Mathematics (JUN10MFP301) General Certificate of Education Advanced Level Examination June Unit Further Pure TOTAL
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1 Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Further Pure 3 Friday 11 June 2010 General Certificate of Education Advanced Level Examination June am to am For this paper you must have: the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. MFP3 Question TOTAL Mark 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 es 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. 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) 6/6/ MFP3
2 2 Answer all questions in the spaces provided. 1 The function yðxþ satisfies the differential equation dy ¼ fðx, yþ dx where and fðx, yþ ¼x þ 3 þ sin y yð1þ ¼1 (a) Use the Euler formula y rþ1 ¼ y r þ hfðx r, y r 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 y rþ1 ¼ y r 1 þ 2hfðx r, y r Þ with your answer to part (a), to obtain an approximation to yð1:2þ, giving your answer to three decimal places. (3 marks) (02)
3 3 Turn over s (03)
4 4 2 (a) Find the value of the constant k for which k sin 2x is a particular integral of the differential equation d 2 y þ y ¼ sin 2x dx2 (3 marks) (b) Hence find the general solution of this differential equation. (4 marks) (04)
5 5 Turn over s (05)
6 6 ð 1 3 (a) Explain why 4xe 4x dx is an improper integral. 1 ð (b) Find 4xe 4x dx. (1 mark) (3 marks) (c) Hence evaluate ð 1 1 4xe 4x dx, showing the limiting process used. (3 marks) (06)
7 7 Turn over s (07)
8 8 4 By using an integrating factor, find the solution of the differential equation dy dx þ 3 x y ¼ðx4 þ 3Þ 3 2 given that y ¼ 1 5 when x ¼ 1. (9 marks) (08)
9 9 Turn over s (09)
10 10 5 (a) Write down the expansion of cos 4x in ascending powers of x up to and including the term in x 4. Give your answer in its simplest form. (2 marks) (b) (i) Given that y ¼ lnð2 e x Þ, find dy dx, d2 y dx 2 and d3 y dx 3. ðyou may leave your expression for d3 y dx 3 unsimplified.þ (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 x x 2 x 3 (2 marks) (c) Find lim x! 0 x lnð2 e x Þ 1 cos 4x (3 marks) (10)
11 11 Turn over s (11)
12 12 (12)
13 13 Turn over s (13)
14 14 6 The polar equation of a curve C 1 is r ¼ 2ðcos y sin yþ, 04y42p (a) (i) Find the cartesian equation of C 1. (4 marks) (ii) Deduce that C 1 is a circle and find its radius and the cartesian coordinates of its centre. (3 marks) (b) The diagram shows the curve C 2 with polar equation r ¼ 4 þ sin y, 04y42p O Initial line (i) Find the area of the region that is bounded by C 2. (6 marks) (ii) Prove that the curves C 1 and C 2 do not intersect. (4 marks) (iii) Find the area of the region that is outside C 1 but inside C 2. (2 marks) (14)
15 15 Turn over s (15)
16 16 (16)
17 17 Turn over s (17)
18 18 7 (a) Given that x ¼ t 1 2, x > 0, t > 0 and y is a function of x, show that: (i) (ii) dy dx ¼ 2t 1 2 dy dt ; d 2 y dx 2 ¼ 4t d2 y dt 2 þ 2 dy dt. (2 marks) (3 marks) (b) Hence show that the substitution x ¼ t 1 2 transforms the differential equation into x d2 y dx 2 ð8x2 þ 1Þ dy dx þ 12x3 y ¼ 12x 5 d 2 y dy 4 þ 3y ¼ 3t dt 2 dt (2 marks) (c) Hence find the general solution of the differential equation x d2 y dx 2 ð8x2 þ 1Þ dy dx þ 12x3 y ¼ 12x 5 giving your answer in the form y ¼ fðxþ. (7 marks) (18)
19 19 Turn over s (19)
20 20 END OF S Copyright Ó 2010 AQA and its licensors. All rights reserved. (20)
(ii) Given that gðxþ ¼ð4x 1Þðax 2 þ bx þ cþ, find the values of the integers a, b and c. (3 marks)
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