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GCE MAY 2010 : AS, F3: Further Pure Mathematics 3

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ADVANCED General Certificate of Education 2010 Mathematics assessing Module FP3: Further Pure Mathematics 3 AMF31 Assessment Unit F3 [AMF31] THURSDAY 27 MAY, MORNING TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all seven 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 5263 Answer all seven questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Using integration by parts, find e cos x d x x 2 [6] The points A (2, 0, 1), B (4, 3, 1) and the origin determine a plane 1 (i) Verify that an equation of the plane is x 2y + 2z = 0 [2] The plane 2 has an equation r . n = d where n = 3i 2j k and d is a constant. (ii) If the point B lies on the plane 2, find the value of d. [2] The planes 1 and 2 intersect in the line L. (iii) Find, in Cartesian form, an equation for the line L. 5263 2 [6] [Turn over 3 For each non-negative integer n, let In = 1 x n x 0 e dx (i) Show that for n > 1 In = nIn 1 e 1 [5] (ii) Hence find the exact value of 1 xe 4 x 0 4 dx By using the substitution u = ex find 5 cosh x + 4 sinh x dx 5 [5] [8] (i) Show that 1 d (cos 1 x ) = dx 1 x2 [4] 1 The tangent at the point where x = on the curve y = cos 1 4x, cuts the y-axis at the point P. 8 (ii) Show that OP = + 3 3 where O is the origin. 5263 [6] 3 [Turn over 6 (i) Using the exponential definitions of sinh x and cosh x show that for | x | < 1 1 1+ x tanh 1 x = ln 2 1 x (ii) Sketch the graph of y = tanh 1 x clearly labelling any asymptotes. (iii) Find d (tanh 1 x ) dx [5] [2] [4] (iv) Solve the equation ( x = tanh ln 6 x ) where 0 < x < 1 7 [5] Relative to an origin O the points A and B have position vectors a and b respectively where a = i + j + 2k and b = i + 2j k (i) Find the area of the triangle OAB. [5] The point C is the intersection of the two lines (r a) m = 0 and where m = i + k and (r b) n = 0 n = i j + 4k (ii) Find c, the position vector of the point C. [7] 4 (iii) Show that the volume of the tetrahedron OABC is 3 [3] 1 [The volume of a tetrahedron is | c . (a b)|] 6 1847-009-1 4 [Turn over

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Additional Info : Gce Mathematics May 2010 Assessment Unit F3 Module FP3 : Further Pure Mathematics 3
Tags : General Certificate of Education, A Level and AS Level, uk, council for the curriculum examinations and assessment, gce exam papers, gce a level and as level exam papers , gce past questions and answer, gce past question papers, ccea gce past papers, gce ccea past papers

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