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Cork Institute of Technology
Bachelor of Engineering in Civil Engineering – Award
(National Diploma in Engineering in Civil Engineering – Award)
(NFQ – Level 7)
Autumn 2005
Mathematics & Computing
(Time: 3 Hours)
FIVE questions are to be answered.
All questions carry equal marks.
Answer one question from Section A and use an
individual answer book for this question.
Answer at least two questions from Section B.
Examiners:
Mr. J. Murphy
Mr. P. L. Anthony
Ms. J. English
Mr. T. O Leary
Section A
1 (a) The following Fortran90 program accepts a sequence of integers, adding the positive integers and the
negative integers separately and then prints the number of, the sum of and the average of each kind. A
sample run is shown. Please rewrite the program so that it does the same analysis but into odd and
even positive integers (rather than positive and negative ones). (8 marks)
PROGRAM q1b implicit none
integer:: theNumber integer:: sum_1 = 0,sum_2 = 0
integer:: count_1 = 0,count_2 = 0
!---------------------------------------------------------------------
do write(*,'(A)',advance='no')'gimmee an integer (+ve or -ve, 0 to END) '
read *, TheNumber
if(TheNumber > 0)then sum_1= sum_1 + TheNumber
count_1 = count_1 + 1
else if (TheNumber < 0)then sum_2= sum_2 + TheNumber
count_2 = count_2 + 1
else exit
end if
end do
if(count_1 > 0) then
print 20, count_1, sum_1, REAL(sum_1)/count_ else
print *,’XXXXXXX’ end if
if(count_2 > 0) then
print 30, count_2, sum_2, REAL(sum_2)/count_
else print *,’ZZZZZZZ’
end if
20 FORMAT (//1x, t10,i5, ” positive integers”,&
/ t20, ” with sum -->> “, i8,&
/ t20, ” and average -->> “, f8.3)
30 FORMAT (//1x, t10,i5, ” negative integers”,&
/ t20, ” with sum -->> “, i8,& / t20, ” and average -->> “, f8.3)
!----------------------------------------------------------------------
stop'q1b.f90 ends ....' END PROGRAM q1b
Sample Run:
(b) Write a FORTRAN program to do the following:
Read in the total number of hours worked by a particular employee in a week, the basic hourly rate of
pay and the overtime-hourly rate of pay. The program should then calculate and print the gross pay for
that employee for that particular week.
(12 marks)
2 (a) Flowcharts are often used as a tool to aid the development of computer code. Discuss the use
of flowcharts and their advantages. (4 marks)
(b) Some of the standard symbols used in flowcharts are shown below. Explain where each one
would be used.
(i) (ii) (iii)
(iv) (v) (vi)
(6 marks)
(c) A flowchart is given below in figure 2(b). Write the FORTRAN code.
(12 marks)
gimmee an integer (+ve or -ve, 0 to END) 4
gimmee an integer (+ve or -ve, 0 to END) -
gimmee an integer (+ve or -ve, 0 to END) 7
gimmee an integer (+ve or -ve, 0 to END) 8
gimmee an integer (+ve or -ve, 0 to END) - gimmee an integer (+ve or -ve, 0 to END) 0
3 positive integers
with sum -->> 19
and average -->> 6.
2 negative integers
with sum -->> -
and average -->> -7. q1b.f90 ends ....
Press any key to continue
Section B
- (a) Using Euler’s Method and the Three Term Taylor Method with a step h=0.
estimate the value of y at x=0.1 where
2 y dx
dy = y(0)=5.
2
k 1 k k y 2!
h Note :y y hy (5 marks)
(b) Between the points x=2m and x=5m along beam of span 5m there is a uniformly
distributed load of 7.2kNm
- . Express the load per length w in term of a step
function. The deflection y at any point along the beam is found by solving the
differential equation
w dx
dy EI 2
2
= −
Solve this differential equation where both ends of the beam are fixed. (8 marks)
(c) The deflection y at any point on a beam of span 4m is found by solving the
differential equation
60 [x 2] R[x- 3] dx
d y EI 2
2
=− − +
where R is a constant and “[x-2]” and “[x-3]” are step functions. Solve this
differential equation where y is zero at x=0 and at x=4. At the point x=4 the slope
of y is zero. (7 marks)
- Select any three of the following:
(a) Solve the differential equation
3y 12 y(0) 1 dx
dy
(b) Solve the differential equation
8x 80 x(0) x(0) 0. dt
dx 6 dt
d x
2
2
(c) Find the general solution of the differential equation
16 x dx
dy 4 dx
d y
2
2
(d) Find the general solution of the differential equation
2y 13 0cos3t. dt
dy 3 dt
d y
2
2
- (a) If f(x)=ln(secx) show that f (x) sec x
2 ′ (^) =. Show that the first two non vanishing terms
of a Maclaurin Series of f(x) is given by
x
x f(x)
2 4
= + +.
Find three terms of a Taylor Series expansion of f(x) about the point x= 4
π
Note: 1 cos 0 1 tan 0 0 4
tan 2
cos cosx
secx = = =
π π (8 marks)
(b) Find a Taylor series expansion of f(x,y)=exp(x
2 -4y) about the values x=2,y=1.
Note : A Taylor Series expansion of f(x,y) about values x=a and y=b is given by
f .... 2!
(y b) f (x a)(y b)f 2!
(x a) f(x, y) f(a,b) (x a)f (y b)f yy
2
xx xy
2
x y +
(8 marks)
(c) In measuring the area of a triangular plot the formula absinC 2
A = is used. Estimate
the area of a plot if the values of a, b and C were measured as 2, 6 and 30
o with maximum
errors of 0.01, 0.02 and 0, respectively. (4 marks)
(c) The number of cars N passing between two observation points is believed to be related to
the transit time t by the formula N=at+b. Using the Least Squares Method find the best
values of the constants a and b for the data below.
N 2 5 4 3 1
t 1 2 3 4 5
(6 marks)
- (a) In the manufacture of items it is found that 0.2% of the items produced fail to satisfy
tolerance requirements and are deemed to be defectives. By using both the Binomial
and Poisson Distributions calculate the probability that a sample of 100 items contains
two or three faulty items. (7 marks)
(b) The weights of concrete blocks are assumed to be Normally distributed with a mean value
of 3.50kg and with a variance of 0.006kg
2
. Calculate the percentage of blocks that weigh
between 3.48 and 3.54kg. If 99.9% of blocks have a weight less than a particular weight
W find the value of W. (5 marks)
(c) A quality control chart is used as an aid in controlling the level of a certain chemical in
a product.
Sample No. 1 2 3 4 5 6
Mean 89 88 89 89
Range 3 4 4 4
The table above is incomplete.
For the data above set up a control chart for sample means. Plot the chart and comment
on whether or not the process is under control. (8 marks)
S TANDARD DERIVATIVES
f(x) f ′ (x)^ a=constant
n x
n 1 nx
−
sinx cosx
cosx -sinx
secx secxtanx
tanx sec
2 x
e
ax ae
ax
lnx
x
uv
dx
du v dx
dv u +
v
u
2 v
dx
dv u dx
du v −
STANDARD INTEGRALS
f(x)
f(x)dx a=constant
n x
n 1
x
n 1
x
(^1) ln x
