paper reference(s) edexcel gce...2011/06/22 · f( ),xx x =+ − − ≠2 xx 5 2 31 0 leave blank...

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Centre No.

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced/Advanced SubsidiaryWednesday 22 June 2011 – MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) Nil

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 or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them.

Instructions to CandidatesIn 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 CandidatesA 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 CandidatesYou 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.

Printer’s Log. No.

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

*P38168A0132*

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*P38168A0232*

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 of width 0.25 which contains .

(3)

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f ( ) 3 3 7xx x= + −

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___________________________________________________________________________ Q1

(Total 5 marks)

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*P38168A0432*

2.

(a) Find the modulus of 1.z(1)

(b) Find, in radians, the argument of 1,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) Find 2z and 3,z giving your answers in the form i p q± , where p and q are integers.(3)

(d) Show, on an Argand diagram, the points representing your complex numbers 1 2,z z and 3 .z

(2)

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z1 2= − + i

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___________________________________________________________________________ Q2

(Total 8 marks)

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*P38168A0832*

3. (a) Given that1 22 1

⎛ ⎞=⎜ ⎟−⎝ ⎠

A

(i) find 2 ,A

(ii) describe fully the geometrical transformation represented by 2.A(4)

(b) Given that

B =−

−⎛⎝⎜

⎞⎠⎟

0 101

describe fully the geometrical transformation represented by B.(2)

(c) Given that1 12

9k

k+⎛ ⎞

= ⎜ ⎟⎝ ⎠

C

where k is a constant, find the value of k for which the matrix C is singular.(3)

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___________________________________________________________________________ Q3

(Total 9 marks)

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*P38168A01232*

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

x x= + − − ≠2 52

3 1 0

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(Total 6 marks)

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*P38168A01432*

5. where a and b are constants.

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

(a) find the value of a and the value of b.(4)

A quadrilateral R has area 30 square units. It is transformed into another quadrilateral S by the matrix A. Using your values of a and b,

(b) find the area of quadrilateral S.(4)

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A =−

−⎛⎝⎜

⎞⎠⎟

42

ab

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(Total 8 marks)

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*P38168A01832*

6. Given that find the value of x and the value of y such that

z z+ = − +∗3 1 13 i i

where z∗ is the complex conjugate of z.(7)

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i ,z x y= +

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(Total 7 marks)

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*P38168A02032*

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

n

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(Total 10 marks)

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*P38168A02432*

8. The parabola C has equation

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

(a) Find the equation of the directrix of C.(2)

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

2 0− +y t 1x t =2

(4)

The tangent to C at the point ( )3, 12 meets the directrix of C at the point X.

(c) Find the coordinates of X.(4)

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(Total 10 marks)

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*P38168A02832*

9. Prove by induction, that for

(a) 3 06 1

3 03 3 1 1

⎛⎝⎜

⎞⎠⎟

=−

⎛⎝⎜

⎞⎠⎟

n n

n( ),

(6)

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

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TOTAL FOR PAPER: 75 MARKS

END

Q9

(Total 12 marks)

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