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GCE JUN 2007 : AS, F1: Further Pure Mathematics 1

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ADVANCED SUBSIDIARY (AS) General Certificate of Education 2007 Mathematics assessing Module FP1: Further Pure Mathematics 1 AMF11 Assessment Unit F1 [AMF11] TUESDAY 26 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 AMFP1S7 2021 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 equation p 8 x q = 2 p y 7 (i) Show that the equation has a unique solution if p 4 (ii) If p = 4 find the value of q for which the equation has infinitely many solutions. 2 [4] [2] The matrix A is given by 1 0 2 A = 0 2 0 2 0 1 (i) Show that = 1 is one of the eigenvalues of A and find the other two eigenvalues. [7] (ii) Find a unit eigenvector corresponding to = 1 [4] AMFP1S7 2 [Turn over 3 (a) (i) Write down the matrix A which represents a reflection in the line y = x [2] 1 0 The matrix B = represents a reflection in the x-axis. 0 1 The matrix C is given by C = AB (ii) Find the matrix C and interpret the equation C = AB geometrically. [4] 2 2 (b) The matrix M = represents a linear transformation of the x-y plane. 5 3 Find the equations of the straight lines through the origin O which are rotated 90 about O under the transformation given by M 4 [7] (a) Given that z1 = 22 + 16i and z2 = 3 i find: (i) z1 + z2; [1] z1 (ii) , z2 [3] giving your answers in the form a + bi, where a and b are real numbers. (b) (i) Sketch on an Argand diagram the locus of those points z which satisfy z ( 4 + 5i) = 2 [3] (ii) On the same diagram, sketch the locus of those points w which satisfy |w (4 + 5i)| = |w + (2 + i)| [3] (iii) Hence or otherwise find the minimum value of |z w| where z, w are complex numbers satisfying the equations in parts (i) and (ii) respectively. AMFP1S7 3 [4] [Turn over 5 Let M be the set of all matrices of the form cos sin sin cos where is any real number. Show that M forms a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.) 6 [9] (i) Find the centre and radius of the circle C1 given by the equation x2 + y2 4x 8y + 15 = 0 [2] (ii) Find the equation of the tangent to C1 at the point (3, 6) [5] (iii) Show that this tangent passes through the point (7, 4) [1] (iv) Find the equation of the other tangent from the point (7, 4) to the circle C1 [5] The circle C2 is given by the equation x2 + y2 7x 7y + 22 = 0 (v) Find the coordinates of the points of intersection of the circles C1 and C2 THIS IS THE END OF THE QUESTION PAPER S 1/06 0000 7-008-1 [Turn over [9]

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Additional Info : Gce Mathematics June 2007 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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