Please check the examination details below before entering your candidate information

Candidate surname Other names

Centre Number Candidate Number

Pearson Edexcel International Advanced Level

Time 1 hour 30 minutes
Paper
reference WFM02/01
Mathematics
 

International Advanced Subsidiary/Advanced Level

Further Pure Mathematics F2

You must have: Total Marks

Mathematical Formulae and Statistical Tables (Yellow), calculator

Candidates may use any calculator allowed 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
•
•
Use black ink or ball‑point pen.
If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
• Fill in the boxes at the top of this page with your name, centre number and
candidate number.
• clearly labelled.
Answer all questions and ensure that your answers to parts of questions are

• Answer the questions in the spaces provided

– there may be more space than you need.
• You should show sufficient working to make your methods clear.
Answers without working may not gain full credit.
• Inexact answers should be given to three significant figures unless otherwise stated.
Information
•
•
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
There are 8 questions in this question paper. The total mark for this paper is 75.
• The marks for each question are shown in brackets
– use this as a guide as to how much time to spend on each question.
Advice
•
•
Read each question carefully before you start to answer it.
Try to answer every question.
• Check your answers if you have time at the end.
• Ifandyouanychange your mind about an answer, cross it out and put your new answer
working underneath.
Turn over

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1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(a) Express the complex number

–4 – 4 3i

in the form  r(cos θ + i sin θ), where  r > 0  and  –π < θ  π

(3)

(b) Solve the equation

z 3 + 4 + 4 3i  =  0

giving your answers in the form reiθ, where  r > 0  and  –π < θ  π
(4)
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(Total 7 marks)

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2. Determine the general solution of the differential equation

d2 y dy
2 2 −5 − 3 y = 2e 3 x
dx dx
(6)
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(Total 6 marks)

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3. y

y  =  x 2 – 8x

O P x
4x
y=
4− x

Figure 1

Figure 1 shows a sketch of the curve C1 with equation

4x
y=
4− x

and the curve C2 with equation

y  =  x2 – 8x

4x
For  x > 0, C1 has equation  y =
4− x

(a) Use algebra to show that C1 touches C2 at a point P, stating the coordinates of P
(5)

(b) Hence or otherwise, using algebra, solve the inequality

4x
x 2 – 8x >
4− x
(6)
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(Total 11 marks)

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4. π / 2

3π / 4 π / 4

π Initial line
0 2 4

5π / 4 7π / 4

3π / 2

Figure 2

Figure 2 shows part of the curve with polar equation

3
r  =  4 –  cos 6θ 0  θ < 2π
2

(a) Sketch, on the polar grid in Figure 2,

3
(i) the rest of the curve with equation r  =  4 –  cos 6θ 0  θ < 2π
2

(ii) the polar curve with equation r  =  1 0  θ < 2π

A spare copy of the grid is given on page 15.

(3)

In part (b) you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(b) Determine the exact area enclosed between the two curves defined in part (a).
(7)
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Only use this grid if you need to redraw your answer to part (a)

π / 2

3π / 4 π / 4

π Initial line
0 2 4

5π / 4 7π / 4

3π / 2
Q4
Copy of Figure 2
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1
5. y  =   4 + ln x x>
2

(a) Show that

d2 y 9 + 2 ln x
= −
dx2 3
4 x (4 + ln x)
2 2

(5)

(b) Hence, or otherwise, determine the Taylor series expansion about  x  =  1  for y, in
ascending powers of (x  –  1), up to and including the term in (x  –  1)2, giving each
coefficient in simplest form.
(3)
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(Total 8 marks)

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6. Given that  A > B > 0, by letting  x = arctan A  and  y = arctan B

(a) prove that

 A− B
arctan A – arctan B = arctan 
 1 + AB 
(3)

(b) Show that when  A  =  r + 2  and  B  =  r

A−B 2
=
1 + AB (1 + r ) 2
(2)

(c) Hence, using the method of differences, show that

n

∑r =1
 2 
arctan  2  = arctan( n + p ) + arctan( n + q ) − arctan 2 −
 (1 + r ) 
π
4

where p and q are integers to be determined.

(4)

(d) Hence, making your reasoning clear, determine

∑ r =1
 2 
arctan 
 (1 + r ) 
2

giving the answer in the form  kπ – arctan 2, where k is a constant.

(2)
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(Total 11 marks)

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7. A transformation from the z‑plane to the w‑plane is given by

(1 + i) z + 2(1 − i)
w= z≠i
z−i

The transformation maps points on the imaginary axis in the z‑plane onto a line in the
w‑plane.

(a) Find an equation for this line.

(2)

The transformation maps points on the real axis in the z‑plane onto a circle in the w‑plane.

(b) Find the centre and radius of this circle.

(6)
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(Total 8 marks)

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8. (a) Show that the transformation  v = y – 2x  transforms the differential equation

dy
+ 2yx( y – 4x) = 2 – 8x 3 (I)
dx

into the differential equation

dv
= –2xv2 (II)
dx
(4)

(b) Solve the differential equation (II) to determine v as a function of x

(4)

(c) Hence obtain the general solution of the differential equation (I).
(1)

(d) Sketch the solution curve that passes through the point (–1, –1).

On your sketch show clearly the equation of any horizontal or vertical asymptotes.

You do not need to find the coordinates of any intercepts with the coordinate axes
or the coordinates of any stationary points.
(5)
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(Total 14 marks)
TOTAL FOR PAPER: 75 MARKS
END
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