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Pearson Edexcel Centre Number Candidate Number

Level 3 GCE
Further Mathematics
Advanced Subsidiary
Paper 1: Core Pure Mathematics

Monday 14 May 2018 – Afternoon Paper Reference

Time: 1 hour 40 minutes 8FM0/01

You must have: Total Marks
Mathematical Formulae and Statistical Tables, calculator

Candidates may use any calculator allowed by the regulations of the

Joint Council for Qualifications. 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).
• centre
Fill in the boxes at the top of this page with your name,
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.
• Answers should be given to three significant figures unless otherwise stated.
Information
•• AThere
booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
are 9 questions in this question paper. The total mark for this paper is 80.
• 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. Turn over

P58302A
©2018 Pearson Education Ltd.

1/1/1/
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Muhammed Ali Mehmood

Answer ALL questions. Write your answers in the spaces provided.

1.

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⎛ 2 1 −3⎞
M = ⎜ 4 −2 1⎟
⎜ ⎟
⎝ 3 5 −2⎠

(a) Find M–1 giving each element in exact form.

(2)
(b) Solve the simultaneous equations
2x + y – 3z = –4
4x – 2y + z = 9
3x + 5y – 2z = 5
(2)
(c) Interpret the answer to part (b) geometrically.
(1)
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By calculator
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a
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Ea
f I
M
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K1
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E s
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L's s fast
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i
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I
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The coordinates of the point where

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c
all three meet
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planes
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2. The cubic equation

z3 – 3z2 + z + 5 = 0

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has roots , and .
Without solving the equation, find the cubic equation whose roots are (2 + 1), (2 + 1)
and (2 + 1), giving your answer in the form w3 + pw2 + qw + r = 0, where p, q and r are
integers to be found.
(5)
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2
z3 32 2 5 0
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et w 122 1
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2
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wz
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i
substituting
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I 4 Ielts 0
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t
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3
343 l www ti t w 1 1
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5 0
I
w
3g
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5 w3 3543W l 6672W 1 4 w c 1
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40 0
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w3_ G 4 40 0
wt2q
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w3 9w2tl9wt29
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3. (a) Shade on an Argand diagram the set of points

⎧ π 3π ⎫
{z ∈ C : z − 1 − i " 3} ∩ ⎨ z ∈ C : " arg ( z − 2) " ⎬

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⎩ 4 4⎭
(5)
The complex number w satisfies
π
w − 1 − i = 3 and arg ( w − 2) =
4
(b) Find, in simplest form, the exact value of w 2
(4)

IZ I i 1 E3 is circle with
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a
Centre Ci 1 radius 3
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L half line starting

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argot 2 is a
toi making
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at the point an angle
2 with the Real axis
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AIM
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argue 2 argcz 2
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I
4 4
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F
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2
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b finding carie imeq.no arglw 4 LyTi

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passes through 2,07 gradient tmI L

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y X
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Z
i
y
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Cartesian eqn of circle x Atty y 9

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Question 3 continued
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x 2 into circle
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subbing y
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x l t 3 9
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x
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x 2 11 tx 6 9 9
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2
2 8 1 0
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X 4 54
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value
we want
bigger so
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the
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4
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14
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Z
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X 2 14
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y
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4114
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i w ti
IL
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4
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142 14 t
E
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Question 3 continued
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16 8514 t
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14
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144
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44 8514
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4
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11t2T
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4. Part of the mains water system for a housing estate consists of water pipes buried beneath the
ground surface. The water pipes are modelled as straight line segments. One water pipe, W, is
buried beneath a particular road. With respect to a fixed origin O, the road surface is modelled

