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GCE JUN 2010 : AS, F2: Further Pure Mathematics 2

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ADVANCED General Certificate of Education 2010 Mathematics assessing Module FP2: Further Pure Mathematics 2 AMF21 Assessment Unit F2 [AMF21] TUESDAY 22 JUNE, 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 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 where it is noted that ln loge 5217 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 de Moivre s theorem to find an expression for sin 3 in the form sin 3 a sin3 + b sin where a and b are integers to be determined. 2 [5] Show that 2n (2k 1)k (2k + 1) 15n + 14n + 2 n 2 n 4 3 3 2 1 [6] k = n+1 3 (i) Use Maclaurin s theorem to derive the series expansion for (1 + x)n up to and including the term in x 3 [5] (ii) Express in partial fractions 1+ x (1 + 2 x 2 )(1 2 x) [6] (iii) Hence find the series expansion of 1+ x (1 + 2 x 2 )(1 2 x) up to and including the term in x 3 5217 [5] 2 [Turn over 4 Consider the differential equation d2 y dy + p + 16 y = f(x) 2 dx dx where p is a constant. (i) If f(x) e3x and p = 10, find the general solution of the differential equation. (ii) If instead f(x) 0 and the general solution is of the form y = (Ax + B)ekx, write down the possible values of p and k. 5 [7] [4] (i) Use mathematical induction to prove 2n + 1 sin x cos x cos (2x) cos (4x) cos (2nx) sin (2n + 1x) where n is a non-negative integer. [7] (ii) Hence find in radians the general solution to the equation sin x cos x cos 2x cos 4x = 5217 3 2 24 [7] [Turn over 6 A parabola has the y-axis as directrix and focus at S(6, 0) as shown in Fig. 1 below. y T (3, 0) S(6, 0) x R Fig. 1 (i) Show that the equation of the parabola is y2 = 12(x 3) [3] (ii) Verify that any point T with parametric coordinates (3t 2 + 3, 6t) lies on the parabola. [2] (iii) Show that the equation of the tangent at T can be written as ty x = 3t 2 3 [6] 1 The point R, at the opposite end of the focal chord through T, has parameter t (iv) Show that the tangents at T and R meet on the y-axis. 5217 4 [4] [Turn over 7 4 (i) Illustrate on an Argand diagram the roots of the equation 8 The roots of the equation 1=0 [2] 1 = 0 are illustrated in the Argand diagram in Fig. 2 below. Im C D B A E F Re H G Fig. 2 (ii) Find the root represented by the point B in Fig. 2 above in the form reik , where r and k are positive numbers. [2] (iii) Find a complex equation whose roots are B, D, F and H. [4] THIS IS THE END OF THE QUESTION PAPER 5217 5 [Turn over 1847-011-1 [Turn over

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Additional Info : Gce Mathematics June 2010 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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