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Paper Reference(s)
6665/01Edexcel GCECore Mathematics C3Advanced Friday 6 June 2008 – 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 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 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 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 5 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2008 Edexcel Limited.
Printer’s Log. No.
N30745AW850/R6665/57570 3/3/3/3
*N30745A0124*
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1. The point P lies on the curve with equation
The y-coordinate of P is 8.
(a) Find, in terms of ln 2, the x-coordinate of P.(2)
(b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found.
(4)
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y e x= +4 2 1.
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(Total 6 marks)
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2.
Given that where R > 0 and
(a) find the value of R and the value of α to 3 decimal places.(4)
(b) Hence solve the equation
for 0 x < 2π.(5)
(c) (i) Write down the maximum value of (1)
(ii) Find the smallest positive value of x for which this maximum value occurs.(2)
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f ( ) cos sinx x x= +5 12
f ( ) cos( ),x R x= −α 0 < <α π2
,
5 12 6cos sinx x+ =
5 12cos sin .x x+
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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 point P. The graph cuts the y-axis at the point Q and the x-axis at the points (–3, 0) and R. 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 and R,(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=12
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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 f –1 (x). State the domain of this inverse function.(3)
The function g is defined by
(d) Solve (3)
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f : ( ) , .x xx x x
x2 12 3
13
32
−− −
−−
>
f ( ) , .xx
x=+
>1
13
g : , .x x x2 32 − ∈
fg( ) .x =18
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5. (a) Given that sin2 θ + cos2 θ ≡ 1, show that 1 + cot2 θ ≡ cosec2 θ .(2)
(b) Solve, for 0 θ < 180°, the equation
2 cot2 θ – 9 cosec θ = 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 to x,
(i) (3)
(ii) (3)
Given that
(b) show that (5)
(c) Hence find and the real values of x for which
(3)
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e3 2x x x(sin cos ) ,+
y x xx
x=+ −+
≠ −3 6 7
11
2
2( ), ,
ddyx x=
+20
1 3( ).
dd
2 yx2
dd
2 yx2
154
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7.
(a) Show that f (x) = 0 has a root, α, between x = 1.4 and x = 1.45(2)
(b) Show that the equation f (x) = 0 can be written as
(3)
(c) Starting with x0=1.43, use the iteration
xn+1 xn
2 23( )= +
to calculate the values of x1, x2 and x3, giving your answers to 4 decimal places.(3)
(d) By choosing a suitable interval, show that α = 1.435 is correct to 3 decimal places.(3)
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f ( )x x x= − −3 2 63
xx
x ≠2 23
0, .( )= +
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TOTAL FOR PAPER: 75 MARKSEND
Q7
(Total 11 marks)