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Examiners use only

Team Leaders use only

Surname Initial(s)

Signature

Centre

No.

Turn over

Candidate

No.

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1

2

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6

7

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Paper Reference(s)

6665/01

Edexcel GCECore Mathematics C3

Advanced

Friday 6 June 2008 Afternoon

Time: 1 hour 30 minutes

Materials required for examination Items included with question papers

Mathematical Formulae (Green) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical 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 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 Candidates

A booklet Mathematical Formulae and Statist ical 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 7 questions in this question paper. The total mark for this paper is 75.There are 24 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 5 0 1

This publication may be reproduced only in accordance withEdexcel Limited copyright policy.

2008 Edexcel Limited.

Printers Log. No.

N30745AW850/R6665/57570 3/3/3/3

*N30745A0124*

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1. The pointPlies on the curve with equation

They-coordinate ofPis 8.

(a) Find, in terms of ln 2, thex-coordinate ofP.

(2)

(b) Find the equation of the tangent to the curve at the pointPin the form y=ax + b,

where a and b are exact constants to be found.

(4)

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y ex

=+

42 1

.

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(Total 6 marks)

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2.

Given that whereR> 0 and

(a) find the value ofR and the value of to 3 decimal places.

(4)

(b) Hence solve the equation

for 0 x< 2.

(5)

(c) (i) Write down the maximum value of

(1)

(ii) Find the smallest positive value ofx for which this maximum value occurs.

(2)

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f ( ) cos sinx x x= +5 12

f ( ) cos( ),x R x= 0 <

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Question 2 continued

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(Total 12 marks)

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3.

Figure 1

Figure 1 shows the graph of

The graph consists of two line segments that meet at the pointP.

The graph cuts they-axis at the point Q and thex-axis at the points (3, 0) andR.

Sketch, on separate diagrams, the graphs of

(a)

(2)

(b) y = f (x).

(2)

Given that

(c) find the coordinates of the points P, Q andR,

(3)

(d) solve

(5)

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xR

P

Q

y

3

y x x= f ( ), .\

y x= f ( ) ,

f ( ) ,x x= +2 1

f ( ) .x x=1

2

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(Total 12 marks)

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4. The function f is defined by

(a) Show that

(4)

(b) Find the range of f.

(2)

(c) Find f1 (x). State the domain of this inverse function.

(3)

The function g is defined by

(d) Solve

(3)

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f :( )

, .xx

x x xx6

2 1

2 3

1

33

2

>

f ( ) , .xx

x=+

>1

13

g : , .x x x6 2 32 \

fg( ) .x =1

8

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5. (a) Given that sin2 + cos2 1, show that 1 + cot2cosec2.(2)

(b) Solve, for 0 < 180, the equation

2 cot2 9 cosec= 3,giving your answers to 1 decimal place.

(6)

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(Total 8 marks)

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6. (a) Differentiate with respect tox,

(i)

(3)

(ii)

(3)

Given that

(b) show that

(5)

(c) Hence find and the real values ofx for which

(3)

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e3 2x x x(sin cos ) ,+

yx x

xx=

+

+

3 6 7

11

2

2( ), ,

d

d

y

x x=

+

20

1 3( ).

d

d

2y

x2d

d

2y

x215

4= .

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(Total 14 marks)

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7.

(a) Show that f (x) = 0 has a root, , betweenx = 1.4 andx = 1.45

(2)

(b) Show that the equation f (x) = 0 can be written as

(3)

(c) Starting withx0=1.43, use the iteration

xn+1 x

n

2 2

3( )= +

to calculate the values ofx1,x

2andx

3, giving your answers to 4 decimal places.

(3)

(d) By choosing a suitable interval, show that = 1.435 is correct to 3 decimal places.

(3)

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f ( )x x x= 3 2 63

xx

x 2 2

30, .( )= +

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Question 7 continued

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Question 7 continued

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TOTAL FOR PAPER: 75 MARKS

END

Q7

(Total 11 marks)