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GCE JUN 2009 : AS, F1: Further Pure Mathematics 1

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1 2 3 4 5 6 ADVANCED SUBSIDIARY (AS) General Certificate of Education 2009 7 8 9 Mathematics 11 Assessment Unit F1 12 assessing 13 Module FP1: Further Pure Mathematics 1 14 AMF11 10 [AMF11] 15 TUESDAY 23 JUNE, MORNING 16 17 18 19 20 21 TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES 22 Write your Centre Number and Candidate Number on the Answer Booklet provided. 23 Answer all six questions. Show clearly the full development of your answers. 24 Answers should be given to three significant figures unless otherwise stated. 25 You are permitted to use a graphic or a scientific calculator in this paper. 26 27 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 29 question or part question. A copy of the Mathematical Formulae and Tables booklet is provided. 30 Throughout the paper the logarithmic notation used is ln z where it is noted that 31 ln z loge z 28 32 33 34 35 4164 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 (a) Describe the transformation given by the matrix Q= 1 2 1 2 1 2 1 2 [2] 1 1 (b) The matrix S = represents a linear transformation of the x y plane. 6 2 Find the equations of the straight lines through the origin O which are invariant under the transformation given by S. 1 1 M= 2 4 2 1 0 1 1 Let M = and I = 0 1 2 4 (i) Show that M2 = 3M + 2I. 1 0 I= 0 1 (ii) Hence, or otherwise, express the matrix M4 in the form aM + bI where a, b are integers. 3 [6] [4] [4] A binary operation * is defined on the set of all ordered pairs (x, y) of real numbers, where x 0, y 0 The operation is given as (a, b)*(c, d) = (ad + bc, bd) (i) Show that * is associative. (ii) Find the identity element. [4] (iii) Find the inverse of (a, b). 4164 [4] [3] 2 [Turn over 4 2 0 6 The matrix N is given by 3 1 4 1 0 1 (i) Show that l = 4 is one of the eigenvalues of N and find the other two eigenvalues. (ii) Find a unit eigenvector corresponding to l = 4 5 [7] [4] (a) Find all the real values of a, b such that (a + bi)2 = 21 20i [8] 3 (b) (i) Sketch on an Argand diagram the locus of all points z such that | z 3 i| = 2 [3] 2 (ii) Hence, or otherwise, show that for all points z on the locus arg z < 6 5 12 [5] The circle C1 has equation x2 + y2 + 2x 14y + 40 = 0 (i) Find the equation of the tangent to the circle C1 at the point (2, 6). [6] (ii) Find the equation of the other tangent from the origin to the circle C1 [7] The circle C2 has equation x2 + y2 10x 8y + 16 = 0 (iii) Find the points of intersection of the circles C1 and C2 4164 3 [8] S 11/07 938-002-1 [Turn over

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Additional Info : Gce Mathematics June 2009 Assessment Unit F1 Module FP1 : Further Pure Mathematics 1
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