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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development &
Computer Networking - Stage 2
(KDNET_8_Y2)
Autumn 2008
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
Q1. (a) You are given that (^)
A^21 and
= (^) − t t 2
B^2
(i) Evaluate the matrices A^2 and AB. (ii) Find the value of t for which det( AB ) =− 8. [10 Marks]
(b) Find the inverse of the matrix
A.
Hence, or otherwise, find the solution of the simultaneous equations
2 3
1 2
1 2 3
− =
x x
x x
x x x [10 Marks]
Q2. (a) The lengths of 50 mass-produced components have been measured, correct to the nearest 0.1 mm, as follows: 54.0 54.9 54.9 55.6 56.7 57.0 57.2 57.5 57.7 57. 58.1 58.3 58.3 58.4 58.6 58.6 58.7 59.2 60.0 60. 60.1 60.1 60.2 60.3 60.3 60.4 60.4 60.6 60.6 60. 60.8 60.8 60.9 60.9 61.0 61.4 61.5 61.5 61.6 61. 61.9 62.2 62.5 62.8 63.0 63.1 63.3 63.7 64.5 65. Organise the data into 6 classes, beginning with 54.0-55.9 etc. Calculate the mean length and the standard deviation from this mean. [12 Marks]
(b) A consignment of memory sticks consists of three batches. Batch A contains 100 sticks of which 4 are defective; batch B contains 200 sticks with 4 defective; batch C contains 250 sticks with 6 defectives. A memory stick is chosen and found to be defective. Find the probability that it came from batch C. [8 Marks]
Q3. (a) The time taken to repair a certain fault is normally distributed with a mean of 8. minutes and a standard deviation of 2.0 minutes. Find the proportion of faults that can be repaired in less than 10 minutes. [5 Marks] (b) The number of defects per metre of manufactured cable is Poisson distributed with a mean of 0.05 per m. Calculate the probability that there will be at most one defect in a 10 m run of this cable. [5 Marks] (c) The time in days between breakdowns in a computer system is believed to be a random variable t with probability distribution f t ( ) = 0.2 e −0.2 t. Find the probability that the time between breakdowns will be less than 4 days. Find the mean time between breakdowns. [5 Marks] (d) The defective rate for disks produced by a certain company is 0.5%. Calculate the probability that in a batch of ten such disks there will be more than one defective. [5 Marks]
(c) Find the preorder, inorder and postorder traversals of the graph shown in Figure 2
[6 Marks]
Figure 2
Section B
Q6. (a) Find
100 1
k ∑ = k k (^ +1).^ [6 Marks]
(b) Given that the series ∑
∞ = 1 2
k k^
is convergent, test for convergence the series ∑
∞ = 1 ( +^1 )^3
k k^
[6 Marks] (c) Find the Maclaurin series expansion of f t ( ) = et. Hence, or otherwise, write down the expansions for g t ( ) = e −0.1 t and h t ( ) = t e −0.1 t.
Evaluate the integral
0.2 0. 0
∫ t e^ −^ tdt.^ [8 Marks]
Q7. (a) Evaluate the integrals ∫ −^ ππ t^2 sin(4 ) t dt and ∫− 22 ππ t cos(4 ) t dt. [4 Marks]
(b) Show that the functions f ( t )= t and g ( t )= 6 − t are orthogonal on the interval [0,9]. [6 Marks] (c) A function f ( t )is defined by
( ) 2,^0 2, 2 f t t t
π π π
= ≤^ <
−^ ≤^ < and^ f t ( )^^ =^ f t (^ +^2 π^ )
Find its Fourier series representation. [10 Marks]
Q8. (a) Solve the differential equation 0.01 y ′( ) t + y t ( ) = 20. Solve the equation subject to the initial condition y (0) = 0. Find the steady state value of y t ( ) and the time taken to reach this value. [10 Marks] (b) The distance x ( t )in m travelled by a particle at time t s is given by x ′′( ) t + 8 x t ′( ) + 15 ( ) x t = 90. Find an expression for the distance travelled given that x (0) = 0, x ′(0) = 0 Draw a rough sketch of the distance travelled. [10 Marks]
