Page 1
Centre Number Candidate Number
Write your name hereSurname Other names
Total Marks
Paper Reference
*S54439A0117*S54439A©2017 Pearson Education Ltd.
1/1/1/1/
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Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic 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.• Answer all questions and ensure that your answers to parts of questions are clearly labelled.• 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• A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.• There are 7 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.
Further MathematicsAdvanced Paper 2: Core Pure Mathematics 2
Sample Assessment Material for first teaching September 2017
Time: 1 hour 30 minutes 9FM0/02You must have:Mathematical Formulae and Statistical Tables, calculator
Pearson Edexcel Level 3 GCE
43Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
Solutions
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Answer ALL questions. Write your answers in the spaces provided.
1. The roots of the equation
x3 − 8x2 + 28x − 32 = 0
are α, β and γ
Without solving the equation, find the value of
(i) 1
α + 1
β + 1
γ
(ii) (α + 2)(β + 2)(γ + 2)
(iii) α2 + β 2 + γ 2(8)
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(Total for Question 1 is 8 marks)
44 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
Ex 8 Exp 28 apt 32
ily pity B8txt 21 L
Lpf 32 8
ii at 2 ptz 8 2
xptzxt2 pt 4 8 2
apt t 2 8 2 Ptt at Zap 42 tap 8
apt t 2 Exp 42 8
32 t 2 28 t 4 8 8 128
iii int ptt x xp txtp tap pt
t 82 txt tpr
Itp't t t 2 Expftp 2182 fzx 2Eap
8Z 2x28
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2. The plane 1 has vector equation
r.(3i − j + 2k =
(a) Find the perpendicular distance from the point (6, 2, 12) to the plane 1(3)
The plane 2 has vector equation
r = (2i + j + k) + (i − j − 2k)
where and µ are scalar parameters.
−i − 3j + k is perpendicular to 2(2)
(c) Show that the acute angle between 1 and 2 2 (3)
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(Total for Question 2 is 8 marks)
46 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
aIT 3x 4g t 22 5 0
1 distance of 6 2,12 from IT
I 316 412 t 2112 5132 t f 4 2 t 22
22 529 or 5.39 to 3sit
b
t II 2 3 5 0
Iz II I 3 2 0
i 3J K is therefore 1 to two non parallelvectors in the planet and so is 1 to the planet
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2. The plane 1 has vector equation
r.(3i − j + 2k =
(a) Find the perpendicular distance from the point (6, 2, 12) to the plane 1(3)
The plane 2 has vector equation
r = (2i + j + k) + (i − j − 2k)
where and µ are scalar parameters.
−i − 3j + k is perpendicular to 2(2)
(c) Show that the acute angle between 1 and 2 2 (3)
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(Total for Question 2 is 8 marks)
47Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
c Angle between planes angle betweentheir normals
normal to IT
µ2normal to Tz I
I
oa.s.us ee
I'If fitill
Coso 3 t 121 2
15 14522n C15 1332 12
cos O 11 ITF 29
O cos JIT 51.980
I angle between planes is also 520 to nearestdegree
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3. (i)
M = 21 1 11 2 1
a− −
− −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
where a is a constant.
(a) For which values of a does the matrix M have an inverse?(2)
Given that M is non-singular,
(b) find M− in terms of a(4)
(ii) Prove by induction that for all positive integers n,
3 06 1
3 03 3 1 1
⎛⎝⎜
⎞⎠⎟
=−( )
⎛⎝⎜
⎞⎠⎟
n n
n
(6)
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(Total for Question 3 is 12 marks)
48 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
a deer 21 I all l tat ll6 t Za t 4
10 t Za
Det Me to provided a t 5
Me has inverse for a E 1131955
2 Ig cofactor Matrix
ftpgIzIHtfata6tf2 a
78 22
a 4 6 2 a
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3. (i)
M = 21 1 11 2 1
a− −
− −
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
where a is a constant.
