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GCE MAY 2008 : AS, F3: Further Pure Mathematics 3

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ADVANCED General Certificate of Education 2008 Mathematics assessing Module FP3: Further Pure Mathematics 3 AMF31 Assessment Unit F3 [AMF31] WEDNESDAY 21 MAY, AFTERNOON TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. You are permitted to use a graphic or a scientific calculator in this paper. INFORMATION FOR CANDIDATES The total mark for this paper is 75 Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. A copy of the Mathematical Formulae and Tables booklet is provided. Throughout the paper the logarithmic notation used is ln z where it is noted that ln z loge z AMFP3S8 3221 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Show that d {sin 1(tanh x) tan 1(sinh x)} = 0 dx [8] [You may find the identity tanh2 x + sech2 x 1 useful.] 2 For each non-negative integer n, let In = (i) Show that if n 2 xn cos x dx In = xn sin x + nx n 1 cos x n(n 1)In 2 [7] (ii) Hence find 3 x4 cos x dx [4] By using the substitution u = tan 1 x or otherwise: (i) show that 3 tan 1 x 1 + x2 1 dx = 7 2 288 [5] (ii) find AMFP3S8 3221 dx (1 + x 2 ) tan 1 x 2 [4] [Turn over 4 (i) Write in Cartesian form the vector equations: (a) [r (i + 2j k)] (i j + k) = 0 (b) [r (j + 2k)] . (i + j 2k) = 0 stating in each case whether the equation is that of a line or a plane. [5] (ii) Calculate the acute angle between (a) and (b) above. 5 [7] Consider the lines 1: 2 (i) Show that the lines 1 and 2 : x 2 y 3 z 4 = = 2 1 2 x +1 y 3 z = = 1 1 2 intersect and find their point of intersection. [7] (ii) Find the equation of the line that passes through the point of intersection of the lines 1 and 2 and is perpendicular to both. [6] (iii) Find the equation of the plane containing the lines [4] AMFP3S8 3221 3 1 and 2 [Turn over 6 (i) From the definitions of the hyperbolic functions in terms of the exponential function prove that 2 sinh2 u + 1 cosh 2u [3] Let I= x 2 + 6 x 55 dx (ii) Using the substitution x + 3 = 8 cosh u, show that I = 32 (cosh 2u 1) du [7] (iii) Hence show that I= 1 ( x + 3) x 2 + 6 x 55 32 ln x + 3 + x 2 + 6 x 55 + c 2 [8] THIS IS THE END OF THE QUESTION PAPER S 2/07 529-045-1 4 [Turn over

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Additional Info : Gce Mathematics May 2008 Assessment Unit F3 Module FP3 : Further Pure Mathematics 3
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