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as a plane with equation 3x – 5y – 18z = 7, and W passes through the points A(–1, –1, –3) and
B(1, 2, –3). The units are in metres.
(a) Use the model to calculate the acute angle between W and the road surface.
(5)
A point C(–1, –2, 0) lies on the road. A section of water pipe needs to be connected
to W from C.
(b) Using the model, find, to the nearest cm, the shortest length of pipe needed to connect
C to W.
(6)

a road r 7
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g
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normal
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I
ABT di of W
Ig Ig
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eanotw
f Hf
i r
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I
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sino Ess
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fingybejameg
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tf
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9
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513 5358
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Question 4 continued
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i D arcsin 7 58
g
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7 580
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or
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b let X closestpoint
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C then CX
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to W
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w c
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filth
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r
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FEED fi
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EYED2 X
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I
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but CIT is perpendicular to the line

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f d I o
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4 9 1 3 0
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at
Question 4 continued
i

I
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EM
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text
f H T
i

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NE 3
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5.
⎛ 1 3⎞
⎜− 2 −
2 ⎟

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A=⎜ ⎟
⎜ 3 1 ⎟
⎜⎝ − ⎟
2 2 ⎠

(a) Describe fully the single geometrical transformation U represented by the matrix A.
(3)
The transformation V, represented by the 2 × 2 matrix B, is a reflection in the line y = –x
(b) Write down the matrix B.
i
(1)
Given that U followed by V is the transformation T, which is represented by the matrix C,
(c) find the matrix C.
(2)
(d) Show that there is a real number k for which the point (1, k) is invariant under T.
(4)

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Rotation 1200 Anti docu
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a wise
about 0
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Inhiasm
and.IE.tation
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fail
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i
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w
f.io
t
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E
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h

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Question 5 continued
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II L L
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d Gus invariant
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s
Ht's 1
YI
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I i 53 tu 2
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k2 53
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i
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1 check this is multiply out

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correct
the 2nd
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row
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4
tff ie U
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IZ
i l t 53 U ZU
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is 2 53 1
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Is
u
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n n
6. (a) Use the standard results for ∑r
r =1
2
and ∑ r to show that
r =1

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n
1
∑ (3r − 2)
r =1
2
=
2
n ⎡⎣6n 2 − 3n − 1⎤⎦

for all positive integers n.

(5)

(b) Hence find any values of n for which

n 28
⎛ rπ ⎞
∑
r =5
(3r − 2) 2
+ 103∑r =1
r cos ⎜ ⎟ = 3n3
⎝ 2⎠
(5)
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a

II Gr 25 L9rH2rt4
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r 12 t I
r
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9 Incntill2ati 12
IHH t4n
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hz 3CntDC2nti
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hz 3 2n2t3ntl 12h 4
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E 6n2t9nt3 12h 4
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6n2 3n D
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Question 6 continued
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25
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b Gr t 103 roos 3
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I
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Gr 25
II Gr 25 712 ref y
r
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6h73m 645314 D
I i
I
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3h3 3 166
Iz
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8
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looking at
rt
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now roos
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notice that when is odd D

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r cos
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this sum follows

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So a
pattern
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ri l r L r 3 r 4 r S r 6 r 28
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O 2 0 4 0 G 28
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adding up all
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such terms manually
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roost
14g
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Question 6 continued
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the

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back to giver equation
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i
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25
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Gr t 103 roos 3 n
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343 3 166 t 103 I 4 3

Iz
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t 1276 0
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3zn2 I n

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Quadratic formula n 29
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x

e
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reject
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n E 2ft
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7.
f (z) = z3 + z2 + pz + q

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where p and q are real constants.
The equation f (z) = 0 has roots z1 , z2 and z3
When plotted on an Argand diagram, the points representing z1 , z2 and z3 form the
vertices of a triangle of area 35
Given that z1 = 3, find the values of p and q.
(7)

f 33 5 3 O
Oi
ptq
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3ptq 36
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let Zz Zz at bi
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3tCatbit a bi I
Sumotts
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i
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3T I Za
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IL da
a

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2 a 4
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c a Z
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Zz n Tem
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y
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e