(a) For which values of a does the matrix M have an inverse?(2)
Given that M is non-singular,
(b) find M− in terms of a(4)
(ii) Prove by induction that for all positive integers n,
3 06 1
3 03 3 1 1
⎛⎝⎜
⎞⎠⎟
=−( )
⎛⎝⎜
⎞⎠⎟
n n
n
(6)
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(Total for Question 3 is 12 marks)
49Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
Transposing gives
n fi l
n 9 9 True for n I
Assume true for n k
then 39 GEE
consider Eg 9 9
k5 Xp
Page 7
Lo
fiwhich is the same form withk replaced by Ktl
Thus if true for n K formula
also true for n let 1
Since true for n l by mathematicalinduction it is true for all positiveintegers n
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4. A complex number z has modulus 1 and argument .
(a) Show that
n + 1zn = 2 , n ∈ ]+
(2)
(b) Hence, show that
cos = 18
+ 2 + 3)(5)
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(Total for Question 4 is 7 marks)
50 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
a z cos O t i since
I cos Oti since cos no tisinnU
by De Moure's theorem
zn cosotisindJ t cos f no tisinfno
by De Moine's theoremz
ncos nd is no
z t I 2 cos no
b z t L 2cosoZ
tf 16 cos o
4cos o Ztt
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4. A complex number z has modulus 1 and argument .
(a) Show that
n + 1zn = 2 , n ∈ ]+
(2)
(b) Hence, show that
cos = 18
+ 2 + 3)(5)
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(Total for Question 4 is 7 marks)
51Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
Z t 42 Iz 62 42
to fit 4 2 t E 6
To 2 cos 40 t 8 cos 20 t 6
Ig Cos 40 t 4 cos 20 3
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5.y = x sinh x
(a) Show that ddyx
= − y
(4)
(b) Hence find the first three non-zero terms of the Maclaurin series for y, giving each coefficient in its simplest form.
(4)
(c) Find an expression for the nth non-zero term of the Maclaurin series for y.(2)
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(Total for Question 5 is 10 marks)
52 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
a did Sinoicoshx t sinhoccosxdoc
dd Sinoisinha t costacosx
Sinha sink t cosx coshoe
2 cosoccosha
d3g 2cosasinhoc 2coshasinxdoes
d g Zcossccoshn ZSinhasinxDI
Zcoshoccosx 2 SinnSinha
4sinx Sinha
4g
b Let f x y sinusinhx
f x fco txt'Co t f o tt
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5.y = x sinh x
(a) Show that ddyx
= − y
(4)
(b) Hence find the first three non-zero terms of the Maclaurin series for y, giving each coefficient in its simplest form.
(4)
(c) Find an expression for the nth non-zero term of the Maclaurin series for y.(2)
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(Total for Question 5 is 10 marks)
53Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
f o so f o so f o 2 f o o f o soI 7 1 I
f o o f o 4Cz 8
fC O 4L 87 32
f x 21 2 t g x 8 t x 32
x _x isac nth non zero term
n l4h 2 2 4
can 2
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6. (a) (i) Show on an Argand diagram the locus of points given by the values of z satisfying
z − − 3i =
Taking the initial line as the positive real axis with the pole at the origin and given that
∈ [α, α + ], where α = − 3
⎛⎝⎜
⎞⎠⎟ ,
(ii) show that this locus of points can be represented by the polar curve with equation
r = 8 + 6 sin (6)
The set of points A is defined by
{ }: 0 arg : 4 3 5
3
πA z z z z⎧ ⎫= ∩ − −⎨ ⎬⎩ ⎭
i! ! !
(b) (i) Show, by shading on your Argand diagram, the set of points A.
(ii) Find the exact area of the region defined by A, giving your answer in simplest form.(7)
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(Total for Question 6 is 13 marks)
54 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
a I z 4 311 5In
Circle centre 4 3
Radius 5 so passesRe through originCi che
j g 35 52
For polar curve a rcoso y r since
r coso 4 rsince 3 25
cos20 8rcoso 16 t V'sin'll Corsino 9 25
r 8r cos O Gr since o
r 8 cosa 6since O
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6. (a) (i) Show on an Argand diagram the locus of points given by the values of z satisfying
z − − 3i =
Taking the initial line as the positive real axis with the pole at the origin and given that
∈ [α, α + ], where α = − 3
⎛⎝⎜
⎞⎠⎟ ,
(ii) show that this locus of points can be represented by the polar curve with equation
r = 8 + 6 sin (6)
The set of points A is defined by
{ }: 0 arg : 4 3 5
3
πA z z z z⎧ ⎫= ∩ − −⎨ ⎬⎩ ⎭
i! ! !
