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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development and
Computer Networking – Stage 2
(NFQ Level 8)
Autumn 2007
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) You are given that (^)
A^21 and
t
t 2
B^2
(i) Evaluate the matrix A^2. (ii) Find the values of t for which det( B ) = 0. (iii) Find the values of t for which the matrix A B^2 is non-invertible. [10 marks]
(b) Find the inverse of the matrix
= ^ −
A
Hence find the solution of the simultaneous equations
2 3
1 2 3
1 2
− =
x x
x x x
x x [10 marks]
- (a) The times taken, correct to the nearest minute, for 100 students to complete a standard test have been compiled in the following table: Time (min) 40-44 45-49 50-54 55-59 60-64 65-
Number of Students 6 17 30 23 14 10 (i) Find the mean time taken and the standard deviation from that mean. (ii) Use these values to find the time within which 95% of values will lie. (Assume a normal distribution) [12 marks] (b) Box 1 contains 2 red balls and 4 black balls; box 2 contains 3 reds and 5 blacks; box 3 contains 6 reds and 2 blacks. A ball is selected and found to be black. Find the probability that it came from box 2. [8 marks]
- (a) Three percent of the DVD writers produced by a certain company are defective. Calculate the probability that, in a batch of 25 such writers, there will be more than one faulty. [5 marks] (b) Breakdowns in a network are known to be Poisson distributed with a mean of 0.5 per day. Find the probability that in a particular 7-day week there will be at most two breakdowns. [5 marks] (c) The time that a password will be remembered by an employee is exponentially distributed with density function f ( t )= 0. 2 e −^0.^2 t , where t is the time elapsed in days. Find the probability that the password will be remembered longer than ten days. [5 marks] (d) The resistances of mass produced resistors are normally distributed with a mean of 80Ω. If 95% of resistors lie between 79.4Ω and 80.6Ω, find the standard deviation. [5 marks]
- (a) State, giving detailed reasons, whether ( Q , ×), where Q is the set of rational
numbers, is a group. [5 marks] (b) State the properties of an integral domain. Is ( Z (^) 4 ,+ 4 ,× 4 )an integral domain? Note: Z (^) 4 ={ 0 , 1 , 2 , 3 }and + 4 and × 4 represent addition modulo 4 and multiplication modulo 4 respectively. [9 marks] (c) State the condition under which an integral domain is a field. Does ( Z , + ,× ) constitute a field? [6 marks]
7. (a) Evaluate the integrals ∫ −
π π t^ sin(^10 t ) dt
(^2) and
π π
2 2 t^ cos(^2 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
π
π t
f t t 5 , 0
( )^5 ,^0 and f ( t )= f ( t + 2 π)
Find its Fourier series representation. [10 marks]
- (a) Use the Method of Undetermined Coefficients to solve the differential equation y ′′ ( t )+ 8 y ′( t )+ 15 y ( t )= 36 subject to the initial conditions y ( 0 )= y ′( 0 )= 0. Draw a rough sketch of the solution. [10 marks] (b) A voltage v ( t )in mV is described by the differential equation
- 01 v ′(^ t )+ v ( t )= 12 Solve the equation subject to the initial condition v ( 0 )= 0. What is the steady state voltage? What is the time taken to reach this steady state? [10 marks]
Some formulae:
i
i i
i
i i
f
A c fu
f
fx x
i
i i
i
i i
f
A c fu
f
fx
∑ (∑^ )
2 2
2
i i
i i i
i i
i
i i
f f
fu f
fu c
f
f x x s
2 2
( )^2
i
i i i
i i
i
i i
f
fu f
fu c
f
f x μ
= (^) n
i i i
i i i PH PE H
PH E PH PE H
1
n i i^ i
E X XPX
1
=
n i i^ i
X PX
1
σ^2 ( μ)^2 ( )
P X
n X p^ q ( ) = X^ n^ X ^
^
− !
X
PX =λ X^ e −^ λ
f ( t )= ke −^ kt^2
2
2
f ( z )=^1 e −^ z
where
z = x −^ μ
