E UNIVERSITY OF CAMBRIDGE NATURAL SCIENCES TRIPOS Part IA Wednesday, 15 June, 2011 9:00 am to 12:00 pm MATHEMATICS (2) Before you begin read these instructions carefully: The paper has two sections, A and B. Section A contains short questions and carries 20 marks in total. Section B contains ten questions, each carrying 20 marks. You may submit answers to all of section A, and to no more than five questions from section B. The approximate number of marks allocated to a part of a question is indicated in the right hand margin. Write on one side of the paper only and begin each answer on a separate sheet. (For this purpose, your section A attempts should be considered as one single answer.) Questions marked with an asterisk (*) require a knowledge of B course material. At the end of the examination: Tie up all of your section A answer in a single bundle, with a completed blue cover sheet. Each section B question has a number and a letter (for example, 11S). Answers to each question must be tied up in separate bundles and marked (for example, 11S, 12X etc) according to the number and letter affixed to each question. Do not join the bundles together. For each bundle, a blue cover sheet must be completed and attached to the bundle, with the correct number and letter written in the section box. A separate green master cover sheet listing all the questions attempted must also be completed. (Your section A answer may be recorded just as A: there is no need to list each individual short question. ) Every cover sheet must bear your examination number and desk number. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS 6 blue cover sheets and treasury tags None Green master cover sheet Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 SECTION A 1 Determine the constant c so that the vectors u = 3i - 2j and v = 4i + cj are perpendicular. [2] 2 Evaluate the indefinite integral Z x2+3x-2 2x +3 dx. [2] 3 Find the stationary points of the function 2x2 - 5x - 25 y = x2+x-2 . [2] Locate the points of discontinuity. 1] 4 Verify that if A =( / 4 3 ) 1 2 ), then A2 - 4A - 5I = 0, where I is the identity matrix. [2] 5 Find the real and imaginary parts of the complex number . [2] 3i + 4 5 - 2i Natural Sciences IA, Paper 2
UNIVERSITY OF CAMBRIDGE 3 6 Give the first three non-zero terms of the Taylor expansion of the function f (x) = Vx about the point x = 1. [2] 7 Solve the differential equation dx dy x2 + 1 xey with y(0) = 0. [2] 8 Sketch the curve represented by the equation |z - 1| = 2 for complex z. [2] 9 Determine the