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 WMA14/01
Mathematics
 

International Advanced Level

Pure Mathematics P4

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.

• Fill
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,

• labelled.
centre number and candidate number.
Answer all questions and ensure that your answers to parts of questions are clearly

• Answer the questions in the spaces provided

• You
– there may be more space than you need.
should show sufficient working to make your methods clear. Answers without

• Inexact answers should be given to three significant figures unless otherwise stated.
working may not gain full credit.

Information
•• AThere
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.

• – use this asfora guide

are 9 questions in this question paper. The total mark for this paper is 75.
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.

•• Check
Try to answer every question.
your answers if you have time at the end.
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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©2022 Pearson Education Ltd.

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1. The curve C has equation

xy2 = x2 y + 6      x ≠ 0   y ≠ 0

Find an equation for the tangent to C at the point P (2, 3), giving your answer in the form
ax + by + c = 0 where a, b and c are integers.
(6)
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(Total 6 marks)

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2. (a) Find, in ascending powers of x, the first three non-zero terms of the binomial series
expansion of

1
3
1 + 4 x3       | x | < 3
4
giving each coefficient as a simplified fraction.
(4)

1
(b) Use the expansion from part (a) with x = to find a rational approximation to 3
31
3
(3)
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(Total 7 marks)

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3. The curve C has parametric equations

6 π π
x = 3 + 2 sin t      y = − t
7 + cos 2t 2 2

(a) Show that C has Cartesian equation

12
y=       p  x  q
(7 − x)(1 + x)

where p and q are constants to be found.

(6)

(b) Hence, find a Cartesian equation for C in the form

a c
y= +       p  x  q
x+b x+d
where a, b, c and d are constants.
(3)
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(Total 9 marks)

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4.

Figure 1

A regular icosahedron of side length x cm, shown in Figure 1, is expanding uniformly.

The icosahedron consists of 20 congruent equilateral triangular faces of side length x cm.

(a) Show that the surface area, A cm2, of the icosahedron is given by

A = 5 3x2
(2)

Given that the volume, V cm3, of the icosahedron is given by

V=
5
12
(3 + 5 x3 )

(b) show that

dV
=
(
3+ 5 x )
dA 8 3
(3)

The surface area of the icosahedron is increasing at a constant rate of 0.025 cm2 s−1

(c) Find the rate of change of the volume of the icosahedron when x = 2 , giving your
answer to 2 significant figures.
(3)
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(Total 8 marks)

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5.
y

9
t=
4

t=0
O x

Figure 2

Figure 2 shows a sketch of the curve with parametric equations

t3 9
x= 9 − 4t       y =        0  t 
9 + 4t 4
9
The curve touches the x-axis when t = 0 and meets the y-axis when t =
4
The region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the y-axis.

(a) Show that the area of R is given by

9

∫
4 t3
K dt
0 81 − 16t 2

where K is a constant to be found.

(4)

(b) Using the substitution u = 81 − 16t 2 , or otherwise, find the exact area of R.

(Solutions relying on calculator technology are not acceptable.)

(6)
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(Total 10 marks)

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6. Three consecutive terms in a sequence of real numbers are given by

k, 1 + 2k and 3 + 3k

where k is a constant.

Use proof by contradiction to show that this sequence is not a geometric sequence.
(5)
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(Total 5 marks)

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7.
y
1
f (x) = (4 − x)e x
4
2 cm
1

O 4 x

4 cm

Figure 3 Figure 4

Figure 3 shows the design of a doorknob.

The shape of the doorknob is formed by rotating the curve shown in Figure 4 through 360°
about the x-axis, where the units are centimetres.

The equation of the curve is given by

1
f (x) = (4 − x)e x       0  x  4
4

(a) Show that the volume, V cm3, of the doorknob is given by

∫(
4

V= K x 2 − 8 x + 16) e 2 x d x
0

where K is a constant to be found.

(3)

(b) Hence, find the exact value of the volume of the doorknob.

Give your answer in the form pπ (e q + r) cm3 where p, q and r are simplified rational
numbers to be found.
(5)
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(Total 8 marks)

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8. With respect to a fixed origin O the points A and B have position vectors

 6  6
 6 and  0
   
 2  7

respectively.

The line l1 passes through the points A and B.

(a) Write down an equation for l1

Give your answer in the form r = p + λq , where λ is a scalar parameter.

(2)

The line l2 has equation

 3  1
r =  1 + μ  5
   
 4  9

where μ is a scalar parameter.

(b) Show that l1 and l2 do not meet.

(4)

The point C is on l2 where μ = −1

(c) Find the acute angle between AC and l2

Give your answer in degrees to one decimal place.

(5)
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(Total 11 marks)

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9. (a) Find the derivative with respect to y of

1
(1 + 2 ln y )2
(2)

(b) Hence find a general solution to the differential equation

dy π π
= y (1 + 2 ln y)       y > 0    − < x <
3
3cosec(2x)
dx 2 2
(4)

π
(c) Show that the particular solution of this differential equation for which y = 1 at x =
6
is given by

1
A sec x −
y=e 2

where A is an irrational number to be found.

(5)
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(Total 11 marks)
TOTAL FOR PAPER IS 75 MARKS
END
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