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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 3
(Bachelor of Engineering in Mechanical Engineering - Stage 3)
(NFQ - Level 8)
Summer 2005
Mathematics and Statistics
(Time: 3 ½ Hours)
Answer FIVE questions, at least TWO questions from each Section. Use separate answer books for each Section. All questions carry equal marks. Statistical tables are available.
Examiners: Mr. T. O’Leary Mr. D. O’Hare Prof. J. Monaghan Mr. J. Hegarty
Section A
- (a) Components of a certain type are shipped to a supplier in batches of twenty. Three
components are randomly selected from a batch and tested. If one such batch contains three defective components, (i) what is the probability that none of the selected components is defective? (ii) what is the probability that at least two of the selected components are defective? (6 marks) (b) A discrete random variable X is distributed as follows:
r 100 200 300 400 P( X = r ) 0.35 0.45 0.1 0. Calculate the expected value and the variance of X. (6 marks) (c) A random sample of 100 items is selected from a large batch of items. Find the probability that: (i) the sample proportion defective is greater than 0.07 if the batch proportion defective is 0.05; (ii) there is no more than one defective in the sample if the proportion defective in the batch is 0.03. (4 marks) (d) A loom experiences breakdowns at a rate of one every 10 hours, on average, with breakdowns occurring according to the pattern of a Poisson distribution. A particular style of cloth that is being produced will take 30 hours on this loom. If four or more breakdowns occur during the production run, the cloth will be unsatisfactory. What is the probability that the cloth is finished with acceptable quality? (4 marks)
- (a) The content, X , of magnesium in an alloy is a random variable, distributed according to the following probability density function
=^ ≤ ≤
0 , otherwise
,for 0 6 ( ) 18 f x x x
(i) Verify that this is a well-defined probability density function. (ii) Find the mean value of X. Is the mean greater than or less than the median in this distribution? Justify your answer. (8 marks) (b) A continuous random variable, X , has moment generating function ( ). t
M (^) X t −
λ
λ
Show that the mean and the standard deviation of X are both equal to 1. λ
(6 marks)
(c) The Rockwell hardness of a particular alloy is normally distributed with a mean of 68 and a standard deviation of 3. (i) If a specimen is acceptable only if its hardness is between 62 and 75, what is the probability that a randomly chosen specimen has an acceptable hardness? (ii) 90% of specimens have a hardness value greater than k. What is the value of k? (6 marks)
- (a) (i) Write a note on the distribution of sample means.
(ii) A random sample of size 40 is taken from a population which is exponentially distributed, with probability density function f ( x )= 0. 05 e −^0.^05 x , x > 0.
What is the probability that the sample mean is less than 25? (6 marks)
(b) Twelve alloy filaments are selected and their melting points are determined, with the following results: 320 326 325 318 322 320 320 317 329 312 308 314 (i) Produce a normal probability plot for these data, and comment. (ii) Calculate the sample mean and variance, and establish a 95% confidence interval for the population mean melting point. Also establish a 95% upper confidence limit on the population variance. (7 marks)
Section B
- (a) Plot two cycles of the triangular wave below and express these cycles in terms of
the Heaviside Unit Step Function. Find the Laplace Transform of one cycle.
f(t)=
8t- 8 if 1 t 2
8 8t if 0 t (^1) f(t+2)=f(t).
By using Laplace Transform solve the differential equation
4y f(t) y(0) y(0) 0 dt
d y 2
2
where f(t) is one cycle of the wave above. (12 marks)
(b) By using Laplace Transform solve the differential equations
ky f(t) y(0) y(0) 0 dt
cdx dt
m d x 2
2
where (i) m=1, c=0, k=9 and f(t)=36sin3t, (ii) m=1, c=2, k=1 and f(t)=3δ(t-2)+6δ(t-4)+…. (8 marks)
- (a) Find the eigenvalues and the corresponding eigenvectors of the matrices
A B =
(i) Show that the eigenvectors of the matrix A are linearly independent and mutually orthogonal. Find an orthogonal matrix P where PTAP is diagonal.
