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GCE JUN 2006 : AS, F3: Further Pure Mathematics 3

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ADVANCED General Certificate of Education 2006 Mathematics assessing Module FP3: Further Pure Mathematics 3 AMF31 Assessment Unit F3 [AMF31] FRIDAY 23 JUNE, 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. 1 29/05/05EA 2 24/10/05ES 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 AMFP3S6 906 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 7 cosh x 3 sinh x = 11 show that x = ln 1 or ln 5 2 2 [6] Find in degrees the acute angle between the line x 7 y 3 z+4 = = 2 1 1 and the plane 2 x + 3y + z = 4 3 [7] Show that the lines with equations {r (2i j + 3k)} (i 2 j k) = 0 and and {r (i 8 j 5k)} (i + j + 2k) = 0 4 1 26/05/04EA [10] (i) If 1 f( x ) = sin 1 + tan 1 x 2 1+ x 2 18/11/04PEG 3 10/03/05AC intersect and find their point of intersection. find f'(x), simplifying as far as possible. (ii) What can be deduced about f(x)? AMFP3S6 906 [7] [1] 2 [Turn over 5 For each non-negative integer, n, let In = 3 tan n d 0 (i) Show that if n 2, 1 1 3(2 n 2 ) In = In 2 n 1 [6] (ii) Hence show that 3 tan d = + ln 2 4 3 5 4 1 26/05/04EA 2 18/11/04PEG 3 10/03/05AC 0 [5] AMFP3S6 906 3 [Turn over 6 Consider the tetrahedron VABC illustrated in Fig. 1 below. V A C B Fig. 1 Two of the planes which form VABC are given by VAC x + y + z = 2 VBC 3 x + 9 y 7z = 30 and (i) Show that the Cartesian equation of the line VC which is the intersection of the above two planes is x+4 y+2 z = = 4 1 3 [7] (ii) If the base of the tetrahedron is given by ABC z=0 find the coordinates of C. [2] (iii) If V, A and B have coordinates (0, 1, 3), (1, 3, 0) and (2, 4, 0) respectively use the volume formula 3 10/03/05AC 1 6 p q r [8] 1 26/05/04EA 2 18/11/04PEG to find the volume of the tetrahedron. AMFP3S6 906 4 [Turn over 7 (i) Show that ( ) sinh ln x + 1 + x 2 x [4] (ii) Show that 25 x 2 + 30 x + 25 = (5 x + 3)2 + 4 2 [1] (iii) Hence, using the substitution, 5x + 3 = 4 sinh u, show that 1 0 ( dx 1 1 = ln 1 + 1 5 2 5 (25 x 2 + 30 x + 25) 5 ) [11] 1 29/05/05EA 2 24/10/05ES THIS IS THE END OF THE QUESTION PAPER AMFP3S6 906 5 [Turn over 1 29/05/05EA 2 24/10/05ES 1 29/05/05EA 2 24/10/05ES 2 24/10/05ES 1 29/05/05EA urn over S 5/05 4000 302507(47)

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Additional Info : Gce Mathematics June 2006 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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