Paper Reference(s) Edexcel GCE - Pearson …. x 0 12 π π 6 π 4 π 3 y 1.3229 1.2973 1 (2) (b) Use...
Transcript of Paper Reference(s) Edexcel GCE - Pearson …. x 0 12 π π 6 π 4 π 3 y 1.3229 1.2973 1 (2) (b) Use...
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Paper Reference(s)
6666/01Edexcel GCECore Mathematics C4Advanced Friday 18 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 32 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 6 0 1
This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2010 Edexcel Limited.
Printer’s Log. No.
H35386AW850/R6666/57570 4/5/3/3
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1.
Figure 1
Figure 1 shows part of the curve with equation 2(0.75 cos ).y x= +√ The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-axis, the x-axis and the line with equation x = π
3.
(a) Complete the table with values of y corresponding to x = π6
and x = π4
.
x 0 12π π
6π4
π3
y 1.3229 1.2973 1
(2)
(b) Use the trapezium rule
(i) with the values of y at 0,x = x = π6
and x = π3
to find an estimate of the area of R.
Give your answer to 3 decimal places.
(ii) with the values of y at 0,x = x = 12π , x = π
6, x = π
4 and x = π
3 to find a
further estimate of the area of R. Give your answer to 3 decimal places.(6)
O
y
x
R
3π
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(Total 8 marks)
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2. Using the substitution cos 1,u x= + or otherwise, show that
2 cos 1
0
e sin d e(e 1)x x xπ
+ = −∫(6)
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3. A curve C has equation22 2x y xy+ =
Find the exact value of ddyx
at the point on C with coordinates (3, 2).
(7)
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(Total 7 marks)
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4. A curve C has parametric equations
2sin , 2 tan , 02
x t y t t π= = <
(a) Find ddyx
in terms of t.
(4)
The tangent to C at the point where 3
t π= cuts the x-axis at the point P.
(b) Find the x-coordinate of P.(6)
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π
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(Total 10 marks)
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5. 22 5 10
( 1)( 2) 1 2x x B CAx x x x
+ − ≡ + +− + − +
(a) Find the values of the constants A, B and C.(4)
(b) Hence, or otherwise, expand 22 5 10
( 1)( 2)x xx x
+ −− +
in ascending powers of x, as far as the
term in 2.x Give each coefficient as a simplified fraction.(7)
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(Total 11 marks)
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6. f(θ) = 4 2cos θ – 3 2sin θ
(a) Show that f(θ) 1 7 cos2 2
= + 2θ.(3)
(b) Hence, using calculus, find the exact value of 2
0
π
∫ θ f(θ) dθ.
(7)
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7. The line 1l has equation
= +⎛⎜⎜⎝
⎞⎟⎟⎠
⎛⎜⎜⎝
⎞⎟⎟⎠
λr121
23
−4, where λ is a scalar parameter.
The line 2l has equation
= +⎛⎜⎜⎝
⎞⎟⎟⎠
⎛⎜⎜⎝
⎞⎟⎟⎠
μr502
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−3, where μ is a scalar parameter.
Given that 1l and 2l meet at the point C, find
(a) the coordinates of C.(3)
The point A is the point on 1l where λ = 0 and the point B is the point on 2l where μ = –1.
(b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places.(4)
(c) Hence, or otherwise, find the area of the triangle ABC.(5)
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8.
Figure 2
Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is 6 m. Water is flowing into the tank at a constant rate of 0.48π 3 1m min .− At time t minutes, the depth of the water in the tank is h metres. There is a tap at a point T at the bottom of the tank. When the tap is open, water leaves the tank at a rate of 0.6πh 3 1m min .−
(a) Show that t minutes after the tap has been opened
d75 (4 5 )dh ht
= −(5)
When 0,t = 0.2h =
(b) Find the value of t when 0.5h =(6)
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h metres
6 metres
T
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TOTAL FOR PAPER: 75 MARKS
END
Q8
(Total 11 marks)