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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1

Advanced/Advanced Subsidiary

Wednesday 22 June 2011 Morning

Time: 1 hour 30 minutes

Materials required for examination Items included with question papers

Mathematical Formulae (Pink) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

Instructions to Candidates

In the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer to each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates

A booklet Mathematical Formulae and Statistical Tables is provided.

Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75.There are 32 pages in this question paper. Any blank pages are indicated.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner.Answers without working may not gain full credit.

Paper Reference

6 6 6 7 0 1

This publication may be reproduced only in accordance with

Edexcel Limited copyright policy.2011 Edexcel Limited.

Printers Log. No.

P38168AW850/R6667/57570 5/5/5/3

*P38168A0132*

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1.

(a) Show that the equation f ( ) 0x = has a root between 1x = and 2.x =(2)

(b) Starting with the interval [ ]1, 2 , use interval bisection twice to find an interval ofwidth 0.25 which contains .

(3)

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f ( ) 3 3 7xx x= +

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Question 1 continued

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___________________________________________________________________________ Q1

(Total 5 marks)

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2.

(a) Find the modulus of1.z

(1)

(b) Find, in radians, the argument of1,z giving your answer to 2 decimal places.

(2)

The solutions to the quadratic equation

2 10 28 0z z + =

are 2z and 3.z

(c) Find2

z and3,z giving your answers in the form ip q , wherep and q are integers.

(3)

(d) Show, on an Argand diagram, the points representing your complex numbers1 2,z z

and3.z

(2)

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z1

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Question 2 continued

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Question 2 continued

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___________________________________________________________________________ Q2

(Total 8 marks)

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3. (a) Given that

1 2

2 1

=

A

(i) find 2 ,A

(ii) describe fully the geometrical transformation represented by 2.A

(4)

(b) Given that

B =

0 1

01

describe fully the geometrical transformation represented by B.

(2)

(c) Given that

1 12

9

k

k

+ =

C

where kis a constant, find the value ofkfor which the matrix C is singular.

(3)

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Question 3 continued

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(Total 9 marks)

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4.

(a) Use differentiation to find f ( ).x(2)

The root of the equation f ( ) 0x = lies in the interval [ ]0.7, 0.9 .

(b) Taking 0.8 as a first approximation to ,apply the Newton-Raphson process once to

f ( )x to obtain a second approximation to . Give your answer to 3 decimal places.

(4)

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f ( ) ,x xx

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23 1 0

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Question 4 continued

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(Total 6 marks)

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5. where a and b are constants.

Given that the matrix A maps the point with coordinates (4, 6) onto the point withcoordinates (2 , 8) ,

(a) find the value ofa and the value ofb. (4)

A quadrilateralR has area 30 square units.

It is transformed into another quadrilateral Sby the matrix A.

Using your values ofa and b,

(b) find the area of quadrilateral S.

(4)

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A =

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2

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Question 5 continued

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6. Given that find the value ofx and the value ofy such that

z z+ = +3 1 13i i

where z is the complex conjugate ofz.

(7)

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Question 6 continued

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(Total 7 marks)

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7. (a) Use the results for and to show that

2

1

1(2 1) (2 1)(2 1)

3

n

r

r n n n

=

= +

for all positive integers n.(6)

(b) Hence show that

( )3

2 2

1

2(2 1)

3

n

r n

r n an b

= +

= +

where a and b are integers to be found.

(4)

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1

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Question 7 continued

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Question 7 continued

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Question 7 continued

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(Total 10 marks)

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8. The parabola Chas equation

The point P t t12 242 ,( ) is a general point on C.

(a) Find the equation of the directrix ofC.

(2)

(b) Show that the equation of the tangent to Cat P t t12 242 ,( ) is

2 0 +y t1x t =2

(4)

The tangent to Cat the point ( )3, 12 meets the directrix ofCat the pointX.

(c) Find the coordinates ofX.

(4)

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Question 8 continued

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Question 8 continued

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Question 8 continued

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(Total 10 marks)

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9. Prove by induction, that for

(a)3 0

6 1

3 0

3 3 1 1

=

n n

n( ),

(6)

(b) 2 1f ( ) 7 5nn = + is divisible by 12.(6)

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Question 9 continued

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Question 9 continued

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*P38168A03132* Turn over

Question 9 continued

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Question 9 continued

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TOTAL FOR PAPER: 75 MARKS

Q9

(Total 12 marks)