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GCE JUN 2006 : AS, F1: Further Pure Mathematics 1

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ADVANCED SUBSIDIARY (AS) General Certificate of Education 2006 Mathematics assessing Module FP1: Further Pure Mathematics 1 AMF11 Assessment Unit F1 [AMF11] THURSDAY 1 JUNE, AFTERNOON TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all six 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. 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 AMFP1S6 1533 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Consider the matrix 2 4 A= 5 3 (i) Find the eigenvalues of A (ii) For each eigenvalue, find a corresponding eigenvector. 2 [6] [5] A binary operator * on the set of all real numbers is defined by a * b = a + b + 2ab (i) Show that * is associative. (ii) Find the identity element. [2] (iii) Find the inverse of a, stating a necessary restriction on the set of allowable values of a. 3 [4] [4] (i) Find the inverse of the matrix 3 1 p 0 2 5 6 0 5 leaving your answer in terms of p. [9] (ii) For which values of p does the inverse exist? AMFP1S6 1533 2 [1] 4 (a) (i) Find the real values of a, b such that (a + ib)2 = 5 12i [6] (ii) Hence write down the two solutions of the equation v = 5 12i [2] (b) (i) On an Argand diagram draw the locus of arg (z 4 + 3i) = 4 (ii) Find the coordinates of the point P where this locus cuts the real axis. 5 [3] [3] A circle A has equation x2 + y2 + 2x 4y + 1 = 0 (i) Find the equations of the two tangents from the origin to this circle. [7] A circle B has equation x2 + y2 + 6x + 2y + 1 = 0 (ii) Find the points of intersection of the circles A and B. AMFP1S6 1533 3 [8] [Turn over 6 2 3 (a) The matrix A = 3 5 (i) The point P is mapped to R( 7, 11) by the transformation represented by A Find the coordinates of P. [4] (ii) The triangle OPQ, where O is the origin and Q is the point (0, 6), is mapped onto the triangle ORS by the transformation represented by A Find the area of the triangle ORS. [4] (b) The set of points which form the curve whose equation is x2 + 5y2 + 4xy + 6x + 4y = 0 1 2 is mapped by the matrix B = 0 1 Show that the curve formed by the image points has the equation x2 + y2 + 6x 8y = 0 S 6/05 4000 302507(45) [7] [Turn over

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Additional Info : Gce Mathematics June 2006 Assessment Unit F1 Module FP1 : Further Pure Mathematics 1
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