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Cork Institute of Technology
Bachelor of Engineering in Building Services Engineering -
Stage 2
(EBSEN_7_Y2)
Summer 2008
Mathematics
(Time: 3 Hours)
Instructions Answer FIVE questions. All questions carry equal marks.
Examiners: Ms. H. Lordan Mr. D. Leonard Dr. M. Smyth
Q1 (a) The parametric equations for a curve are 1 1 2 1 2 1 x t^ y t t t
= −^ = −
Find the value of dydx when x = 1 and y = 1. (8 marks) (b) A function is described by the equation 2 x^3^ − y^3^ − 3 xy^2 = 6. Find dydx for the function at any point and in particular at the point (1, 3) (6 marks)
(c) Show that the equation f ( ) x = x^3 + 4 x − 1 has a root between x = 0 and x = 1. Use Newton’s method twice to get a closer approximation to this root. (6 marks)
Q2 (a) Investigate the function V = 5 te −^2 t for its turning point. Sketch the curve. What is the initial value of V and what happens as t → ∞? (8 marks) (b) Given z = 7 x^3^ + 4 xy^2 − 3 y^3 , find ∂∂^ zx^^ ,∂∂ zy and
2 2
z x
(6 marks)
(c) An equation for heat H generated is given by H = i Rt^2 (watts) where variables i R t , , denote current, resistance and time respectively. Determine the percentage error in the calculated value of H if the error made in measuring i is +4%, the error made in measuring R is -5% and the error made in measuring t is +7%. (6 marks)
Q3 Determine the following:
(a) 2
(^13 ) 0
∫^2 te^ t − dt
(5 marks)
(b) ∫ 5 x e^4 xdx
(5 marks) (c)
1 0 2
∫ 9 + 4 x dx
(5 marks)
(d)
(^5 ) 3
x x (^) dx x x x
(5 marks)
Q6 (a) The velocity, v , in metres per second, of a body t seconds after a
certain instance is given by the formula v = 40(1 − e −0.05^ t ) m sec−^1. Find the distance traveled by the object in the interval from t = 0 to t = 10 seconds. (8 marks) (b) Find the solution of the following initial value problem:^ dydx^ + 2 y = 4 given that y = 5 at x = 0. (6 marks)
(c) Solve the initial value problem
2 2 cos
d y (^) x x dx +^ =^ given that^ x^ =^ 0,^ y =^1 and dydx = 3. (6 marks)
Q7 (a) The mean height of 200 engineering students is 179cm with a standard deviation of 5cm. Assuming that heights are normally distributed, calculate (i) the number of students who have heights greater than 185cm (ii) the number of students who have heights less than 170cm (iii) the number of students who have heights between 170cm and 185cm. (7 marks) (b) The probability that a new car needs a warranty repair in the first 90 days is 0.05. If a sample of 10 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)
(c) The number of accidents occurring in a certain factory follows a Poisson distribution with a mean of one accident per week. Calculate (i) the probability that exactly 2 accidents occur in a given week (ii) more than two accidents occur in a given week. (6 marks)
Newton’s Method: x 1 = x − f^ f ( )'( ) xx
Root mean square: 1 ( ( ))^2
b a
Rms = b − a ∫ f x dx
Centroid:
b ab
a
xydx x ydx
and
b
ba a
y dx y ydx
Volume of revolution / centre of gravity: 2 2 2
xy dx
v = π∫ y dx x = ∫ y dx
Second moment of area about a tangent
π r radius of gyration 5
2 r
Binomial distribution: P r ( ) = n^ C p qr r^ n^ − r
Poisson Distribution: ( ) (^)! e a^ ar P r (^) r
−
Normal Distribution: Z = x σ^ −^ μ
