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GCE MAY 2009 : AS, F3: Further Pure Mathematics 3

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ADVANCED General Certificate of Education 2009 Mathematics assessing Module FP3: Further Pure Mathematics 3 AMF31 Assessment Unit F3 [AMF31] FRIDAY 22 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 4085 Answer all seven questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Use the substitution x = 5 sin u to find 2 2 dx 25 4 x 2 Straight lines l1 and l2 have equations x 3 y p z 1 = = 2 3 1 x 3 y +1 z 4 = = 1 2 1 l1 l2 where p is a scalar constant. The lines intersect at the point A. Find the value of p and the coordinates of the point A. 3 [6] [8] (i) Show that d 1 1 2 2 sin x + x 1 x = 1 x dx 2 (ii) Write 4x x2 3 in the form a (x b)2 [4] [1] (iii) Hence find the exact value of 3 4 x x 2 3 dx 2 4085 2 [5] [Turn over 4 (i) Using the definition of the hyperbolic functions in terms of the exponential function, prove that cosh2 2x + sinh2 2x cosh 4x [4] (ii) Hence solve the equation cosh2 2x + sinh2 2x = 2 leaving your answers in logarithmic form. [4] A plane passes through the points A (5, 3, 1), B ( 3, 2, 3) and C (2, 3, 2). (i) Find AC BC. [4] (ii) Hence or otherwise find in Cartesian form an equation for . 5 [3] The perpendicular from the point Q(6, 6, 4) to meets the plane at the point P. (iii) Find the coordinates of P. (iv) Show that the perpendicular distance from Q(6, 6, 4) to the plane is 2 14 6 [5] [2] (a) Find the coordinates of the stationary points on the curve with equation y = x 2 sinh 1 x and determine their nature. [7] (b) Evaluate 0 2 x 2 sinh 1 x dx correct to 2 decimal places. 4085 [7] 3 [Turn over 7 (i) Differentiate with respect to x x5 (ln x)n 5 [3] For each non-negative integer n, let In = e 1 x 4 (ln x )n dx (ii) Using your answer to (i) or otherwise, show that if n 1, then In = 1 e5 n In 1 5 5 [5] The shaded region in Fig. 1 below is bounded by the curve with equation y = x2 ln x, the line x = e and the x-axis. y e 1 x Fig. 1 y The region is rotated through 2 about the x-axis. (iii) Show that the volume of the solid formed is [17e5 2]. 125 1 e [7] x THIS IS THE END OF THE QUESTION PAPER 937-003-1 4 [Turn over

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Additional Info : Gce Mathematics May 2009 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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