Download - FP2 QP Jun 2008
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Examiner’s use only
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Centre No.
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Candidate No.
Paper Reference(s)
6675/01Edexcel GCEFurther Pure Mathematics FP2Advanced/Advanced SubsidiaryWednesday 18 June 2008 – MorningTime: 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.
Question Leave Number Blank
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Paper Reference
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This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2008 Edexcel Limited.
Printer’s Log. No.
N30746AW850/R6675/57570 3/3/6/3
*N30746A0128*
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. Write your answers in the spaces provided in this question paper.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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1. Show that
(4)
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ddx
x[ ]ln(tanh ) = 22 cosech 2x, x > 0.
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(Total 4 marks)
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2. Find the values of x for which
8 cosh x – 4 sinh x = 13,
giving your answers as natural logarithms. (6)
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(Total 6 marks)
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3. Show that
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3 + x dx = 31n 2 + 3 + 3 3 – 4.(x2 – 9) 3 5
6
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(Total 7 marks)
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4. The curve C has equation
y = arsinh (x3), x 0.
The point P on C has x-coordinate √2.
(a) Show that an equation of the tangent to C at P is
y = 2x – 2√2 + ln (3 + 2√2). (5)
The tangent to C at the point Q is parallel to the tangent to C at P.
(b) Find the x-coordinate of Q, giving your answer to 2 decimal places. (5)
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(Total 10 marks)
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5. Given that
,
(a) show that, for n 2,
(8)
(b) Find the exact value of I4. (4)
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In n
nIn n= −
+ −( ) .1
12 2
I x x nnx n= ∫ e dsin
00
Àπ
0x dx,
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(Total 12 marks)
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6.
Figure 1
Figure 1 shows the curve C with equation
The shaded region R is bounded by C, the x-axis and the line x = 2.
(a) Find . (5)
(b) Hence show that, to 2 significant figures, the area of R is 0.34 (2)
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y
xO 2
C
R
cosh arctan(sinh )x x x∫ dcosh x arctan (sinh x) dx .
y = 110
cosh x arctan (sinh x) , x 0.
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(Total 7 marks)
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7. The hyperbola H has equation
(a) Show that an equation for the normal to H at a point P (4 sec t, 3 tan t) is
4x sin t + 3y = 25 tan t. (6)
The point S, which lies on the positive x-axis, is a focus of H. Given that PS is parallel to the y-axis and that the y-coordinate of P is positive,
(b) find the values of the coordinates of P. (5)
Given that the normal to H at this point P intersects the x-axis at the point R,
(c) find the area of triangle PRS. (3)
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x y2 2
16 91− = .
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(Total 14 marks)
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8. The curve C has parametric equations
(a) Show that (3)
The arc length s of C is measured from the origin O.
(b) Show that (4)
(c) Hence write down the intrinsic equation of C in the form . (1)
The point P lies on C and the arc OP of C has length L. The arc OP is rotated through 2π radians about the x-axis.
(d) Show that the area of the curved surface generated is given by
(7)
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ddy
x
t= tan .2
st=122
sin .
s = f (ψ)
L 3
36.
π
π.x = 3(t + sin t), y = 3(1 – cos t), 0 t <
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Question 8 continued
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 15 marks)
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