AQA MFP3 p QP JUN15

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 2015 Mark Mathematics MFP3 Unit Further Pure Wednesday 13 May 2015 9.00 am to 10.30 am For this paper you must have: * the blue AQA booklet of formulae and statistical tables You may use a graphics calculator TOTAL Time allowed * 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 each question in the space provided for that question If you require extra space, use an AQA supplementary answer book; 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 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 * You not necessarily need to use all the space provided (JUN15MFP301) P88829/Jun15/E3 MFP3 Do not write outside the box Answer all questions Answer each question in the space provided for that question It is given that yðxÞ satisfies the differential equation dy ¼ f ðx, yÞ dx where and f ðx, yÞ ¼ x þ y2 x yð2Þ ¼ Use the Euler formula (a) yrþ1 ¼ yr þ hf ðxr , yr Þ with h ¼ 0:05 , to obtain an approximation to yð2:05Þ [2 marks] Use the formula (b) yrþ1 ¼ yrÀ1 þ 2hf ðxr , yr Þ with your answer to part (a), to obtain an approximation to yð2:1Þ, giving your answer to three significant figures [3 marks] QUESTION PART REFERENCE Answer space for question (02) P/Jun15/MFP3 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (03) Turn over P/Jun15/MFP3 Do not write outside the box By using an integrating factor, find the solution of the differential equation dy þ ðtan xÞy ¼ tan3 x sec x dx given that y ¼ when x ¼ QUESTION PART REFERENCE p [9 marks] Answer space for question (04) P/Jun15/MFP3 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (05) Turn over P/Jun15/MFP3 Do not write outside the box (a) (i) Write down the expansion of lnð1 þ 2xÞ in ascending powers of x up to and including the term in x [1 mark] (ii) Hence, or otherwise, find the first two non-zero terms in the expansion of ln½ð1 þ 2xÞð1 À 2xފ in ascending powers of x and state the range of values of x for which the expansion is valid [3 marks] Find xlim fi0 (b) pffiffiffiffiffiffiffiffiffiffiffi ! 3x À x þ x ln½ð1 þ 2xÞð1 À 2xފ [4 marks] QUESTION PART REFERENCE Answer space for question (06) P/Jun15/MFP3 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (07) Turn over P/Jun15/MFP3 Do not write outside the box ð1 Explain why (a) À x À 2ÞeÀ2x dx is an improper integral ð1 Evaluate (b) À [1 mark] x À 2ÞeÀ2x dx , showing the limiting process used [6 marks] QUESTION PART REFERENCE Answer space for question (08) P/Jun15/MFP3 Do not write outside the box QUESTION PART REFERENCE Answer space for question s (09) Turn over P/Jun15/MFP3 Do not write outside the box 10 Find the general solution of the differential equation (a) d2 y dy þ 9y ¼ 36 sin 3x þ dx dx [7 marks] It is given that y ¼ f ðxÞ is the solution of the differential equation (b) d2 y dy þ 9y ¼ 36 sin 3x þ dx dx such that f ð0Þ ¼ and f ð0Þ ¼ (i) 00 Show that f ð0Þ ¼ [1 mark] (ii) Find the first two non-zero terms in the expansion, in ascending powers of x, of f ðxÞ [3 marks] QUESTION PART REFERENCE Answer space for question (10) P/Jun15/MFP3 Do not write outside the box 11 QUESTION PART REFERENCE Answer space for question s (11) Turn over P/Jun15/MFP3 Do not write outside the box 12 QUESTION PART REFERENCE Answer space for question (12) P/Jun15/MFP3 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question s (13) Turn over P/Jun15/MFP3 Do not write outside the box 14 A differential equation is given by pffiffiffiffiffi d2 y pffiffiffi pffiffiffi x þ ð2 xÞy ¼ xðln xÞ2 þ 5, x > dx Show that the substitution x ¼ e2t transforms this differential equation into (a) d2 y dy À þ 2y ¼ 4t þ 5eÀt dt dt Hence find the general solution of the differential equation (b) pffiffiffiffiffi d2 y pffiffiffi pffiffiffi x þ ð2 xÞy ¼ xðln xÞ2 þ 5, x > dx QUESTION PART REFERENCE [7 marks] [10 marks] Answer space for question (14) P/Jun15/MFP3 Do not write outside the box 15 QUESTION PART REFERENCE Answer space for