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GCE JAN 2011 : AS, F2: Further Pure Mathematics 2

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3rd CLEAN PROOF: 18 AUGUST 2010 ADVANCED General Certificate of Education January 2011 Mathematics assessing Module FP2: Further Pure Mathematics 2 AMF21 Assessment Unit F2 [AMF21] WEDNESDAY 2 FEBRUARY, 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 scientific calculator in this paper. INFORMATION FOR CANDIDATES 111594 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 1n z loge z 6229 Answer all seven questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Show that the sum of the series 1 2 5 + 2 3 6 + . . . + n (n + 1) (n + 4) is given by 1 n ( n + 1) ( n + 2) (3n + 17) 12 2 [6] Write 2x 2 x + 1 ( x 2 + 1) ( x 2 + 2) in partial fractions. 3 [6] Find the general solution of the differential equation d2 y dx 4 2 dy 6 y = sin x dx [12] (i) Using Maclaurin s theorem, derive a series expansion for cos up to and including the [5] term in 4 (ii) Hence, and using a binomial expansion, find a series expansion for cos 3x 1 x2 up to and including the terms in x 4 6229 [8] 2 [Turn over 5 Prove by mathematical induction that an = 5n + 3 is divisible by 4 for each non-negative integer n. [7] 6 Fig. 1 Fig. 1 above shows an ellipse with equation x2 17 2 + y2 82 =1 The foci of the ellipse are F , F and its directrices are D and D. (i) Show that the equation of the directrix D is x = 289 15 (ii) Find the coordinates of the focus F. [3] [2] (iii) Derive the equation of the tangent to the ellipse at a general point (17 cos , 8 sin ). [5] PP is a latus rectum of the ellipse. (iv) Show that the tangent at P meets the x-axis on the directrix D. 6229 3 [6] [Turn over 7 (i) If z = cos + i sin is a complex number, show that cos = 1 2 (z + z 1) [2] (ii) Hence find numbers a, b and c such that cos4 = a cos 4 + b cos 2 + c [7] (iii) Hence, or otherwise, find the general solution of 2 cos 4 + 8 cos 2 + 5 = 0 6229 111594 [6] [Turn over

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Additional Info : Gce Mathematics January 2011 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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