(b) (i) Show, by shading on your Argand diagram, the set of points A.
(ii) Find the exact area of the region defined by A, giving your answer in simplest form.(7)
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(Total for Question 6 is 13 marks)
55Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
r 8 cosO t 6 since
b i Shaded green
Area f t r do0
Iscost 6since da
0
Lz 64 cos o t 96sincecos 0 36since do
Iz 36 t 28 cos O t 48 sinza do
t fo 36 t 14C It cos 20 48since do
tz 50 t 14 cos 20 48 sin 20 do
500 t sin20 24 cos201O
I t 7 t iz 0 to 24
Page 14
E 50ft t 7 t 36
25Gt t t 184
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7. 2000 island.
At time t years after the survey began, the number of foxes, f, and the number of rabbits, r, on the island are modelled by the differential equations
ddft = 0 2 f + 0 r
ddrt = 0 2 f + 0 r
(a) Show that dd
2
2f
t 0 ddft + 0 f = 0
(3)
(b) Find a general solution for the number of foxes on the island at time t years.(4)
(c) Hence find a general solution for the number of rabbits on the island at time t years.(3)
2000 20
(d) (i) According to this model, in which year are the rabbits predicted to die out?
(ii) According to this model, how many foxes will be on the island when the rabbits die out?
(iii) Use your answers to parts (i) and (ii) to comment on the model.(7)
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56 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
a From O l r DI 0.2 fdt
o Ide da o 2date
Sub fordirdt
0 If o.zfto.IR da 0.2 duty
0 If 0.2ft Adat 0.8 f dd 0.2 date
O If t 0.4 ddt dd 0.2date
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7. 2000 island.
At time t years after the survey began, the number of foxes, f, and the number of rabbits, r, on the island are modelled by the differential equations
ddft = 0 2 f + 0 r
ddrt = 0 2 f + 0 r
(a) Show that dd
2
2f
t 0 ddft + 0 f = 0
(3)
(b) Find a general solution for the number of foxes on the island at time t years.(4)
(c) Hence find a general solution for the number of rabbits on the island at time t years.(3)
2000 20
(d) (i) According to this model, in which year are the rabbits predicted to die out?
(ii) According to this model, how many foxes will be on the island when the rabbits die out?
(iii) Use your answers to parts (i) and (ii) to comment on the model.(7)
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57Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
da 0.6 II to If Odt
bAux earn m2 O Gm t 0 I O
mlo
f etAcost t Bsinio
c 0.1 r DI 0.2 fdt
o 3EO.ir e x f Asin to 1 Bcos to.se tfAcusFotBsinto
o.zeQ3tfAcosfotBsintTo
r et f Asin.to Beos.to te 3t AcosFotBsinEo
r co B A sin B A cost
d i Eso f b r 20
r 20 If 0 t A l A TB 20
f 6 AA 6 B 14
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(Total for Question 7 is 17 marks)
TOTAL FOR PAPER IS 75 MARKS
58 Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
r east 8sinto t 20cost
rabbits die out when 8sinfo t 20cost O
8 sin 20Cotto
C an
tTo I 951
E 19.5 yearsso 2019
iif eo3 19.5 6 cos 7o t 14sin t
f 3744.697
f 3745
Iii Model is unrealistic As rabbit population died
out there wouldnot be enough food to sustain
such a large tox population
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Question 7 continued
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(Total for Question 7 is 17 marks)
TOTAL FOR PAPER IS 75 MARKS
59Pearson Edexcel Level 3 Advanced GCE in Further Mathematics Sample Assessment Materials – Issue 1 – July 2017 © Pearson Education Limited 2017
![(+ IGCSE) EXAM QUESTION PRACTICE 1. [2 marks] (+ IGCSE ...homework.m34maths.com/files/documents/6886c86c-6f... · Trena wants to build a sandpit in the shape of a cuboid. The volume](https://static.documents.pub/doc/80x56/5fa9b8fd8ca900043d2bc3e9/-igcse-exam-question-practice-1-2-marks-igcse-homework-trena-wants.jpg)
