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GCE JUN 2009 : AS, F2: Further Pure Mathematics 2

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ADVANCED General Certificate of Education 2009 Mathematics assessing Module FP2: Further Pure Mathematics 2 AMF21 Assessment Unit F2 [AMF21] FRIDAY 19 JUNE, AFTERNOON TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all eight 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 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 logez 4177 Answer all eight questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Express 1 (2x2 + 3)(x 1) in partial fractions. 2 [5] Find in radians the general solution of the equation 3 sin cos = 2 3 [7] Show that the sum of the series 13 + 33 + 53 + . . . + (2n 1)3 is given by n2(2n2 1). 4 [7] Given that one of the roots of z3 z2 + 3z + 5 = 0 is z = 1 2i, find the other 2 roots and plot all 3 roots on an Argand diagram. 5 [6] If u1 = 7 and un+1 = 3 un 2 prove by the method of mathematical induction that un = 2(3n) + 1, where n 4177 2 + [6] [Turn over 6 Solve the differential equation dy d2y 6 + 9 y = 36 e 3x dx 2 dx dy given that y = 2 and = 5 when x = 0 dx 7 [11] (i) Using Maclaurin s theorem, derive a series expansion of sin up to and including the term in 5 [5] (ii) Using de Moivre s theorem, show that sin 3 3 sin 4 sin3 (iii) Hence, find a series expansion for sin3 up to and including the terms in 5 [5] [4] Please turn over for Question 8 4177 3 [Turn over 8 The parabola y2 = 8x is shown in Fig. 1 below. F is the focus and P a point on the parabola. The normal to the parabola at P cuts the x-axis at G, and PP is a line parallel to the x-axis. y P P O G F x Fig. 1 (i) Write down the co-ordinates of F [1] (ii) Verify that the point P is given parametrically by (2t2, 4t). [2] (iii) Show that the equation of the normal PG is given by y + t x = 2t3 + 4t (iv) Show that FP = FG [6] [7] (v) Prove that FP G = GPP This proves that light rays parallel to the axis of a parabolic mirror illuminate the focus. [3] S 12/07 938-007-1 4 [Turn over

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Additional Info : Gce Mathematics June 2009 Assessment Unit F2 Module FP2 : Further Pure Mathematics 2
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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