −  .

3
(3)
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Q5
(Total for Question 5 is 9 marks)
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(a) Find the general solution of the differential equation
Given that y = 0 and
(I)
(8)
dy
= 1 when x = 0
dx
(b) find the particular solution of differential equation (I).
(5)
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d2 y
dy
−2
− 3 y = 2 sin x
2
dx
dx
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Q6
(Total for Question 6 is 13 marks)
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7.
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R
O
Initial line
Figure 1
Figure 1 shows the two curves given by the polar equations
r = 3 sinθ,
0θπ
r = 1 + cosθ,
0θπ
 3 π
(a) Verify that the curves intersect at the point P with polar coordinates  ,  .
 2 3
The region R, bounded by the two curves, is shown shaded in Figure 1.
(
)
(b) Use calculus to find the exact area of R, giving your answer in the form a π − 3 ,
where a is a constant to be found.
(6)
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(2)
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Q7
(Total for Question 7 is 8 marks)
*S59763A02732*
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8.
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(a) Show that
3
3
where k is a constant to be found.
(3)
Given that z = cosθ + isin θ, where θ is real,
(b) show that
1
= 2cosnθ
zn
1
n
(ii) z − n = 2isin nθ
z
n
(i) z +
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1
1
1 
1

 2
6
 z +   z −  = z − 6 − k  z − 2 
z
z
z
z
(3)
(c) Hence show that
1
(3sin 2θ – sin 6θ)
32
(4)
(d) Find the exact value of
π
8
0
∫ cos3 θ sin3 θ dθ
(4)
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cos3 θ sin3 θ =
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*S59763A03132*
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Q8
(Total for Question 8 is 14 marks)
TOTAL FOR PAPER IS 75 MARKS
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