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Cork Institute of Technology
Bachelor of Engineering in Biomedical Engineering
(Bachelor of Engineering in Biomedical Engineering)
(NQF Level – 7)
Summer 2005
Mathematics & Computing
(Time: 3 Hours)
Answer question 1 from Section A.Answer two questions from Section B. Examiners: Dr. N. DunneMr. D. Tallon Answer two questions from Section C.Use a separate answer book for each section. Ms. M. BrennanDr. T. Creedon The weighting of Computing:Maths is 15:55 Mr. G. McSweeney
Section A
Q1. (a) A palindrome is a word or phrase which spells the same way backwards as forwards. Write a program which determines if a given string is a palindrome. Sample Output Please enter word This is not a palindrome cork Please enter word This is a palindrome navan The bold text is entered by the user, the italic text is generated by the program. (10 marks) (b) The following are prototypes for functions which return the area and circumference of a circle respectively. double areaCircle (double radius) double circumferenceCircle(double radius) (i) Write the code to implement each function. (ii) Write a simple driver program to test both functions.
Area of Circle = π r^2
Circumference of Circle = 2 π r
π = 3. 1415 (10 marks)
Section B
Q2. (a) For what values of t do the given set of simultaneous equations not have a unique solution?
3
2
1 x
x
x t
t (4 marks)
(b) If A =
and B =
determine AB and hence write down
the matrix BT^ A T. (4 marks)
(c) Find the inverse of the matrix A =
and hence solve the system of equations:
2 3 17
x y z
x y z
x y z
Use Cramer’s rule to verify the value of z. (12 marks)
Q3. (a) A force F is of magnitude 28 N and acts in the direction AB , where A is the point (2,3,4) and B is the point (5,1,-2). Find the component of F in the direction − 3 i + 6 j + 2 k. (6 marks) (b) A , B , C are three points with Cartesian coordinates A (2,-1,5), B (3,1,2) and C (1,2,4). Find the vectors AB , BC , and AC. Verify that they form the sides of the triangle ABC. Determine the angle opposite AC. (7 marks) (c) A force F 1 of magnitude 34 kN lies in the direction 8 i − 15 k , while a force F 2 is of
magnitude 56 kN and acts in the direction of the vector 6 i + 12 j − 4 k.
The resultant F of these two forces acts through the point P (-1,3,5). Find its moment about the point A (6,-2,4). (7 marks)
Q6. (a) Find the Laplace transform of each of the following functions, expressing your answer as a single fraction. (i) f ( t )= 5 sin 3 t − te −^2 t (ii) f ( t )= 5 t^4 e −^0.^3 t (5 marks) (b) Find the inverse Laplace transform of (i) F ( s )= (^) ( s +^73 ) 5 (ii) F ( s ) = (^3) s 2^ s (^) +^ − 51 (5 marks) (c) Use Laplace transforms to solve the differential equation x ''^ ( t )+ 4 x '( t )+ 3 x ( t )= 10 , x ( 0 )= 0 , x '( 0 )= 1 (10 marks)
Q7. (a) Customers arrive randomly at a department store at an average rate of 3.4 per minute. Assuming the customer arrivals form a Poisson distribution, calculate the probability that (i) two or more customers arrive in any particular minute; (ii) one or more customers arrive in any 30-second period. (5 marks) (b) The lifetime of a particular type of electronic component is exponentially distributed with a mean of 2000 hours (i) Calculate the probability that a component lasts less than 1000 hours; (ii) Find the percentage of components that last longer than 6000 hours. (5 marks) (c) A new machine fills boxes of cereal by weight. Five observations on amount of fill are taken every hour until 10 such samples are obtained. The data are given in the table. SampleReadings (^1) 16.1 (^2) 16.0 (^3) 16.5 (^4) 16.1 (^5) 15.7 (^6) 16.2 (^7) 16.7 (^8) 16.1 (^9) 17.1 (^10) 15. 16.215.9 16.115.7 16.116.4 16.916.2 16.716.1 16.315.8 16.216.4 16.216.1 16.916.2 15.115. 16.016.1 15.916.4 16.416.2 16.516.5 16.416.5 16.216.1 15.816.6 15.915.8 16.016.1 15.214. Set up a control chart for sample means. Plot the chart and comment on the process. (10 marks)
Short table of Laplace Transforms
'' 2 '
'
0
L f t s F s sf f
L f t sF s f
L e f t F s a
L f t F s e f t dt
at
st
∞ −
f(t) F(s)
(^1 1) s
t n snn! + 1
e −^ α^ t s +^1 α
sin ω t s 2 ω + ω 2
cos ω t s 2 +^ s ω 2
te −^ α^ t ( s +^1 α) 2
e −α^ t^^ − e −^ β^ t ( s +αβ)(− s α+ β)
e −α t^ sin ω t ( s +αω) 2 + ω 2
e −α t^ cos ω t ( s +α s +) 2 α+ ω 2
