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Cork Institute of Technology
Bachelor of Engineering in Civil Engineering - Award
(NFQ Level 7)
Spring 2007
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. Lapthorne Mr. J. Kindregan Ms. J. English Mr. T. O Leary
Section A
1 (a) List the order of precedence of the arithmetic operators. (2 marks)
(b) What is the output of the following Fortran90 program?
NOTE: Trace your working so that partial credit may be given for incomplete or incorrect work.
(8 marks)
(c) Write a program that will read in the coordinates of two points ( x 1 , y 1 )and ( x 2 , y 2 )and
calculate the distance between them. The program should allow more than one set of data to be entered if the user wishes.
Note : The distance between the two points (x 1 ,y 1 ) and (x 2 ,y 2 ) is given by
( x 2 −x 1 ) 2 +( y 2 −y 1 )^2
and the sqrt( ) function returns the square root of a real. (10 marks)
PROGRAM q1_b implicit none integer::limit = 3 integer::a, b do a = limit,0, - do b = limit, 0, - if( (0 == a) .or. (limit == a) )then write(,'(i1)',advance='no')a else if( (b - a == 0))then write(,'(i1)',advance='no')a else if( (b + a == limit))then write(,'(i1)',advance='no')b else write(,'(a)',advance='no')'#' end if end do print* end do STOP'q1_b.f90 ends ....' END PROGRAM q1 b
(b) Write a menu-driven program that allows the user to convert measurements either from miles to kilometres (1 mile=1.60935 km), from feet to meters ( 1 foot=0.3048m), or from degrees Fahrenheit to degrees Celsius {C= 95 (F-32)}. A sample run of the program should proceed as follows:
Available options are:
- Display this menu.
- Convert miles to kilometres.
- Convert feet to meters.
- Convert degrees Fahrenheit to degrees Celsius.
- Quit.
Enter an option (0 to see menu): 3 Enter degrees Fahrenheit: 212 This is equivalent to 100.00 degrees Celsius.
Enter an option (0 to see menu): 0 Available options are:
- Display this menu.
- Convert miles to kilometres.
- Convert feet to meters.
- Convert degrees Fahrenheit to degrees Celsius.
- Quit.
Enter an option (0 to see menu): 1 Enter miles: 10 This is equivalent to 16.1 kilometers
Enter an option (0 to see menu): 2 Enter number of feet: 1 This is equivalent to 0.31 meters
Enter an option (0 to see menu): 4 (12 marks)
Section B
- (a) A light beam of span 5m is simply supported at its endpoints. At the point x=3m there is a load of 36kN. Between the points x=2m and x=5m there is a U.D.L. of 24kNm-1^. Express the Bending Moment M in terms of step functions. By solving the differential equation
EI (^) dxd y 2 M
2 = y(0)=y(5)= find the deflection y at any point on the beam. (8 marks)
(b) Solve the differential equation
EI (^) dxd y 2 R[x-1]-12[x-4]
2
where y is zero at x=1 and at x=5. At x=5 the slope of y is zero. In this differential equation R is a constant and “[x-1]” and “[x-4]” are step functions (7 marks)
(c) A light beam of span 4m is simply supported at its end points and carries a load w that varies in a linear fashion with the distance x from one end of the beam. The Bending Moment M can be found by solving the differential equation ddx M 2 w 2 = −. Solve this differential equation where w=0 at x=0 and w=12kNm-1^ at x=4m. Also find the maximum value of the Bending Moment. (5 marks)
- (a) If f(x)=ln(1+cos2x) show that f (x)=-2tanx′^.
With the aid of this result find the first three non vanishing terms of a Maclaurin Series and the first three terms of a Taylor Series expansion of f(x) about the point x= π 4.
Note: 2cos 2 A=1+cos2A sin2A=2sinAcosA secA =cosA^1
cos π^1 tan π^1 cos π 1 cos 0 1 tan 0 0 4 2 4 2
. (9 marks)
(b) Find the Taylor series expansion of f(x,y)=ln(3x-4y) about the values x=3,y=2. Note : A Taylor Series expansion of f(x,y) about values x=a and y=b is given by
f(x, y) f(a,b) (x a)f (y b)f (x2!a) f (x a)(y b)f (y2!b) fyy ....
2 xx xy
2 x y + = + − + − + − + − − + − (6 marks)
(c) In estimating a quantity V the formula V= 2x+y 2x-y is used. Find the partial derivatives of V with respect to x and to y. Hence estimate the value of V where the values of x and y were measured as 2 and 3 with maximum errors of 0.01 and 0.02, respectively. (5 marks)
Section C
- (a) Simplify as much as possible the matrix expressions (i) ( A -2 B )( A +2 B ) (ii) ( A T^ -2 I ) T( A -1 + I ) (3 marks)
(b) Find the inverse of the matrix A below. If they exist find the matrices BC and CB.
A =
B = ( 1 2 3 ) =^ ^11
C (9 marks)
(c) By using Gaussian Elimination without partial pivoting solve the set of linearly dependent equations (i). By using Cramer’s Rule for the set of simultaneous equations (ii) find the value of z.
(i)
1 1 1 1 x 1 2 4 3 4 y (^) = 2 3 7 4 5 z 3 5 3 6 7 w 5
(ii)
5x-2y-3z= 3x- y-2z= 2x+3y-4z=
(8 marks)
- (a) (i) After production of items by a manufacturer the items are packed into batches of 40 and in a number of these batches the number of defective items were counted
No. of defectives 0 1 2 3 > No. of batches 17 27 5 1 0
If an item is picked at random calculate the probability that it is defective. By using the Binomial Distribution calculate the probability that a random sample of 80 items contains 3 or more defective items. (ii) In a computer laboratory on average each student sent three files to a printer during any hour. By using the Poisson Distribution calculate the probability that a student will send two or three files to the printer during any ten minute period. (7 marks)
(b) The lengths of concrete blocks of a large batch are assumed to be Normally distributed with a mean value of 450mm and with a standard deviation of 0.16mm. What percentage of blocks have lengths between 450.3mm and 450.6mm? If 99.8% of lengths are less than some critical value L find the value of L. Calculate the probability that the mean length of a sample of 16 blocks will be greater than 449.4mm. (6 marks) (c) To monitor the breaking strength of precast concrete 6 samples of size 3were taken and the measurements are recorded below. The table is incomplete.
Sample No 1 2 3 4 5 6 Strength Nmm -2^ 54.
Mean 54.7 54.9 54.7 55. Range 0.5 0.4 0.4 0.
Set up and plot a control chart for samples means. Include on the chart the mean values of the samples above and comment on the control of the process. (7 marks)
S TANDARD DERIVATIVES & INTEGRALS
f(x) f^ ′ (x)^ a=constant x n nx n−^1 sinx cosx cosx -sinx secx secxtanx tanx sec^2 x e ax^ ae ax lnx x
uv dx vdu dx u dv+
v
u
v^2 dx
udv dx v du−
f(x) ∫ f(x)dx^ a=constant
xn n 1
x n^1
x
(^1) ln x
