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GCE JAN 2011 : AS, F1: Further Pure Mathematics 1

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ADVANCED SUBSIDIARY (AS) General Certificate of Education January 2011 Mathematics assessing Module FP1: Further Pure Mathematics 1 AMF11 Assessment Unit F1 [AMF11] wEDNESDAY 19 JANUARY, 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 scientific calculator in this paper. INFoRMAtIoN FoR CANDIDAtES 111533 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 1n z where it is noted that 1n z loge z 6228 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 The matrix A is given by 7 4 A= 2 5 (i) Show that the eigenvalues of A are 3 and 9 (ii) Find a unit eigenvector corresponding to the eigenvalue 9 2 [5] [4] Two circles have equations x2 + y2 + 2x 6y + 8 = 0 x2 + y2 4x 28 = 0 (i) Find the point where these circles meet. (ii) Determine whether the circles touch internally or externally. 6228 [8] [4] 2 [Turn over 3 (a) Explain why the set {1, 2, 3, 4, 5, 6, 7} cannot form a group under multiplication modulo 8 [3] (b) (i) Copy and complete the group table for addition modulo 8 0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 0 2 2 3 4 5 6 7 0 1 3 3 4 5 4 4 5 6 5 5 6 7 6 6 7 0 7 7 0 1 [5] (ii) Using the group table in (i), or otherwise, write down the two values of x which satisfy x3 = x (iii) For this group, write down a subgroup of order 4 6228 3 [2] [3] [turn over 4 (a) Describe fully the transformation given by the matrix 3 M= 5 4 5 4 5 3 5 [5] (b) The set of points which form the circle x2 + y2 = 25 is mapped under a transformation given by the matrix 3 2 N= 2 1 Show that the equation of the curve formed by the image points is 5X 2 + 13Y 2 16XY = 25 6228 4 [8] [turn over 5 A matrix M is given by 3 2 a M = 1 2 1 a 0 3 (i) Find, in terms of a, the determinant of M. [3] A system of linear equations is given by 3x + 2y + az = 7 x 2y z = 1 ax + 3z = 11 (ii) Find the values of a for which the system has a unique solution. (iii) If a = 1, find the inverse of M. [6] (iv) Hence, for a = 1 find the unique solution of the system of equations. 6 [3] [3] (a) Find the complex roots of the equation 2z2 2iz 5 = 0 [4] (b) (i) Sketch, on an Argand diagram, the locus of those points w which satisfy |w 3| = 5 [3] (ii) On the same diagram, shade the region which represents the locus of those points w which satisfy and |w 3| 5 arg(w 3) 6 4 6228 5 [6] [turn over THIS IS THE END OF THE QUESTION PAPER 6228 6 [Turn over 6228 7 [Turn over 111533 6228 8 [Turn over

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Additional Info : Gce Mathematics January 2011 Assessment Unit F1 Module FP1 : Further Pure Mathematics 1
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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