Paper Reference(s) 6676/01 Edexcel GCE Level... · 3. Given that 0 1 −1 ⎛ ⎝ ⎜ ⎜ ⎞ ......

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Paper Reference

6 6 7 6 0 1 Paper Reference(s)

6676/01Edexcel GCEFurther Pure Mathematics FP3Advanced/Advanced SubsidiaryTuesday 26 June 2007 – AfternoonTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Green) 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, differentiation and integration, or have retrievable mathematical formulas stored in them.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initial(s) and signature.Check that you have the correct question paper.You must write your answer for 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 8 questions in this question paper. The total mark for this paper is 75.There are 28 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.

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*N22336A0128*Turn over

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This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited.

Printer’s Log. No.

N22336AW850/R6676/57570 3/3/3/3/3/

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*N22336A0228*

1. 2d e .

dxy y

x=

It is given that y = 0.2 at x = 0.

(a) Use the approximation y y

hyx

1 0

0

− ≈⎛⎝⎜

⎞⎠⎟

dd

, with h = 0.1, to obtain an estimate of

the value of y at x = 0.1.(2)

(b) Use your answer to part (a) and the approximation y y

hyx

2 0

12− ≈

⎛⎝⎜

⎞⎠⎟

dd

, with

h = 0.1, to obtain an estimate of the value of y at x = 0.2.

Give your answer to 4 decimal places.(3)

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Question 1 continued

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___________________________________________________________________________ Q1

(Total 5 marks)

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*N22336A0428*

2. 2

22

d d(1 ) 2 0.d d

y yx x yx x

− − + =

At x = 0, y = 2 and

(a) Find the value of 3

3

dd

yx

at x = 0.(3)

(b) Express y as a series in ascending powers of x, up to and including the term in 3.x(4)

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ddyx= −1.

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(Total 7 marks)

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*N22336A0628*

3. Given that 0

1

1−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

is an eigenvector of the matrix A, where

A =⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

3 4

−1 −4

1 1 3

pq ,

(a) find the eigenvalue of A corresponding to 0

1

1−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

,

(2)

(b) find the value of p and the value of q.(4)

The image of the vector lmn

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ when transformed by A is

10

4

3

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

.

(c) Using the values of p and q from part (b), find the values of the constants l, m and n.(4)

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(Total 10 marks)

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*N22336A01028*

4. (a) Given that z i= +cos sinθ θ , use de Moivre’s theorem to show that

zz

nnn+ =

1 2cos .θ

(2)

(b) Express 326

cos θ in the form p q r scos cos cos6 4 2θ θ θ+ + + , where p, q, r and s are integers.

(5)

(c) Hence find the exact value of

cos .6

0

3 θ θπ

d∫(4)

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*N22336A01428*

5. Prove by induction that, for n +∈ ,

(5)

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( )2 1 2

1r

r

n

− ==∑ 1

3 ( )( ).2 1 2 1n n n− +

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(Total 5 marks)

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*N22336A01628*

6. Given that

(a) show that, for k +∈ , f (k + 1) – f (k) is divisible by 15,(4)

(b) prove that, for n +∈ , f (n) is divisible by 5,(3)

(c) show that it is not true that, for all positive integers n, f (n) is divisible by 15.(1)

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f n n n( )= + +3 2

4 4 2,

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(Total 8 marks)

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*N22336A02028*

7. The points A, B and C have position vectors, relative to a fixed origin O,

respectively. The plane Π passes through A, B and C.

(a) Find AB AC× .(4)

(b) Show that a cartesian equation of Π is 3 2 7.x y z− + =(2)

The line l has equation (r – 5i – 5j – 3k) × (2i – j – 2k) = 0. The line l and the

plane Π intersect at the point T.

(c) Find the coordinates of T.(5)

(d) Show that A, B and T lie on the same straight line.(3)

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a = 2i – j,

b = i + 2j + 3k,

c = 2i + 3j + 2k,

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*N22336A02428*

8. The transformation T from the z-plane, where z = x + iy, to the w-plane, where

w = u + iv, is given by

i , 0.zw zz+= ≠

(a) The transformation T maps the points on the line with equation y x= in the z-plane, other than (0, 0), to points on a line l in the w-plane. Find a cartesian equation of l.

(5)

(b) Show that the image, under T, of the line with equation 1 0x y+ + = in the z-plane is a circle C in the w-plane, where C has cartesian equation

2 2 0.u v u v+ − + =(7)

(c) On the same Argand diagram, sketch l and C.(3)

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 15 marks)

Paper Reference(s) 6676/01 Edexcel GCE Level... · 3. Given that 0 1 −1 ⎛ ⎝ ⎜ ⎜ ⎞ ......

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Transcript of Paper Reference(s) 6676/01 Edexcel GCE Level... · 3. Given that 0 1 −1 ⎛ ⎝ ⎜ ⎜ ⎞ ......