(ii) Does a non-singular matrix H exist where H-1^ BH is diagonal? Justify your answer. If they exist write down the matrix (^) H and the diagonal matrix.
(iii) Find the general solution of the set of simultaneous differential equations x (^) = Ax dt
d (^) (14 marks)
(b) Two masses are attached to two springs and the displacements of these masses
x 1 and x 2 are found by solving the system of differential equations
2 1 2
1 1 2 3x 16 x 11 x
3x 19 x 8x ″= −
By assuming periodic solutions of the form xi =Ri cos(ωt-α) find the general solution of this set of simultaneous differential equations. (6 marks)
- (a) A function f(t) is even in t , is periodic with a period of 2 and is defined over one
half of a cycle by
f(t)=1-t where 0 ≤ t≤ 1. Sketch this function over the interval [-2,2]. Find the equation of the line that represents the second half of the cycle. Find the Fourier Series for f(t).
( ) (^ )^ (^ )
∫ (^ )^ (^ )^ (^ )
2 2
2 2
n π
sinnπx nπ
(1-x)sinnπx (1-x)cosnπ^ x
n π
cosnπx nπ
Note: (1-x)cosnπx (1-x)sinnπ^ x (8 marks)
(b) The temperature u(x,t) at any point on the rod of length L is found by solving the partial differential equation
2
2 x
k u t
u ∂
Both ends of the rod are insulated, that is, u (^) x (0,t)=ux (L,t)=0. The initial temperature distribution is given by u(x,0)=f(x). Solve this partial differential equation. In particular find the solution where (i) f(x)=20 and (ii) f(x)=20x.
Note: (^)
∫ L
sin nπx n π
L
L
cos nπx nπ dx^ Lx L xsin nπx 2 2
2
∫ L
cos nπx n π
L
L
sin nπx nπ
dx^ Lx L
xcos nπx 2 2
2 (12 marks)
8 (a) (i) Find the first two sampled values of the function whose z Transform is given by
STATISTICAL FORMULAE
z = x σ− μ z x
n
z
p
n
π ( 1 π) z^
x x
n n
= −^ −^ −
12 1
22 2
z p^ p n n
= −^ −^ −
1 2 1 2 1 1 1
2 2 2
π π π π^ χ^
2 = ∑ (^ O^ − E )^2
E
t x n − (^) s n
1 =^ −^ μ
/ t^
x x s (^) n n
n n p
1 2 2
1 2 1 2
1 2
= −^ −^ −
χ n σ
n s − =^
12
2 2
Fn n (^2111) ss^1
2 12 − 22 22
σ σ
Regression:
Simple linear model: y = β 0 + β 1 x +ε.
V (^) S e S x x xx xx
( β∃ 1 ) σ ( ).
(^22)
= , where = ∑ −
2 −
n
s SSE e
Confidence interval for mean value of y at x = x 0 : ( β∃ 0 β∃ 1 0 ) (^0 )
- x ± t sc e n + x^ S − x xx
Prediction interval for (^) y at x = x 0 : xx
c e S
x x x ts n 0 2 0 1 0
( βˆ +βˆ )± 1 +^1 +( − ).
Z-TRANSFORMS
For a sequence f(n) the Z-Transform is defined by ∑
∞
n 0
F(z) f(n)zn
f(t) F(z) U(n)= z 1
z − a^ N z a
z − n (^2) (z 1)
z − n 2 (z 1)^3
z(z 1) −
e bn z eb
z − cosωn z 2 zcos 1
z(z cos ) (^2) − +
sinωn z -2zcos 1
zsin
a n f(n)
a F z
nf(n) -zF(z)
f(n+1) zF(z)-zf(0)
f(n+2) z^2 F(z)−z^2 f(0)−zf(1)
