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GCE JUN 2010 : AS, F1: Further Pure Mathematics 1

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ADVANCED SUBSIDIARY (AS) General Certificate of Education 2010 Mathematics assessing Module FP1: Further Pure Mathematics 1 AMF11 Assessment Unit F1 [AMF11] THURSDAY 24 JUNE, MORNING 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 logez 5209 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three signi cant gures unless otherwise stated. You are permitted to use a graphic or a scienti c calculator in this paper. 1 The matrix A is given by A= 2 3 7 2 (i) Show that the eigenvalues of A are 5 (ii) Find a unit eigenvector corresponding to the eigenvalue 5 2 [5] [4] A system of linear equations is given by 9x + 2y + az = 0 x y 3z = 0 (a 1)x + y + 3z = 0 (i) Find the values of a for which there is a solution other than x = y = z = 0 (ii) For the value a = 6 find the general solution. 5209 [6] [5] 2 [Turn over 3 The permutations p= 123 123 q= 123 132 r= 123 213 s= 123 312 t= 123 231 u= 123 321 form a group under composition. (i) Copy and complete the following group table. Note that q r = s indicates that r followed by q produces s. p q r s t u p p q r s t u q q p s r u t r r t p u q s s s u q t t r u u u s t [6] (ii) Find the period of the element s. (iii) Write down the inverse of the element r. [2] (iv) Write down a subgroup of order 3 5209 [2] [2] 3 [Turn over 4 The equation of a circle is given by x 2 + y2 + 4x 10y + 9 = 0 (i) Find the equation of the tangent to the circle at the point (2, 7). (ii) Show that this tangent passes through the point (3, 5). [1] (iii) Hence, or otherwise, find the equation of the other tangent to the circle from the point (3, 5). 5 [5] [5] (a) (i) Write down the matrix M which represents a rotation of 180 about the origin O. [2] (ii) Find the image of the point (7, 3) under the transformation represented by M. (b) The matrix N = [2] 12 4 7 (i) The transformation represented by N maps a region S, of area 3 cm2, to a new region T. Find the area of T. [3] (ii) Under the transformation represented by N, the line y = mx is reflected in the x-axis. Find the possible values of m. 5209 [8] 4 [Turn over 6 The complex numbers z1 and z 2 are given by z1 = 2 + 2 i and z2 = 1 + 3 i (i) Find the modulus and argument of each of z1 and z2 [6] (ii) On a clearly labelled Argand diagram, plot the points representing z1, z 2 and z1 + z 2 [5] (iii) Hence, without using a calculator, show that tan 5209 7 2+ 3 = 24 1+ 2 5 [6] [Turn over [Turn over 1847-010-1

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Additional Info : Gce Mathematics June 2010 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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