PMT

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 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 9 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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©2022 Pearson Education Ltd.

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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.

2 x3 − 4 x − 15
f (x) = 2
x + 3x + 4

(a) Show that

C ( 2 x + 3)
f(x) ≡ Ax + B +
x 2 + 3x + 4

where A, B and C are integers to be found.

(4)

(b) Hence, find

∫
5

f ( x) d x
3

giving your answer in the form p + ln q , where p and q are integers.

(5)
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(Total 9 marks)

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2. The functions f and g are defined by

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f(x) = 5 −         x  0
3x + 2

 x π
g(x) = 4sin  +     x ∈ 
 3 6

(a) Find the range of f

(2)

(b) (i) Find f −1(x)

(ii) Write down the domain of f −1

(3)

(c) Find fg(−π)

(2)
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(Total 7 marks)

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

Solutions relying entirely on calculator technology are not acceptable.

O (2, 0) x

Figure 1

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

f(x) = (x − 2)2 e3x      x ∈ 

The curve has a maximum turning point at A and a minimum turning point at (2, 0)

(a) Use calculus to find the exact coordinates of A.

(5)

Given that the equation f(x) = k , where k is a constant, has at least two distinct roots,

(b) state the range of possible values for k.

(2)
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(Total 7 marks)

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4. y = log10 (2x + 1)

(a) Express x in terms of y.

(2)

dy
(b) Hence, giving your answer in terms of x, find
dx
(3)
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(Total 5 marks)

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5. P

O t

Figure 2

The profit made by a company, £ P million, t years after the company started trading, is
modelled by the equation

4t − 1 3  t + 1 
P= + ln  
10 4  ( 2t + 1)2 

The graph of P against t is shown in Figure 2.

According to the model,

(a) show that exactly one year after it started trading, the company had made a loss of
approximately £ 830 000
(2)

A manager of the company wants to know the value of t for which P = 0

(b) Show that this value of t occurs in the interval [6, 7]

(2)

(c) Show that the equation P = 0 can be expressed in the form

1 15  ( 2t + 1) 
2

t = + ln  
4 8  t + 1 
(2)

(d) Using the iteration formula

(
1 15  2tn + 1  )
2

t n +1 = + ln   with t1 = 6
4 8  tn + 1 
 

find the value of t2 and the value of t6 , giving your answers to 3 decimal places.
(3)

(e) Hence find, according to the model, how many months it takes in total, from when the
company started trading, for it to make a profit.
(2)

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(Total 11 marks)

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6.
2 + 3 sin x
y=
cos x + sin x

Show that

dy a tan x + b sec x + c
=
dx sec x + 2 sin x

where a, b and c are integers to be found.

(6)
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(Total 6 marks)

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7. y

O x

C1

Figure 3

Figure 3 shows a sketch of the graph of C1 with equation

y = 5 − | 3x − 22 |

(a) Write down the coordinates of

(i) the vertex of C1

(ii) the intersection of C1 with the y-axis.

(2)

(b) Find the x coordinates of the intersections of C1 with the x-axis.

(2)

Diagram 1, shown on page 21, is a copy of Figure 3.

(c) On Diagram 1, sketch the curve C2 with equation

1 2
y= x −9
9

Identify clearly the coordinates of any points of intersection of C2 with the

coordinate axes.
(3)

(d) Find the coordinates of the points of intersection of C1 and C2

(Solutions relying entirely on calculator technology are not acceptable.)
(5)
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O x

C1

Diagram 1

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(Total 12 marks)

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

Solutions relying entirely on calculator technology are not acceptable.

π
(a) Express 8 sin x − 15 cos x in the form R sin (x − α) , where R > 0 and 0 < α <
2

Give the exact value of R, and give the value of α, in radians, to 4 significant figures.
(3)

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f(x) =     x > 0
41 + 16 sin x − 30 cos x

(b) Find

(i) the minimum value of f(x)

(ii) the smallest value of x at which this minimum value occurs.

(4)

(c) State the y coordinate of the minimum points on the curve with equation

y = 2f(x) − 5     x > 0

(1)

(d) State the smallest value of x at which a maximum point occurs for the curve
with equation

y = −f(2x)    x > 0
(1)
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(Total 9 marks)

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

Solutions relying entirely on calculator technology are not acceptable.

Given that cos 2θ − sin 3θ ≠ 0

(a) prove that

cos 2 θ 1 + sin θ
≡
cos 2θ − sin 3θ 1 − 2sin θ − 4sin 2 θ
(4)

(b) Hence solve, for 0 < θ  360°

cos 2 θ
= 2cosec θ
cos 2θ − sin 3θ

Give your answers to one decimal place.

(5)
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*P71379A03132* Turn over
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Question 9 continued
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(Total 9 marks)
TOTAL FOR PAPER IS 75 MARKS
END
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*P71379A03232*