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Cork Institute of Technology
Bachelor of Science (Honours) in Computer Applications – Stage 2
(Bachelor of Science in Computer Applications – Stage 2)
(NFQ – Level 8)
Summer 2005
Mathematics
(Time: 3 Hours)
Instructions Answer any FIVE questions.
Examiners: Mr. J. Mulhare Mr. P. O Connor Mr. E. A. Parslow Dr. D. Chambers
Q1. (a) Compare the two measurements of the standard deviation σ n and sn − 1. State when these
measurements may be used. (10 marks)
(b) A computer engineer finds that the sizes (in Kbytes) of files stored on a disk are as follows. 228 219 205 217 224 258 247 228 208 217 213 209 224 211 237 249 236 223 212 232 246 228 243 235 214 209 200 231 213 211 224 231 233 231 228 218 216 240 213 239 204 211 212 208 252 217 222 217 215 217 247 226 252 237 248 229 248 211 218 237 You are required to: (i) Determine the Mean and standard deviation of the above data; (ii) Arrange the data as a grouped frequency distribution of six groups…(200 to 209, etc); (iii) Construct an ogive from the grouped frequency distribution; (iv) Evaluate the median, first quartile and third quartile of the data; (v) Construct a box-plot of the above data .Do you detect any outliers? (30 marks)
Q2 (a) Solve the following simultaneous equations for x, y, and z using Cramer’s rule (Determinants) at least once.
x y
x y z
x y z
(10 marks) (b) Show that the following matrices, [ P ] and [ Q ] , are inverses of each other:
[ ]
P [ ]
Q^1
(5marks) (c) The vectors (^) A and B are given by A = 2 i + j − 5 k and B = 3 i − 4 j − 6 k. You are required to find; (i) The magnitude of B , (ii) The direction cosines of B (iii) The dot product A • B , (iv) The angle between A and B , (v) The cross-product A × B , (25 marks)
Q3 (a) Consider the function ;
f ( x )= 1 + 11 x −^3 ........... x = 1 , 2 , 3 , 4 , 5
Show that f(x) is a well defined discrete probability density function. Sketch the graphs of the function f(x) and of the Cumulative Distribution Function F(x). Find the mean and the variance of the function. (16 marks)
(b) A member of a sports club enters the club lotto by choosing four numbers, at a cost of €2, from the numbers one through twenty-four. The club draws four numbers at random. Any player who matches these four numbers wins €2000. Any player who matches three numbers receives €50. What is the probability that a player wins €2000? What is the probability that a player wins a prize? What is the players expected return on his investment? (16 marks)
(c) Faults occur on computer tape at a rate consistent with the Poisson distribution, where the average number of faults per meter is four. Find the probability that in a two meter length there are ; (i) three faults, (ii) more than three faults. (8 marks)
(b) In order to test the effectiveness in cars of a new “fuel saving device” six cars are tested for fuel consumption, without this device, under test conditions. These six cars are then fitted with the new device and tested for fuel consumption under the same test conditions. The resulting fuel consumptions (km. per litre) were as follows:
Test the Hypothesis that “the new device gives an improvement in fuel consumption” @ 1% level of significance.
Q6 The table below shows the activities involved in a project to expand production at a factory and their associated costs.
You are required to; (i) Draw up a network for the project, at normal durations; (ii) Determine the critical path of the project; (iii) Evaluate the float time for each non-critical chain; (iv) Calculate the cost of the project at normal activity costs; (v) Determine the most cost efficient way to crash the project time by one week.
Car model A B C D E F Cars without new device 15.5^ 13.6^ 13.5^ 13.0^ 13.3^ 12. Cars with new device fitted 16.1^ 14.3^ 13.2^ 13.4^ 16.4^ 16.
Activit y
Preceded by Duration(Weeks ) Normal Crash
Cost(€’000) Normal Crash
A - 10 8 2 4 B A 8 4 3 5 C B 2 2 1 1 D C,H 10 8 2 3 E D 6 4 5 7 F B 3 2 1 2 G C,F 4 3 2 3 H A,E 15 12 12 21 I E,G 4 2 2 5
(5 x 8 marks)
Q7 Write out the six assumptions governing a simple queue. (6 marks)
What is meant by opportunity costs applying to queues? (4 marks) Tourists arrive randomly at an information centre at an average rate of 12 per hour. One receptionist deals with their inquiries at an average time of three minutes per tourist. Assuming simple queue rules apply, evaluate; (i) the probability of having to queue ; (ii) the probability of there being fewer than three people in the queue ; (iii) The average number of customers in the queue; (iv) The average number in the system; (v) The average time per customer spent in the system. If two receptionists were employed to deal with the queue what would the decrease in the average time in the system be? (20 marks) (b) Cars arrive at a car park entry boom at random according to the Poisson distribution at an average rate of two cars per minute. Each car takes twenty seconds to negotiate to boom to enter the car park. Starting at time zero simulate the first six cars to arrive at the boom, giving the time in the queue and time in the system of each car. Use the following values as your six generated random numbers…. 0.712, 0.386, 0.891, 0.301 0.541, 0.238 (10 marks)
Q8 (a) What is a prime number?
Prove, by induction, that any number of the form k^3 + 2 k is divisible by three, where k is an integer. (10 marks) (b) Use the Euclidean Algorithm to: (i) Find the greatest common divisor ( gcd ) of 51 and 39, by subtraction. Find also the gcd (51,39)= gcd ( a,b )in the form ma+nb where m and n are integers. (ii) Find the greatest common divisor of 108 and 42, by division.
(iii) Evaluate the inverse of 32 (mod 59) ……..note 59 is prime. (24 marks)
