Centre Number For Examiner’s Use Candidate Number Surname Other Names Examiner’s Initials Candidate Signature Question General Certificate of Education Advanced Level Examination June 2010 Mark 1 2 3 Mathematics MFP3 5 Unit Further Pure 3 Friday 11 June 2010 4 6 9.00 am to 10.30 am 7 For this paper you must have: * the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. TOTAL Time allowed * 1 hour 30 minutes 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 the questions in the spaces provided. 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. 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. (JUN10MFP301) P27889/Jun10/MFP3 6/6/ MFP3 Do not write outside the box 2 Answer all questions in the spaces provided. The function yðxÞ satisfies the differential equation 1 dy ¼ f ðx, yÞ dx where and (a) f ðx, yÞ ¼ x þ 3 þ sin y yð1Þ ¼ 1 Use the Euler formula yrþ1 ¼ yr þ hf ðxr , yr Þ with h ¼ 0:1 , to obtain an approximation to yð1:1Þ , giving your answer to four decimal places. (3 marks) (b) Use the formula yrþ1 ¼ yr1 þ 2hf ðxr , yr Þ with your answer to part (a), to obtain an approximation to yð1:2Þ , giving your answer to three decimal places. (3 marks) QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. (02) P27889/Jun10/MFP3 Do not write outside the box 3 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (03) Turn over P27889/Jun10/MFP3 Do not write outside the box 4 2 (a) Find the value of the constant k for which k sin 2x is a particular integral of the differential equation d2 y þ y ¼ sin 2x dx 2 (b) Hence find the general solution of this differential equation. (3 marks) (4 marks) QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. (04) P27889/Jun10/MFP3 Do not write outside the box 5 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (05) Turn over P27889/Jun10/MFP3 Do not write outside the box 6 ð1 3 (a) Explain why 4xe4x dx is an improper integral. (1 mark) 1 ð (b) Find 4xe4x dx . ð1 (c) Hence evaluate (3 marks) 4xe4x dx , showing the limiting process used. 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(06) P27889/Jun10/MFP3 Do not write outside the box 7 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (07) Turn over P27889/Jun10/MFP3 Do not write outside the box 8 4 By using an integrating factor, find the solution of the differential equation 3 dy 3 þ y ¼ ðx 4 þ 3Þ2 dx x 1 given that y ¼ 5 when x ¼ 1 . (9 marks) QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. (08) P27889/Jun10/MFP3 Do not write outside the box 9 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (09) Turn over P27889/Jun10/MFP3 Do not write outside the box 10 Write down the expansion of cos 4x in ascending powers of x up to and including (2 marks) the term in x 4 . Give your answer in its simplest form. 5 (a) (b) (i) Given that y ¼ lnð2 e x Þ , find dy d2 y d3 y , 2 and 3 . dx dx dx ðYou may leave your expression for d3 y unsimplified.Þ dx3 (6 marks) (ii) Hence, by using Maclaurin’s theorem, show that the first three non-zero terms in the expansion, in ascending powers of x, of lnð2 e x Þ are (c) x x 2 x 3 (2 marks) x lim x lnð2 e Þ x ! 0 1 cos 4x (3 marks) Find QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. (10) P27889/Jun10/MFP3 Do not write outside the box 11 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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................................................................................................................................................................. ................................................................................................................................................................. s (11) Turn over P27889/Jun10/MFP3 Do not write outside the box 12 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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(12) P27889/Jun10/MFP3 Do not write outside the box 13 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (13) Turn over P27889/Jun10/MFP3 Do not write outside the box 14 The polar equation of a curve C1 is 6 r ¼ 2ðcos y sin yÞ, (a) (i) 0 4 y 4 2p Find the cartesian equation of C1 . (4 marks) (ii) Deduce that C1 is a circle and find its radius and the cartesian coordinates of its centre. (3 marks) The diagram shows the curve C2 with polar equation (b) r ¼ 4 þ sin y, 0 4 y 4 2p O (i) Initial line Find the area of the region that is bounded by C2 . (6 marks) (ii) Prove that the curves C1 and C2 do not intersect. (4 marks) (iii) Find the area of the region that is outside C1 but inside C2 . 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(14) P27889/Jun10/MFP3 Do not write outside the box 15 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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................................................................................................................................................................. ................................................................................................................................................................. s (15) Turn over P27889/Jun10/MFP3 Do not write outside the box 16 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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(16) P27889/Jun10/MFP3 Do not write outside the box 17 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. s (17) Turn over P27889/Jun10/MFP3 Do not write outside the box 18 1 Given that x ¼ t 2 , x > 0 , t > 0 and y is a function of x, show that: 7 (a) (i) 1 dy dy ¼ 2t 2 ; dx dt (2 marks) (ii) d2 y d2 y dy ¼ 4t þ2 . 2 2 dx dt dt (3 marks) 1 (b) Hence show that the substitution x ¼ t 2 transforms the differential equation x d2 y dy 2 þ 12x 3 y ¼ 12x 5 ð8x þ 1Þ 2 dx dx into d2 y dy 4 þ 3y ¼ 3t 2 dt dt (c) (2 marks) Hence find the general solution of the differential equation x d2 y dy 2 ð8x þ 1Þ þ 12x 3 y ¼ 12x 5 2 dx dx giving your answer in the form y ¼ f ðxÞ . (7 marks) QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. (18) P27889/Jun10/MFP3 Do not write outside the box 19 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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................................................................................................................................................................. ................................................................................................................................................................. s (19) Turn over P27889/Jun10/MFP3 Do not write outside the box 20 QUESTION PART REFERENCE ................................................................................................................................................................. ................................................................................................................................................................. ................................................................................................................................................................. 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END OF QUESTIONS Copyright Ó 2010 AQA and its licensors. All rights reserved. (20) P27889/Jun10/MFP3
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