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Paper Reference(s)

6669/01Edexcel GCEFurther Pure Mathematics FP3Advanced/Advanced SubsidiaryMonday 24 June 2013 – AfternoonTime: 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 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.

This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. ©2013 Pearson Education Ltd.

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P43143AW850/R6668/57570 5/5/5/5

*P43143A0128*

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1. A hyperbola H has equation

xa

y2

2

2

251− = , .where is a positive constanta

The foci of H are at the points with coordinates (13, 0) and (–13, 0).

Find

(a) the value of the constant a,(3)

(b) the equations of the directrices of H. (3)

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(Total 6 marks)

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*P43143A0428*

2. (a) Find

∫ 14 92√ ( )x

x+

d

(2)

(b) Use your answer to part (a) to find the exact value of

−∫

3

3 14 92√ ( )x

x+

d

giving your answer in the form k ln(a + b �5), where a and b are integers and k is a constant.

(3)

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(Total 5 marks)

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*P43143A0828*

3. The curve with parametric equations

x = cosh 2�, y = 4 sinh �, 0 �� 1

is rotated through 2� radians about the x-axis.

Show that the area of the surface generated is �(cosh3 �� 1), where �� = 1 and � is a constant to be found.

(7)

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(Total 7 marks)

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*P43143A01028*

4.

Figure 1

Figure 1 shows part of the curve with equation

y = 40 arcosh x – 9x, x ��1

Use calculus to find the exact coordinates of the turning point of the curve, giving your

answer in the form pq

r s, ln ,3 +⎛⎝⎜

⎞⎠⎟

where p, q, r and s are integers. (7)

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(Total 7 marks)

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*P43143A01228*

5. The matrix M is given by

M =−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

1 121 0 1

ab c , .where , and are constantsa b c

(a) Given that j + k and i – k are two of the eigenvectors of M,

find

(i) the values of a, b and c,

(ii) the eigenvalues which correspond to the two given eigenvectors.(8)

(b) The matrix P is given by

P =−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

≠ −1 1 02 11 0 1

1d d d, where is constant,

Find

(i) the determinant of P in terms of d,

(ii) the matrix P–1 in terms of d.(5)

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(Total 13 marks)

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*P43143A01628*

6. Given that

In =0

4

∫ x x x nn�( ) , ,16 02− d �

(a) prove that, for n ��2,

( ) ( )n I n In n+ = − −2 16 1 2

(6)

(b) Hence, showing each step of your working, find the exact value of I5(5)

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(Total 11 marks)

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*P43143A02028*

7. The ellipse E has equation

xa

yb

a b2

2

2

2 1 0+ = > >,

The line l is a normal to E at a point ( cos , sin ), 02πP a θ b θ θ< <

(a) Using calculus, show that an equation for l is

axsin� – bycos� = (a2 – b2) sin��cos�(5)

The line l meets the x-axis at A and the y-axis at B.

(b) Show that the area of the triangle OAB, where O is the origin, may be written as ksin2�, giving the value of the constant k in terms of a and b.

(4)

(c) Find, in terms of a and b, the exact coordinates of the point P, for which the area of the triangle OAB is a maximum.

(3)

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(Total 12 marks)

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*P43143A02428*

8. The plane 1 has vector equation

r.(3i – 4j + 2k) = 5

(a) Find the perpendicular distance from the point (6, 2, 12) to the plane 1

(3)

The plane 2 has vector equation

r = �(2i + j + 5k) + �(i – j – 2k), where � and � are scalar parameters.

(b) Find the acute angle between 1 and 2 giving your answer to the nearest degree.(5)

(c) Find an equation of the line of intersection of the two planes in the form r × a = b, where a and b are constant vectors.

(6)

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 14 marks)

paper reference(s) edexcel gce - pearson … that you have the correct question paper. answer all...

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