General Certificate of Education Advanced Level Examination June 2012
Mathematics
MFP3
Unit Further Pure 3 Thursday 14 June 2012
9.00 am to 10.30 am
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For this paper you must have: * the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator.
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Time allowed * 1 hour 30 minutes
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Instructions * Use black ink or black ball-point pen. Pencil should only be used for drawing. * Fill in the boxes at the top of this page. * Answer all questions. * Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. * You must answer each question in the space provided for that question. If you require extra space, use an AQA supplementary answer book; do not use the space provided for a different question. * Do not write outside the box around each page. * Show all necessary working; otherwise marks for method may be lost. * Do all rough work in this book. Cross through any work that you do not want to be marked.
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Information * The marks for questions are shown in brackets. * The maximum mark for this paper is 75. Advice * Unless stated otherwise, you may quote formulae, without proof, from the booklet. * You do not necessarily need to use all the space provided.
P50013/Jun12/MFP3 6/6/6/
MFP3
2
The function yðxÞ satisfies the differential equation
1
dy ¼ f ðx, yÞ dx pffiffiffiffiffiffiffiffiffi pffiffiffi f ðx, yÞ ¼ ð2xÞ þ y
where
yð2Þ ¼ 9
and Use the improved Euler formula
yrþ1 ¼ yr þ 1ðk1 þ k2 Þ 2
where k1 ¼ hf ðxr , yr Þ and k2 ¼ hf ðxr þ h, yr þ k1 Þ and h ¼ 0:25 , to obtain an approximation to yð2:25Þ , giving your answer to two decimal places. (5 marks)
2 (a)
(b)
Write down the expansion of sin 2x in ascending powers of x up to and including (1 mark) the term in x 5 . Show that, for some value of k, 2x sin 2x lim ¼ 16 x ! 0 x 2 lnð1 þ kxÞ and state this value of k.
3
(4 marks)
The diagram shows a sketch of a curve C, the pole O and the initial line.
O
Initial line
The polar equation of C is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ 2 1 þ tan y ,
p p 4y4 4 4
Show that the area of the shaded region, bounded by the curve C and the initial line, p is ln 2 . (4 marks) 2
(02)
P50013/Jun12/MFP3
3 4 (a)
By using an integrating factor, find the general solution of the differential equation dy 4 þ y ¼ 4ð2x þ 1Þ5 dx 2x þ 1 giving your answer in the form y ¼ f ðxÞ .
(b)
(7 marks)
The gradient of a curve at any point ðx, yÞ on the curve is given by the differential equation dy 4 ¼ 4ð2x þ 1Þ5 y 2x þ 1 dx The point whose x-coordinate is zero is a stationary point of the curve. Using your answer to part (a), find the equation of the curve. (3 marks) ð
5 (a)
Find
x 2 ex dx .
(4 marks)
ð1 (b)
Hence evaluate
x 2 ex dx , showing the limiting process used.
(3 marks)
0
It is given that y ¼ lnð1 þ sin xÞ .
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dy . dx
(2 marks)
(a)
Find
(b)
Show that
(c)
Express
(d)
Hence, by using Maclaurin’s theorem, find the first four non-zero terms in the expansion, in ascending powers of x, of lnð1 þ sin xÞ . (3 marks)
d4 y dy and ey . in terms of 4 dx dx
(3 marks)
(3 marks)
Turn over
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(03)
d2 y ¼ ey . dx 2
P50013/Jun12/MFP3
4
Show that the substitution x ¼ et transforms the differential equation
7 (a)
d2 y dy x 4x þ 6y ¼ 3 þ 20 sinðln xÞ 2 dx dx 2
d2 y dy þ 6y ¼ 3 þ 20 sin t 5 dt 2 dt
into
(7 marks)
Find the general solution of the differential equation
(b)
d2 y dy 5 þ 6y ¼ 3 þ 20 sin t dt 2 dt
(11 marks)
Write down the general solution of the differential equation
(c)
d2 y dy x 4x þ 6y ¼ 3 þ 20 sinðln xÞ 2 dx dx 2
(1 mark)
A curve has cartesian equation xy ¼ 8 . Show that the polar equation of the curve is (3 marks) r 2 ¼ 16 cosec 2y .
8 (a)
The diagram shows a sketch of the curve, C, whose polar equation is
(b)
r 2 ¼ 16 cosec 2y ,
O
0
