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Cork Institute of Technology
Higher Certificate in Engineering in Mechanical Engineering – Award
(NFQ Level 6)
Summer 2007
Mathematics
(Time: 3 Hours)
Answer FIVE questions Examiners: Ms. J. English Dr. D. Cremin Mr. J. Connelly Dr. P. Delassus
Q1. (a) A curve is described by the parametric equations x = 3 sin θ+ 8 , y = 2 cos θ.
Find the equation of the tangent to the curve at θ = 2.6radians.
[7 marks]
(b) If 4 x − 2 x sin( y ) − 3 y = 0 , find dydx at the point (2, 1). [6 marks]
(c) Show that the equation f ( ) x = x^3 − 1 + 4 x has a root between x=0 and x=1. Use the Newton-Raphson method with three iterations to find the root correct to three decimal places. [7 marks]
Q2. (a) Given r = -6x^3 -3x 2 y+5y 3 find
∂∂ rx (^) ,∂∂ ry and 2 2
r x
∂ [6 marks]
(b) You are given that
3 2 y k^ t = (^) 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 –3.5% in l. [8 marks] (c) Locate the turning points on the curve y = 53 x^3^ + 2 x^2 − x and establish whether they are maximum or minimum points. [6 marks]
Q3. Determine each of the following integrals:
(i) ∫ x^3^ ln( ) x dx (ii)
6 4 2
x (^) dx x x
(iii) ∫ e^3^ x^ (2 − e^3 x )^4 dx (iv)
(^4 ) 2 3 4
x (^) dx
∫ x^ − x
[20 marks]
Q4. (a) Find the position of the centroid of the figure bounded by the curve y = x^2 –5x, the x-axis, the y-axis and the ordinate at x = 2 and x= 4. b ab
a
X
xydx
ydx
=
b
ba a
Y
y dx
ydx
=
[10 marks]
(b) Calculate the area bounded by the curve y = 3x 2 + 9, the x-axis and the ordinates x = -2 and x = 3. [4 marks]
(c) Find the root mean square of the function y = 5 x − 7 over the interval 3 ≤ x ≤ 7. [6 marks]
Q5. (a) The curve y = x^2 + 7 x is rotated about the x-axis between the limits x = 0.5 and
x = 2. (i) Find the volume of the solid produced. (ii) Find the ordinate X of the center of gravity of the solid.
b 2 a
Vol = π∫ y dx
2
2
b ab
a
X
xy dx
y dx
=
[10 marks]
Q5 contd…/
Q7. (a) Warranty records show that the probability that a new car needs a warranty repair in the first 90 days is 0.05. If a sample of 5 cars is selected, what is the probability that in the first 90 days (i) 2 need a warranty repair? (ii) at least one needs a warranty repair? (iii) Three or more need a warranty repair? [7 marks]
(b) A chemical manufacturer produces aspirin tablets having a mean mass of 5g and a standard deviation of 0.2g. What is the probability that a tablet chosen at random will have a mass (i) less than 5.3g? (ii) greater than 4.6g? (iii) between 4.9g and 5.3g? [7 marks]
(c) Cork Airport has recently analysed its passenger listings and found that, on average, out of every thousand, two are known to be French citizens. Calculate the probability that if a random sample of 2,500 travellers were interviewed, that: (i) three (ii) more than two would be French citizens. [6 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, Z = x σ^ − X
