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GCE JUN 2007 : AS, F2: Further Pure Mathematics 2

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ADVANCED General Certificate of Education 2007 Mathematics assessing Module FP2: Further Pure Mathematics 2 AMF21 Assessment Unit F2 [AMF21] FRIDAY 22 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 20.03.06HF 2 29.6.06EA 3 25-09-06RR 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 AMFP2S7 2253 Answer all seven questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 x2 The graphs of + y2 = 9 and y2 = 4(x 2) are shown in Fig. 1 below. 4 B A C A B Fig. 1 (i) Copy Fig. 1 inserting the coordinates of the 5 points A, A , B, B and C on the x- and y-axes. Also show on your diagram the focus and directrix of the parabola, indicating the coordinates of the focus and the equation of the directrix. [5] 1 20.03.06HF 2 29.6.06EA 3 25-09-06RR (ii) Calculate the equation of the common chord of the parabola and ellipse in surd form. [4] AMFP2S7 2253 2 [Turn over 2 (i) Express 2 in partial fractions. r r [4] 3 (ii) Hence, using the method of differences, prove that n r 2 r = 1 1 + n 1 1 + 2n 3 n 2 [4] r =2 (iii) Deduce the sum of the infinite series 11 1 1 + + + + 3 + 6 24 60 r r 3 [2] One of the complex roots of z4 2z3 + 4z2 4z + 4 = 0 is 1 + i, where i2 = 1 (i) State one other complex root. (ii) Find the remaining two roots and plot all four roots on an Argand diagram. 4 [1] [8] (i) By using de Moivre s theorem prove that cos 3 4 cos 3 3 cos [5] 1 20.03.06HF 2 29.6.06EA 3 25-09-06RR (ii) Hence, or otherwise, find the general solution of the equation cos 3 + cos = 0 5 [6] Find the general solution to the differential equation d2 y dy + 5 y = sin x 2 +2 dx dx AMFP2S7 2253 3 [12] [Turn over 6 Prove by induction that ( ) n dn x e cos x = 2n e x cos x + 4 dx n for every non-negative integer n Note 7 [10] d0 x e cos x = e x cos x dx0 ( ) (i) Use Maclaurin s theorem to derive the first 4 terms in the series expansion of 1 (1 + x)2 where | x | < 1 [5] (ii) Hence sum the series 1 2 (iii) Given that 3 3 3 ( 4 ) + 3( 4 )2 4 ( 4 )3 + ... 1 has a series expansion of the form (1 6 x )(1 + kx )2 |x < 1 x| 6 1 + px2 + . . . , 1 20.03.06HF 2 29.6.06EA 3 25-09-06RR find k and p. THIS IS THE END OF THE QUESTION PAPER S 3/06 7-044-1 [2] [7]

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Additional Info : Gce Mathematics June 2007 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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