centre number candidate number edexcel gce further pure … · 2020. 3. 6. · mathematics fp2...
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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). Coloured pencils and highlighter pens must not be used.• 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.• When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information• 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.
You must have:Mathematical Formulae and Statistical Tables (Pink)
Centre Number Candidate Number
Write your name hereSurname Other names
Total Marks
6668/01Paper ReferenceWednesday 7 June 2017 – Morning
Time: 1 hour 30 minutes
P48860A©2017 Pearson Education Ltd.
1/1/1/1/1/*P48860A0128*
Pearson Edexcel GCE
Further Pure Mathematics FP2Advanced/Advanced Subsidiary
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1. (a) Show that, for r > 0
1 1
1
2 1
12 2 2 2r rr
r r−
+++( ) ( ) (1)
(b) Hence prove that, for n
2 1
1
2
12 21
2
rr r
n nn
r
n++
= ++
=∑ ( ) ( )( ) (3)
(c) Show that, for n , n > 1
6 3
1 3 12 2
32
2 2
rr r
an bn cn n
r n
n++
= + ++
=∑ ( ) ( )
where a, b and c are constants to be found.(3)
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2. Use algebra to find the set of values of x for which
xx x x−+ +
2
2 2
12
2( ) ( ) �
(9)
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3. Solve the equation
z3 32 32 3 0+ + =i
giving your answers in the form rei where r > 0 and – < (6)
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4.
yx
x=−
⎛⎝⎜
⎞⎠⎟
ln ,1
1 2
1
2 <
(a) Find dd
d
d and
d
d
yx
yx
yx
,2
2
3
3
(4)
(b) Hence, or otherwise, find the series expansion of ln 11 2−
⎛⎝⎜
⎞⎠⎟x
about x = 0, in
ascending powers of x, up to and including the term in x3. Give each coefficient in its simplest form.
(3)
(c) Use your expansion to find an approximate value for ln 32
⎛⎝⎜
⎞⎠⎟
, giving your answer to 3 decimal places.
(3)
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5. (a) Find the general solution of the differential equation
d
d
d
d
2
22 26 3
yx
yx
x− = sin
(8)
(b) Find the particular solution of this differential equation for which y = 0 and dd
yx
= 0 when x = 0
(5)
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6.
O Initial line
Figure 1
Figure 1 shows a sketch of a curve with polar equation
r = 6 + asin
where 0 < a < 6 and 0 < 2
The area enclosed by the curve is 97
2
π
Find the value of the constant a.(8)
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7. (a) Find, in the form y = f(x), the general solution of the equation
3dcos sin 2cos sin 1, 0 < x <d 2
y πx y x x xx
+ = +(8)
Given that 5 2 when 4
πy x= =
(b) find the value of y when 6
πx = , giving your answer in the form a b+ 3 , where a
and b are rational numbers to be found.(3)
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Question 7 continued
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8. The transformation T from the z-plane to the w-plane is given by
w zz
z= ++
≠31
i
i i,
The transformation T maps the circle z = 1 in the z-plane onto the line l in the w-plane.
(a) Find a cartesian equation of the line l.(5)
The circle z a b c− − =i in the z-plane is mapped by T onto the circle w = 5 in the w-plane.
(b) Find the exact values of the real constants a, b and c.(6)
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OT
WRI
TE IN
TH
IS A
REA
D
O N
OT
WRI
TE IN
TH
IS A
REA
Question 8 continued
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26
*P48860A02628*
D
O N
OT W
RITE IN TH
IS AREA
D
O N
OT W
RITE IN TH
IS AREA
D
O N
OT W
RITE IN TH
IS AREA
Question 8 continued
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27
Turn over*P48860A02728*
D
O N
OT
WRI
TE IN
TH
IS A
REA
D
O N
OT
WRI
TE IN
TH
IS A
REA
D
O N
OT
WRI
TE IN
TH
IS A
REA
Question 8 continued
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Leave blank
28
*P48860A02828*
D
O N
OT W
RITE IN TH
IS AREA
D
O N
OT W
RITE IN TH
IS AREA
D
O N
OT W
RITE IN TH
IS AREA
Question 8 continued
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 11 marks)