question s (15) Turn over P/Jun15/MFP3 Do not write outside the box 16 QUESTION PART REFERENCE Answer space for question (16) P/Jun15/MFP3 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question s (17) Turn over P/Jun15/MFP3 Do not write outside the box 18 The diagram shows the sketch of a curve C1 O Initial line The polar equation of the curve C1 is r ¼ þ cos 2y, p p À 4y4 2 Find the area of the region bounded by the curve C1 (a) [5 marks] The curve C2 whose polar equation is (b) r ¼ þ sin y, p p À 4y4 2 intersects the curve C1 at the pole O and at the point A The straight line drawn through A parallel to the initial line intersects C1 again at the point B (i) Find the polar coordinates of A [4 marks]   pffiffiffiffiffi 13 þ (ii) Show that the length of OB is [6 marks] (iii) Find the length of AB , giving your answer to three significant figures [3 marks] QUESTION PART REFERENCE Answer space for question (18) P/Jun15/MFP3 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question s (19) Turn over P/Jun15/MFP3 Do not write outside the box 20 QUESTION PART REFERENCE Answer space for question (20) P/Jun15/MFP3 Do not write outside the box 21 QUESTION PART REFERENCE Answer space for question END OF QUESTIONS (21) P/Jun15/MFP3 Do not write outside the box 22 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (22) P/Jun15/MFP3 Do not write outside the box 23 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (23) P/Jun15/MFP3 Do not write outside the box 24 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright ª 2015 AQA and its licensors All rights reserved (24) P/Jun15/MFP3 [...]... QUESTIONS (21) P/ Jun15 /MFP3 Do not write outside the box 22 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (22) P/ Jun15 /MFP3 Do not write outside the box 23 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (23) P/ Jun15 /MFP3 Do not write outside the box 24 There are no questions printed on this page DO NOT... s (17) Turn over P/ Jun15 /MFP3 Do not write outside the box 18 The diagram shows the sketch of a curve C1 7 O Initial line The polar equation of the curve C1 is r ¼ 1 þ cos 2y, p p À 4y4 2 2 Find the area of the region bounded by the curve C1 (a) [5 marks] The curve C2 whose polar equation is (b) r ¼ 1 þ sin y, p p À 4y4 2 2 intersects the curve C1 at the pole O and at the point A The straight... s (13) Turn over P/ Jun15 /MFP3 Do not write outside the box 14 A differential equation is given by 6 p ffiffiffiffi d2 y p ffiffi p ffiffi 4 x 5 2 þ ð2 xÞy ¼ xðln xÞ2 þ 5, x > 0 dx Show that the substitution x ¼ e2t transforms this differential equation into (a) d2 y dy À 2 þ 2y ¼ 4t 2 þ 5eÀt 2 dt dt Hence find the general solution of the differential equation (b) p ffiffiffiffi d2 y p ffiffi p ffiffi 4 x 5 2 þ ð2 xÞy ¼ xðln... PAGE ANSWER IN THE SPACES PROVIDED (23) P/ Jun15 /MFP3 Do not write outside the box 24 There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED Copyright ª 2015 AQA and its licensors All rights reserved (24) P/ Jun15 /MFP3 ... s (15) Turn over P/ Jun15 /MFP3 Do not write outside the box 16 QUESTION PART REFERENCE Answer space for question 6 ... (16) P/ Jun15 /MFP3 Do not write outside the box 17 QUESTION PART REFERENCE Answer space for question 6 ... (12) P/ Jun15 /MFP3 Do not write outside the box 13 QUESTION PART REFERENCE Answer space for question 5 ... and at the point A The straight line drawn through A parallel to the initial line intersects C1 again at the point B (i) Find the polar coordinates of A [4 marks]   1 p ffiffiffiffi 13 þ 1 (ii) Show that the length of OB is 4 [6 marks] (iii) Find the length of AB , giving your answer to three significant figures [3 marks] QUESTION PART REFERENCE Answer space for question 7 ... (18) P/ Jun15 /MFP3 Do not write outside the box 19 QUESTION PART REFERENCE Answer space for question 7 ... s (19) Turn over P/ Jun15 /MFP3 Do not write outside the box 20 QUESTION PART REFERENCE Answer space for question 7