2b_ ly
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z
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t 3k
4
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I
z
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x
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23 s
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Area
tbh L 35
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2b s
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i b 353
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71
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Question 7 continued
i
Zz 2 7
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Z 2 7 i
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told
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p
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dpy ad
3 f2t7i fZ7i JE
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3 4 497
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q iS9f
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i
from 3ptq 36
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36 C 159
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p
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T
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P
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Question 7 continued
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+
8. (i) Prove by induction that for n
n
⎛ 5 −8⎞ ⎛ 4n + 1 −8n ⎞

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⎜⎝ 2 −3⎟⎠ = ⎜⎝ 2n 1 − 4n⎟⎠
(6)
+
(ii) Prove by induction that for n

f (n) = 4n + 1 + 52n – 1
is divisible by 21
(6)
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LHS
Ig
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a n
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II I KID
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c
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LHS
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i n true for
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1
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assume true for

a U
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is D
E
ie
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Consider h UH i
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5
k p
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Question 8 continued
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is X
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164 8144117 244
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µ4UtD
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10kt 2 8k 164
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31 124
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4Uts 8k 8
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ZU T2 44 3
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tl Uti
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44Mt
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Uti
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2 l 4 Uti
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i true for nzuti
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we have proved the statement true
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for I
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n
it to be true for
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we have shown
when assumed the torn ak
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n UH
By Mathematical Inductiont it is
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i
true for all n E
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t
Question 8 continued
f n 4M 52N dirt
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42

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b n 5 21 21 l
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true for me 1
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i
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U
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q m
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ie FCK 4kt 5 is dir

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by 21
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consitt
i
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2 1
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HUH 44 524
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4144 4 5624
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fafothtuffzscyutit.su J 4h
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21
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1
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25 f Cle 21 44
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Div
T 21
n
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by D by 21
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feet't divisible 21
is
by
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i
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Question 8 continued
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we have proved the statement true

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for
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n I
it to be true for
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we have shown
n UH assumed the torn
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when k
By Mathematical Induction it is
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i
for all n E
Rft
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true
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(Total for Question 8 is 12 marks)

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9.
2 cm

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A H
10 cm
B G G
4 cm y 18
C F F
y 14

14 cm

E
D E
O 1 4 x
8 cm
Figure 1 Figure 2

A mathematics student is modelling the profile of a glass bottle of water. Figure 1

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shows a sketch of a central vertical cross-section ABCDEFGHA of the bottle with the
measurements taken by the student.
The horizontal cross-section between CF and DE is a circle of diameter 8 cm and the
horizontal cross-section between BG and AH is a circle of diameter 2 cm.
The student thinks that the curve GF could be modelled as a curve with equation

y = ax2 + b 1 x 4
where a and b are constants and O is the fixed origin, as shown in Figure 2.
(a) Find the value of a and the value of b according to the model.
(2)
(b) Use the model to find the volume of water that the bottle can contain.
(7)
(c) State a limitation of the model.
(1)
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The label on the bottle states that the bottle holds approximately 750 cm3 of water.
(d) Use this information and your answer to part (b) to evaluate the model, explaining your
reasoning.
(1)
l
y _18 72 14
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a 7c 4,6
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18 _all tb 14 acid
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Question 9 continued
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4 Isa i a
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2 l
So b 18 f
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e's
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A
b
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Ilo Volume t t
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G
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Off
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Ll
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D E
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8
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i V Tith LOT
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1
it lo
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i trth 4 14 2241T
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IT
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V
TJ x'dy
i
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27
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y x't
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274 x2
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9
415
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Question 9 continued 18
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141274 Isy dy

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V
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14
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18
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Ie 2749 1512
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y
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II 2502 2366 341T

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Volume total

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i lot1 224T t 341T
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268T 842cm
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c The cane may not be suitable

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to
model bottle
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the
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Any sensible comment

regarding
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the simplified
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shape will do
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d
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8427750.1
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842 751
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error 100 12 3 1
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Large y 8427750 i this

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error
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isn't good
a model
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