Mathematics Exam Questions for Software Development and Computer Networking Students, Exams of Mathematics

Download Mathematics Exam Questions for Software Development and Computer Networking Students and more Exams Mathematics in PDF only on Docsity! Cork Institute of Technology Bachelor of Science (Honours) in Software Development & Computer Networking - Stage 1 (NFQ - Level 8) Summer 2006 Mathematics (Time: 3 Hours) Answer FIVE questions, selecting three questions from section A and two questions from section B. All questions carry equal marks. Examiners: Mr. P. Ahern Dr. J. Buckley Dr. A. Kinsella Section A 1. (a) One per cent of the memory sticks produced by a certain company are known to be defective. Find the probability that a batch of 50 such sticks will contain no defectives. What should the defective rate be reduced to in order for this probability to exceed 0.95? [6 marks] (b) Demand for a certain tool in a workshop has been found to follow a Poisson distribution with a mean of 1.2 per day. What is the probability that more than two such tools will be required on a particular day? How many such tools should be kept in stock in order to satisfy demand at least 99% of the time? Assume that, once taken, the tool is not returned until the end of the day. [8 marks] (c) The lifetime (in hours) of a battery is a random variable with probability density function 0.04( ) 0.04 tf t e−= . Find the probability that a battery will last longer than 20 hours. Find the mean lifetime of the battery. [6 marks] 2. (a) The diameters of a sample of 60 cylinders have been measured correct to the nearest 0.1 mm as follows: 61.6 61.5 60.7 62.1 62.1 61.2 60.2 60.4 60.6 61.2 61.6 62.0 61.8 61.7 60.6 60.0 61.1 59.3 60.8 60.2 61.5 60.0 59.9 62.2 60.2 60.6 59.1 60.8 60.3 60.8 61.0 60.3 60.8 60.7 61.2 60.3 61.1 61.0 60.9 61.7 62.4 61.5 60.7 60.9 61.0 61.8 62.4 60.8 60.3 62.3 61.1 59.7 59.6 61.9 59.9 60.3 58.8 60.4 60.0 60.5 Organise the data into a grouped frequency distribution consisting of 5 classes. Calculate the mean diameter and the standard deviation from this mean. In a batch of 200 such cylinders, estimate the number of cylinders with diameter greater than 61.0 mm. Assume a normal distribution. [15 marks] (b) Box A contains 4 red balls, 3 black balls and 1 yellow ball; box B contains 8 red balls, 3 black balls and 3 yellow balls; while Box C contains 5 red balls, 5 black balls and 2 yellow balls. A ball is chosen and is found to be black. Find the probability that it came from box B. [5 marks] 3. (a) Find the values of t for which 1 4 0 1 8 0 0 2 t t = [4 marks] (b) Find the inverse of the matrix 2 1 1 0 2 2 0 1 3 −   −   −  Hence, or otherwise, solve the system of equations 1 2 3 2 3 2 3 2 10 2 2 24 3 15 x x x x x x x − + = − = − = [8 marks] Continued over /… 8. (a) The motion of a point on a monitor screen can be described by the first order differential equation 0.4 ( ) ( ) 200y t y t′ + = . Solve this equation subject to the initial condition (0) 0y = . Find the time at which ( ) 120y t = and the rate at which ( )y t is changing at this time. [10 marks] (b) The angle θ radians through which a door turns is described by the second order differential equation 0.5 ( ) 3 ( ) 4 ( ) 5t t tθ θ θ′′ ′+ + = where t is the time elapsed in seconds. Solve the equation subject to the initial conditions (0) (0) 0θ θ′= = . Find the steady state solution and the approximate time taken to reach this steady state. 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