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Cork Institute of Technology
Higher Certificate in Engineering in Mechanical
Engineering – Award
(NFQ Level 6)
Autumn 2006
Mathematics
(Time: 3 Hours)
Answer FIVE questions Examiners: Ms. J. English Dr. D. Cremin Mr. J. Connelly Mr. R. Simpson
Q1. (a) Find the general solution of
dx
dy
for the parametric equations x = t^2. e t
and y = t e. t. Find
dx
dy
at t = 2.
[7 marks]
(b) A function is described by the equation 4 x^3 − 2 y^3 + 2 xy^2 = 4. Find
dy dx
for
the function at any point and in particular at the point (2,3).
[6 marks]
(c) Show that the function f ( ) x = x^3 + 5 x − 7 = 0 has a root between x=1 and x=3.Use three iterations of the Newton-Raphson method to find the root correct to three decimal places. [7 marks]
Q2. (a) Given z = -7x 3 + 3xy 2 +8y^3 find
z z x y
and
2 2
z x
[6 marks]
(b) You are given that (^2) 3 k t y l
= where k is a constant and t and l are variables. Use a
calculus method to find the approximate percentage error in y due to errors of -2% in t and +1.5% in l. [8 marks]
(c) Locate the turning points on the curve y = x^3 +x^2 -5x+2 and establish whether they are maximum or minimum points. [6 marks]
Q3. Determine each of the following integrals:
(i)
6 2 4
x dx x x
(ii)
(^4 ) 4 2
x dx x x
∫^ −
(iii)
4 2
∫ x^ + xdx (iv)^ ∫ x^3^^ ln( ) x dx
[20 marks]
Q4. (a) Find the position of the centroid of the figure bounded by the curve y = x 2 and the ordinates at x=1 and x=.
b
a b
a
X
xydx
ydx
=
b
a b
a
Y
y dx
ydx
=
[8 marks]
(b) Calculate the area bounded by the curve y = 8 − x^2 , the x-axis and the ordinates at x = -2 and x= [6 marks]
(c) Find the root mean square of the function y = 4 − x^2 over the interval 0 ≤ x ≤ 2
[6 marks]
Q7. (a) The mean diameter of a sample of 450 rollers is 22.2mm and the standard deviation of 0.75mm. How many rollers would be expected to have a diameter (i) less than 21.32mm. (ii) greater than 19.91mm. (iii) between 19.37mm and 23.2mm. [7 marks]
(b) Production of resistors includes, on average, 10 percent defectives. Determine the probability that a sample of 6 resistors contains: (i) 3 defective resistors. (ii) fewer than 4 defective resistors. [6 marks]
(c) The number of times a patrol car passes through a particular neighborhood is on average 2.5 times per nightly shift. Calculate the probability that on any particular night shift
(i) three patrol cars pass through the neighborhood (ii) more than three patrol cars pass through the neighborhood (iii) What is the probability that, over a period of four nightly shifts, exactly 6 patrol cars will pass through the neighborhood? [7 marks]
Probability Distributions
Binomial Distribution: P r ( ) = n^ C p qr r^ n^ − r
Poisson Distribution:
e m^ mr P r r
−
Normal Distribution: Standard units,
x X Z
