FP3-P6 2010-06

Examiner’s use only

Team Leader’s use only

Surname Initial(s)

Signature

Centre No.

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8

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Paper Reference(s)

6669/01Edexcel GCEFurther Pure Mathematics FP3Advanced/Advanced SubsidiaryMonday 28 June 2010 – 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, differentiation and 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 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.

Paper Reference

6 6 6 9 0 1

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2010 Edexcel Limited.

Printer’s Log. No.

N35389RAW850/R6669/57570 4/5/5/3

*N35389RA0128*

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1. The line 8x = is a directrix of the ellipse with equation

and the point (2, 0) is the corresponding focus.

Find the value of a and the value of b.(5)

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2 2

2 2 1, 0,x y aa b

+ = > b >0,

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___________________________________________________________________________ Q1

(Total 5 marks)

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2. Use calculus to find the exact value of

1

22

1 d4 13

x .x x− + +∫ (5)

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(Total 5 marks)

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3. (a) Starting from the definitions of sinh x and cosh x in terms of exponentials, prove that

2cosh 2 1 2sinhx x= +(3)

(b) Solve the equationcosh 2 3sinh 15,x x− =

giving your answers as exact logarithms.(5)

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(Total 8 marks)

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4. I0

a

n ∫ (a – x)n cos x dx, a > 0, n 0

(a) Show that, for 2n , In = nan – 1 – n(n – 1)In

– 2

(5)

(b) Hence evaluate

π

π−

−

2

22

x0

cos x dx.(3)

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(Total 8 marks)

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5. Given that y x= ( )arcosh 32 , where 3 1x > , show that

(a) (9x2 – 1) 2

d36

d

yy

x

⎛ ⎞=⎜ ⎟

⎝ ⎠,

(5)

(b) (9x2 – 1) 2

2

d d9 18

d d

y yx

x x+ = .

(4)

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Question 5 continued

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(Total 9 marks)

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*N35389RA01628*

6. 1 0 3

0 2 1

0 1k

⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

M , where k is a constant.

Given that 6

1

6

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

is an eigenvector of M,

(a) find the eigenvalue of M corresponding to 6

1

6

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

,(2)

(b) show that 3k = ,(2)

(c) show that M has exactly two eigenvalues. (4)

A transformation 3 3:T → is represented by M.

The transformation T maps the line 1l , with cartesian equations 2 1

1 3 4

x y z− += =−

, onto the line 2l .

(d) Taking 3k = , find cartesian equations of 2l .(5)

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Question 6 continued

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(Total 13 marks)

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*N35389RA02028*

7. The plane Π has vector equation

r = 3i + k + λ (–4i + j) + μ (6i – 2j + k)

(a) Find an equation of Π in the form r.n p= , where n is a vector perpendicular to Π and p is a constant.

(5)

The point P has coordinates (6, 13, 5). The line l passes through P and is perpendicular to Π. The line l intersects Π at the point N.

(b) Show that the coordinates of N are (3, 1, –1).(4)

The point R lies on Π and has coordinates (1, 0, 2).

(c) Find the perpendicular distance from N to the line PR. Give your answer to 3 significant figures.

(5)

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Question 7 continued

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___________________________________________________________________________ Q7

(Total 14 marks)

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*N35389RA02428*

8. The hyperbola H has equation 2 2

116 4

x y− = .

The line 1l is the tangent to H at the point P (4 sec t, 2 tan t).

(a) Use calculus to show that an equation of 1l is

2 sin 4cosy t x t= −(5)

The line 2l passes through the origin and is perpendicular to 1l .

The lines 1l and 2l intersect at the point Q.

(b) Show that, as t varies, an equation of the locus of Q is

x y x y2 2 2 2 216 4+( ) = −(8)

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*N35389RA02628*

Question 8 continued

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*N35389RA02728* Turn over

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*N35389RA02828*

Question 8 continued

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TOTAL FOR PAPER: 75 MARKSEND

Q8

(Total 13 marks)