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GCE MAY 2007 : AS, F3: Further Pure Mathematics 3

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ADVANCED General Certificate of Education 2007 Mathematics assessing Module FP3: Further Pure Mathematics 3 AMF31 Assessment Unit F3 [AMF31] THURSDAY 31 MAY, AFTERNOON 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 log e z AMFP3S7 2221 Answer all seven questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 By solving the equation 5 cosh x = 7 + sinh x show that x = ln 1 or x = ln 3 2 2 [7] Find where the line [r (3i + 2k)] (i j + 3k) = 0 meets the (x, y)- plane. 3 [5] (i) Show that ( ) x sinh ln x + x 2 + 1 [4] (ii) Hence show that 30 12 AMFP3S7 2221 dx 256 + x 2 2 = ln 2 [6] [Turn over 4 (i) Prove that d tan 1 x = 1 2 dx 1+ x [5] Fig. 1 below shows the area, A, bounded by the curve y = sech x, the x- and y-axes and the line x = 1 ln 3 2 y 1 A _ 1 ln 2 x 3 Fig. 1 (ii) Show that this area can be written as A= 1 ln 3 2 0 2e x dx 1+ e 2 x (iii) Using the substitution u = e x and part (i), find the exact value of A. AMFP3S7 2221 3 [3] [5] [Turn over 5 The glass pyramid outside the Louvre museum in Paris can be represented by the model illustrated in Fig. 2 below. V C B A D Fig. 2 The plane faces through A, B, V and D, A, V are given by ABV : 24x 18y + 25z = 462 DAV : 18x + 24y 25z = 84 (i) Find the vector equation of the line AV through A and V. [6] The line through B and V is given by BV : x y 16 z 30 = = 7 1 6 (ii) Find in degrees the angle between this line and the plane DAV . AMFP3S7 2221 4 [6] [Turn over 6 Fig. 3 below shows the tetrahedron OABC. O is the origin and A, B and C have position vectors a = OA = i + j b = OB = j + k c = OC = i + k ^ The unit vector n is perpendicular to the plane OAB. B z C ^ n y O x A Fig. 3 (i) Derive an expression for the area of the triangle OAB in terms of a b. [3] The volume of a tetrahedron is given by 1 area of triangular base perpendicular 3 height. (ii) Using part (i), derive the expression for the volume of the tetrahedron OABC in the form pc . a b, where p is a constant to be determined. [4] (iii) Hence calculate the volume of OABC. [3] AMFP3S7 2221 5 [Turn over 7 (a) The integral 4 where n is a non-negative integer. Use integration by parts to establish the reduction formula In = n 1 1 n In 2 2 2 n n [9] (b) Evaluate 1 sin 1 x dx [9] 0 THIS IS THE END OF THE QUESTION PAPER AMFP3S7 2221 6 [Turn over S 3/06 0000 7-045-1

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Additional Info : Gce Mathematics May 2007 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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