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Please check the examination details below before entering your candidate information
Candidate surname Other names

Centre Number Candidate Number

Pearson Edexcel International Advanced Level

Paper
Time 1 hour 30 minutes
reference WMA13/01
Mathematics
International Advanced Level
Pure Mathematics P3

You must have: Total Marks

Mathematical Formulae and Statistical Tables (Yellow), calculator

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
•• 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
Answer all questions and ensure that your answers to parts of questions are
labelled.
• – there may
Answer the questions in the spaces provided
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
stated.
answers should be given to three significant figures unless otherwise

Information
•• AThere
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
are 10 questions in this question paper. The total mark for this paper is 75.
• – use this asfora guide
The marks each question are shown in brackets
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. Turn over

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1. The function f is defined by

5x 5x
f ( x) = 2
+ x>0
x + 7 x + 12 x + 4

5x
(a) Show that f ( x) =
x+3 (3)

(b) Find f –1
(3)

(c) (i) Find, in simplest form, f ′(x).

(ii) Hence, state whether f is an increasing or a decreasing function, giving a reason

for your answer.
(3)
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(Total 9 marks)
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2.
y

O x

Figure 1

Figure 1 shows a sketch of part of the graph with equation y = f (x), where

f(x) = ½3x – 13½ + 5 x∈

The vertex of the graph is at point P, as shown in Figure 1.

(a) State the coordinates of P.

(2)

(b) (i) State the range of f.

(ii) Find the value of ff(4)
(2)

(c) Solve, using algebra and showing your working,

16 – 2x > ½3x – 13½ + 5

(4)
The graph with equation y = f(x) is transformed onto the graph with equation y = af (x + b)
The vertex of the graph with equation y = af (x + b) is (4, 20)
Given that a and b are constants,

(d) find the value of a and the value of b.

(2)
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(Total 10 marks)
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3.
G

O k t
Figure 2

The total mass of gold, G tonnes, extracted from a mine is modelled by the equation

G = 40 – 30e1–0.05t tk G0

where t is the number of years after 1st January 1800.

Figure 2 shows a sketch of G against t.

Use the equation of the model to answer parts (a), (b) and (c).

(a) (i) Find the value of k.

(ii) Hence find the year and month in which gold started being extracted from
the mine.
(3)

(b) Find the total mass of gold extracted from the mine up to 1st January 1870.
(2)
There is a limit to the mass of gold that can be extracted from the mine.

(c) State the value of this limit.

(1)
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(Total 6 marks)
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4. In this question you should show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

(a) Show that the equation

2 sin (θ – 30°) = 5 cos θ

can be written in the form

tan θ = 2 3
(4)

(b) Hence, or otherwise, solve for 0  x  360°

2 sin(x – 10°) = 5 cos(x + 20°)

giving your answers to one decimal place.

(3)
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(Total 7 marks)
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5. (i) Find, by algebraic integration, the exact value of
4


8
dx
2 (2 x  3)3
(4)

(ii) Find, in simplest form,

 x  x 2  3 d x
7

(2)
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(Total 6 marks)
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6. (i) The curve C1 has equation

y = 3 ln(x 2 – 5) – 4x 2 + 15 x> 5

p
Show that C1 has a stationary point at x = where p is a constant to be found.
2
(4)

(ii) A different curve C2 has equation

y = 4x – 12 sin2 x

(a) Show that, for this curve,

dy
= A + B sin 2x
dx

where A and B are constants to be found.

(b) Hence, state the maximum gradient of this curve.

(4)
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(Total 8 marks)
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7 The mass, M kg, of a species of tree can be modelled by the equation

log10 M = 1.93 log10 r + 0.684

where r cm is the base radius of the tree.

The base radius of a particular tree of this species is 45 cm.
According to the model,
(a) find the mass of this tree, giving your answer to 2 significant figures.
(2)

(b) Show that the equation of the model can be written in the form

M = pr q

giving the values of the constants p and q to 3 significant figures.

(3)

(c) With reference to the model, interpret the value of the constant p.
(1)
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(Total 6 marks)
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8. A curve C has equation y = f(x), where

f(x) = arcsin  x 
1 π π
–2  x  2 – y
2  2 2

(a) Sketch C.
(1)

(b) Given x = 2 sin y, show that

dy 1
=
dx A - x2

where A is a constant to be found.

(3)
π
The point P lies on C and has y coordinate
4
(c) Find the equation of the tangent to C at P. Write your answer in the form y = mx + c,
where m and c are constants to be found.
(3)

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(Total 7 marks)
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9.
y
l

y = f(x)

O
x
P

Figure 3

Figure 3 shows a sketch of part of the curve with equation y = f(x), where

1
- x
f (x) = x(x 2 – 4)e 2

(a) Find f ′(x).

(2)
The line l is the normal to the curve at O and meets the curve again at the point P.
The point P lies in the 3rd quadrant, as shown in Figure 3.

(b) Show that the x coordinate of P is a solution of the equation

1
1 x
x=- 16 + e
2
2
(4)
(c) Using the iterative formula

1
1 x
2 n with x1 = –2
xn +1 =- 16 + e
2

find, to 4 decimal places,

(i) the value of x2

(ii) the x coordinate of P.

(3)
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(Total 9 marks)
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10.
y

O a x

Figure 4

Figure 4 shows a sketch of part of the curve with equation

y = (1 + 2 cos 2x)2

(a) Show that

(1 + 2 cos 2x)2 ≡ p + q cos 2x + r cos 4x

where p, q and r are constants to be found.

(2)
The curve touches the positive x‑axis for the second time when x = a, as shown in Figure 4.
The regions bounded by the curve, the y‑axis and the x‑axis up to x = a are shown shaded
in Figure 4.

(b) Find, using algebraic integration and making your method clear, the exact total area
of the shaded regions. Write your answer in simplest form.
(5)
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(Total 7 marks)
TOTAL FOR PAPER IS 75 MARKS
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