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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
- (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 f t ( ) = 0.04 e −0.04 t. Find the probability that a battery will last longer than 20 hours. Find the mean lifetime of the battery. [6 marks]
- (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. 61.6 62.0 61.8 61.7 60.6 60.0 61.1 59.3 60.8 60. 61.5 60.0 59.9 62.2 60.2 60.6 59.1 60.8 60.3 60. 61.0 60.3 60.8 60.7 61.2 60.3 61.1 61.0 60.9 61. 62.4 61.5 60.7 60.9 61.0 61.8 62.4 60.8 60.3 62. 61.1 59.7 59.6 61.9 59.9 60.3 58.8 60.4 60.0 60.
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]
- (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
x x x x x x x
[8 marks]
Continued over /…
Section B
- (a) Test the following series for convergence: 1
k k k
∞
and (^2) 1
3 k k k
∞
. [6 marks]
(b) Given the series S n = 12 + 24 + 38 + ...+ 2 nn , show that
1
2^ n^ 4 8 16 2 n
S n = + + + + (^) + and that (^1)
n (^) 2 n (^) 2 4 8 2 n (^) 2 n S S n − = + + + + − (^) +.
Hence, or otherwise, sum the series. [6 marks]
(c) Find the Maclaurin series expansion of f ( ) t = sin(2 ) t.
Hence, or otherwise, evaluate the integral
0
t sin(2 ) t dt t
∫.^ [8 marks]
- (a) Evaluate the integrals
2 2
t t dt
−∫^
and
4 2
t t dt
−∫^
. [5 marks]
(b) The Fourier series representation of the 2 π -periodic function
f t t t
= ≤^ <
−^ ≤^ <
is 1
( ) 40 sin((2^ 1) ) n^2
f t n^ t
π n
∞
Write down the series for each of the 2 π -periodic functions
1
f t t t
π π π
= ≤^ <
−^ ≤^ <
and (^2)
t f t (^) t
π π π
≤^ <
[5 marks]
(c) Find the half range sine series for the function g t ( ) = 4 , t 0 ≤ t < π.
Note: ∫ t sin( nt dt ) = (sin( nt ) − nt cos( nt )) / n^2 [10 marks]
- (a) The motion of a point on a monitor screen can be described by the first order differential equation 0.4 y ′( )^ t + y t ( ) = 200. Solve this equation subject to the initial condition y (0) = 0. Find the time at which y t ( ) = 120 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 θ ′′( )^ t + 3 θ ′( ) t + 4 ( )θ t = 5 where t is the time elapsed in seconds.
Solve the equation subject to the initial conditions θ (0) = θ ′(0) = 0.
Find the steady state solution and the approximate time taken to reach this steady state. [10 marks]
