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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 2
(NFQ – Level 8)
Summer 2007
Mathematics
(Time: 3 Hours)
Answer FIVE questions. All questions carry equal marks.
Examiners: Mr. P. Clarke Prof. M. Gilchrist Mr. T O Leary
- (a) (i) The differential equation below can be solved by using the Method of Undetermined Coefficients or by using Laplace Transforms.
dxdy^ +3y=18x y(0)=^2 Solve this differential equation by using one of these methods. (ii) By using two numerical method with a step of 0.1 estimate the value of y at x=0.1. (9 marks)
(b) A plate of varying thickness lies in the region in the first quadrant bounded by the line y=2x and the parabola y=x 2. The mass per unit area at any point (x,y) on the plate is given by ρ=(12+0.6x)kgm-2^. By evaluating a double integral find the mass of the plate. (6 marks)
(c) An open rectangular box is to contain 4m^3 of liquid. By using a Lagrangian Multiplier find the dimensions of the box so that the surface area is at a minimum value. (5 marks)
- (a) Find a Taylor Series expansion of the function f(x,y)=ln(x^2 -3y) about the values x=2,y=1. The series is to contain terms deduced from second order partial derivatives. (6 marks)
(b) Find the partial derivatives of u and v with respect to x and y where
u sin 1 y 2x
= −^ ^
v= 4x 2 − y^2. cont//…
(i) If T=f(u) is an arbitrary function in u show that
x T^ y T 0 x y
(ii) Estimate the value of v if the values of x and y were estimated to be 5 and 6 with maximum errors of 0.03 and 0.10, respectively. (8 marks)
(c) Find the maximum/minimum values of the function V=x 3 +y 2 -4xy+4x+3. (6 marks)
- (a) Find the Inverse Laplace transform of the expressions
(i) (^) s +4s +3s 3 122 (ii) (^) (s-2)(s -4)^402 (10 marks)
(b) By using Laplace Transforms solve the differential equation 2 2
d y (^) +4 dy 20y 80 y(0)=y (0)= dt dt
(c) Find the zero and the poles of the transfer function L[f(t)]L[y] where
ddt y 2 4 dydt 9 y 10 0 tydt f(t) y(0) y(0) 0 2
+ + + ∫ = = ′ = (4 marks)
- In answering the following question you are required to use the Method of Undetermined Coefficients. No marks will be awarded if any other method is used. The displacement x of a mass attached to a dashpot and a spring at any instant t is found by solving the differential equation
cdxdt kx f(t) x(0) x(0)^0 dt m d x 2
2
(a) Solve this differential equation when m=1, c=6, k=8, f(t)=32. (6 marks)
(b) Find the general solution of this differential equation when m=1, c=4, k=4, f(t)=16t. (6 marks) (c) Find the general solution of this differential equation when m=1, c=0, k=4, f(t)=16sin2t. (8 marks)
(b) The region R is the region in the first and fourth quadrant bounded by the lines y=x, y=-x and an arc of the circle x 2 +y^2 =4. (i) Evaluate the line integral below where C is the arc of the circle above
2 2 C
∫(4x +4y )dx+4xydy
(ii) By evaluating a double integral locate the centroid of this region. (10 marks)
- (a) A variate can only assume values between x=1 and x=2. The probability density function is given by
p(x)= 10 1 (3x^2 +2x).
Find (i) the mean value and (ii) the median value of the distribution correct to two places of decimal. Find the median value by using Newton’s Method. This value is close to x=1.6. (8 marks)
(b) The diameters of bolts produced by a certain machine are assumed to be normally distributed with a mean value of 45 mm with a variance of 0.02 mm^2. Calculate the probability of a diameter of such a bolt being (i) greater than 45.3mm and (ii) .between 44.6 and 44.8mm. If 0.1% are deemed to be oversize and 0.1% are rejected for being undersize what are the critical limits? (6 marks)
(c) Ten samples of fifty items are taken at random from the output of a machine and the number of defective items were counted and were recorded 2, 0, 1, 0, 2, 1, 0, 0, 1, 1 Calculate the average defective rate. What are the chances of a batch of one hundred of these items contains at most two defectives? Use the Binomial and the Poisson distributions. (6 marks)
f(x) f ′ (x)^ a=constant x n^ nxn- lnx x
e ax^ ae ax sinx cosx cosx -sinx sin-1^ (x) 2
1-x
sin 1 x a
− ^
a -x uv dx vdu dx u dv+
v
u
v^2 dx
udv dx v du−
f(x) ∫ f(x)dx a=constant
sinx -cosx cosx sinx e ax a
(^1) e ax
∫ UdV=^ UV−∫VdU
