UNIVERSITY OF LONDON BSc/MSci/MSc EXAMINATION 2016 For internal students of Royal Holloway DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1100 (MOCK EXAM): FROM EUCLID TO MANDELBROT Time Allowed: TWO hours Attempt all questions. MT and Type A Calculators are permitted. CRoyal Holloway, University of London 2016 Page 1 of 5 2015-16
MT1100 (MOCK EXAM) 2015-16 Page 2 of 5 Section A (60 marks) 1. Let A, B and C be points in the plane with |AB| = a and |AC| = b (see the diagram below for an example). Given A, B and C, show that the length a + b can be constructed by ruler and compass. [Give a diagram, a list of steps, and a justification.] a B A 0 b C (10 marks) 2. Consider the conic C given by the equation x2 + 4y2 - 2x - 8y - 31 = 0. Write the equation for C in normal form. Hence, or otherwise, determine whether C is a hyperbola, parabola or ellipse. (10 marks) NEXT PAGE
MT1100 (MOCK EXAM) Page 3 of 5 2015-16 3. Let f : [0,1] -> [0, 1] be a function. The curve y = f(x) is drawn in the first graph below, along with the line y = x. The second graph shows a typical cobweb diagram for f. (a) What are the fixed points of f (approximately)? (b) Write down the (approximate) values of any stable fixed points and stable limit cycles for f. 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 (10 marks) 4. Let b be an integer, b ? 2. Let x, y € {0, 1, ... , b-1}. Let r = (0.iyi)b = (0.xyxxyxxyx ... )b. Show that r can be written (in decimal) as T = (b2 + 1)x + by b3 - 1 . (10 marks) NEXT PAGE