pearson centre number candidate number edexcel gce further ... · edexcel gce further pure...

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

6667/01Paper ReferenceMonday 14 May 2018 – Afternoon

Time: 1 hour 30 minutes

P51567A©2018 Pearson Education Ltd.

1/1/1/*P51567A0136*

Pearson Edexcel GCE

Further Pure Mathematics FP1Advanced/Advanced Subsidiary

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1. f ( )z z z z= − + −2 4 15 133 2

Given that f ( ) ( )( )z z z az b≡ − + +1 2 2 , where a and b are real constants,

(a) find the value of a and the value of b.(2)

(b) Henceusealgebratofindthethreerootsoftheequationf( z )=0(4)

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Question 1 continued

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___________________________________________________________________________ Q1

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2. f ( ) ,x xx

x x= + + − <3

2

4

32 5 02

Theequationf( x )=0hasasinglerootα .

(a) Show that α lies in the interval [−3, −2.5](2)

(b) Taking−3asafirstapproximationtoα,applytheNewton-Raphsonprocedureoncetof( x )toobtainasecondapproximationtoα.Giveyouranswerto3decimalplaces.

(5)

(c) Uselinearinterpolationonceontheinterval[−3, −2.5]tofindanotherapproximationto α,givingyouranswerto3decimalplaces.

(3)

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___________________________________________________________________________ Q2

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3. (i) Given that

A AB=−

=−

− −

2 3

1 1

1

3

5

5

12

1,

(a) find A−1

(2)

(b) Hence,orotherwise,findthematrixB,givingyouranswerinitssimplestform.(3)

(ii) Given that

C =−

0 1

1 0

(a) describefullythesinglegeometricaltransformationrepresentedbythematrixC.(2)

(b) HencefindthematrixC39

(2)

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___________________________________________________________________________ Q3

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4. (a) Use the standard results for rr

n

=∑

1 and r

r

n2

1=∑ toshowthat,forallpositiveintegersn,

r r n n a n ar

n2

1

81

3− −( ) = −( ) +( )

=∑

where aisapositiveintegertobedetermined.(4)

(b) Hence,orotherwise,statethepositivevalueofn that satisfies

r rr

n2

1

8 0− −( ) ==

∑(1)

Given that

kr r rr

3 2

3

17

8 6710+ − −( ) ==

∑ where k is a constant

(c) findtheexactvalueofk.(4)

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___________________________________________________________________________ Q4

(Total 9 marks)

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5. TherectangularhyperbolaH has equation xy = c2, where cisapositiveconstant.

Given that P ct ct

t, ,

≠ 0 ,isageneralpointonH,

(a) use calculus to show that the equation of the tangent to H at P can be written as

t 2y + x = 2ct(4)

ThepointsA and B lie on H.

The tangent to H at A and the tangent to H at Bmeetatthepoint −

8

5

3

5

c c, .

Given that the x coordinate of Aispositive,

(b) find,intermsofc, the coordinates of A and the coordinates of B.(5)

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___________________________________________________________________________ Q5

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6. M =−

−

8 1

4 2

(a) Find the value of det M(1)

The triangle Thasverticesatthepoints(4,1),(6,k)and(12,1),wherek is a constant.

The triangle TistransformedontothetriangleT'bythetransformationrepresentedbythematrixM.

Given that the area of triangle T'is216squareunits,

(b) findthepossiblevaluesofk.(5)

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___________________________________________________________________________ Q6

(Total 6 marks)

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7. TheparabolaC has equation y 2 = 4 ax, where aisapositiveconstant. ThepointS is the focus of C.

The straight line lpassesthroughthepointSandmeetsthedirectrixofCatthepointD.

Given that the y coordinate of D is 245

a ,

(a) show that an equation of the line l is

12 x + 5 y = 12 a(2)

ThepointP (ak 2, 2ak ), where kisapositiveconstant,liesontheparabolaC.

GiventhatthelinesegmentSPisperpendiculartol,

(b) find,intermsofa,thecoordinatesofthepointP. (6)

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___________________________________________________________________________ Q7

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8. Provebyinductionthat

f (n) = 2n + 2 + 32n+1

isdivisibleby7forallpositiveintegersn.(6)

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___________________________________________________________________________ Q8

(Total 6 marks)

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9. (i) Given that

3 7

5

4

3

w p+ = −−i

i where p is a real constant

(a) expresswintheforma + bi , where a and b are real constants. Giveyouranswerinitssimplestformintermsofp.

(5)

Given that arg w = −π–2

(b) find the value of p.(1)

(ii) Given that

(z + 1 − 2 i)* = 4 i z

find z,givingyouranswerintheformz = x + i y, where x and y are real constants.(6)

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TOTAL FOR PAPER: 75 MARKSEND

Q9

(Total 12 marks)

pearson centre number candidate number edexcel gce further ... · edexcel gce further pure...

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