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Cork Institute of Technology
Bachelor of Engineering in Mechanical Engineering (Design) โ Award
Bachelor of Engineering in Manufacturing Engineering โ Award
(National Diploma in Engineering in Mechanical Engineering โ Award)
(National Diploma in Engineering in Manufacturing Engineering โ Award)
(NFQ โ Level 7)
Autumn 2005
Mathematics and Computing
(Time: 3 Hours)
Instructions Answer FIVE questions, at least ONE question from each Section. Use separate answer books for each Section. All questions carry equal marks.
Examiners: Mr. R. Simpson Mr. J. Connolly Dr. T. Creedon Ms. M. Brennan Ms. J. English Mr. J. Kelleher
Section A
Q1. (a) List the order of precedence of the arithmetic operators. (2 marks)
(b) What is the purpose of the following program? Test it with the values 12, 15, 19, 37 and others if necessary. Find what IT is in each case. Deduce what the code does from your findings.
NOTE 1: The result of mod(a,b) is the integer remainder having divided a by b. Example: The result of mod(13 , 5) is 3
NOTE 2: Trace your working so that partial credit may be given for incomplete or incorrect work.
PROGRAM q1b implicit none
integer::h, j logical::gotIT = .false.
write(*,'(A)',advance='no')'gimmee an integer h --->' read *, h
j = h / 2 do if(j <= 1)exit if(mod(h, j) == 0)then gotIT = .true. exit end if j = j - 1 end do print* if(gotIT)then print, 'so there it is --->', j, ' is IT' else print, 'so there is NO IT at all' end if print* stop'q1b.f90 ends ...' END PROGRAM q1b
(8 marks)
A formula that can be used to find the angle between two vectors
a = (a 1 , a 2 , a 3 ) and b = (b 1 , b 2 , b 3 ) is
cos
a b
ab
Write a program that will read in two vectors a and b into two one dimensional arrays and
then calculate the angle between them according to the equation above.
a.b = a 1 * b 1 + a 2 * b 2 + a 3 * b 3
AND |^ a^ |= a 1^^2 +^ a 22 + a 32 (10 marks)
elapsed time (^) t is thought to be of the form (^) x = x 0 e โ kt. The following experimental data has been obtained: t 0.1 0.2 0.3 0.4 0.5 0. x 12.1 7.4 4.5 2.7 1.6 1.
Use the Least Squares method to find x (^) 0 and k.
(12 marks)
(b) The temperature in a workshop and the time taken to complete a standard task are recorded in the following table: Temperature X
Time Y 10.2 11.8 11.9 9.6 12.6 13.7 10.
Calculate the correlation coefficient and interpret your result.
(8 marks)
Section C
Q5. (a) Use Eulerโs Method to find the approximate value of y ( 0. 2 )for the solution of
y '^ = ( x + y โ 1 )^2 given that h = 0. 1 and y ( 0 )= 2.
(4 marks) (b) Test whether the differential equation
( 1 3 2 2 ) y 2 xy^3 dx
dy
is exact.
Hence determine the general solution of the equation. (7 marks)
(c) Use the method of Undetermined Coefficients to find the general solution of
y ''^ โ 6 y '+ 5 y =โ 10 x^2 โ 6 x + 32 + e^2^ x
(9 marks)
Q6. (a) Find the Laplace transform of each of the following functions.
(i) f ( t )= t^3 ( 6 โ t^4 ) (ii)
t
f t t t e
2 3
( ) 7 cos 5
โ
(5 marks)
(b) Find the inverse Laplace transform of
(i) 4 1
s
s F s (ii) s s
F s 3
(5 marks)
(c) Use Laplace transforms to solve the differential equation
x ''^ ( t )+ 3 x '( t )+ 2 x ( t )= 10 , x ( 0 )= 0 , x '( 0 )= 2
(10 marks)
Q7. (a) The average number of vehicles arriving at a particular junction is 18 per hour. Assuming the vehicle arrivals form a Poisson distribution, calculate the probability that (i) two or more vehicles arrive in any 10 minute period; (ii) one or more vehicles arrive in any single minute.
(5 marks)
(b) A sugar refinery has a processing plant. The amount of raw sugar that can be processed each day has an exponential distribution with a mean of 4.5 tonnes. Find the probability that the plant can process more than 4.5 tonnes in a day. How much raw sugar should be stocked each day so that the chance of the processing plant running out of product is only 0.09?
(5 marks)
(c) In the manufacture of a certain chemical product, three measurements are made each week of the percentage acid content. Data for 8 weeks are available as shown below:
Week 1 2 3 4 5 6 7 8 Readings 5.74 6.21 5.28 5.51 5.19 5.91 5.27 5. 5.54 5.18 5.68 5.81 6.21 6.21 6.22 5. 5.97 5.50 5.00 5.75 4.71 4.71 5.85 5.
Set up a control chart for the sample means. Plot the chart and comment on the process.
(10 marks)
Sample size n '
- A - A '
- 2 1.229 1.
- 3 0.668 1.
- 4 0.476 0.
- 5 0.377 0.
- 6 0.316 0.
- 7 0.274 0.
- 8 0.244 0.
- 9 0.220 0.
- 10 0.202 0.
- 11 0.186 0.
- 12 0.174 0.
