Download Mechanical Engineering Exam - Summer 2005 - Mathematics and more Exams Mathematics in PDF only on Docsity!
Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 2
(Bachelor of Engineering in Mechanical Engineering – Stage 2)
(NFQ – Level 8)
Summer 2005
Mathematics
(Time: 3 Hours)
Answer FIVE questions. All questions carry equal marks.
Examiners: Mr. J. E. Hegarty Prof. J. Monaghan Mr. T. O. Leary
- (a) Solve the differential equation
dxdy^ +^ 2y=4e−2x^ y(0)=^6. (i) Find the maximum/minimum value of the solution. (ii) Using the Three term Taylor Method or the Improved Euler Method with a step of 0.1 estimate the value of y at x=0.1. Find the error in this approximation. (13 marks)
(b) In constructing a closed rectangular tank it costs €12m-2^ to construct the base and €4m- to construct all other sides. Find the dimensions of the tank of maximum volume that can be constructed for €48. You are required to use a Lagrangian Multiplier. No marks will be awarded if any other method is used in answering this question. ( marks)
- (a) Find a Taylor series expansion of f(x,y)=ln(x 2 -4y) about the values x=3,y=2. The series is to contain terms deduced from second order partial derivatives. (6 marks) (b) Find the minimum/minimum values of the function V=8x 3 -6xy+y^2 -6x+4. (6 marks) (c) Variables u and v are related to variables x and y by the formulae
3x u = 4y v= 4x 2 − y^2.
Find the partial derivatives of u and v with respect to x and y. (i) Estimate the value of v if the values of x and y were estimated to be 5±0. and 6±0.04, respectively. (ii) If P=f(u) is an arbitrary function in u show that x (^) ∂∂xP+y∂∂yP= 0
(iii) If T is an arbitrary function in u and v write down the relationships between the partial derivatives of T with respect to x and y and those with respect to u and v. Show that (^) x (^) ∂∂T^ x+y∂∂Ty=v∂∂Tv (8 marks)
- (a) Find the Inverse Laplace transform of
s 4s 20s
(^3) + 2 + (5 marks) (b) Using Laplace Transforms solve any two of the following (i) 2 dxdt 12 x(0) 0 x(0) 3 dt
d x 2
2
(ii) ddt^ x 2 6 dxdt 8 x 80 x(0) 0 x(0) 4
2
(iii) ddt^ x 2 6 dxdt 9x 80e-t x(0) x(0) 0
2
(c) Find the zero and the poles of the transfer function L[f(t)]L[y] where
dtdy 4y^4 ydt f(t) y(0) y(0)^0 dt
d y t (^20)
2
+ + + ∫ = = ′ = (4 marks)
∫∫ R (^12 x^2 +^12 y^2 )dA.
where (i) R is the triangular region with vertices (0,0), (1,0) and (0,2) and (ii) R is the region in the first quadrant bounded by the x-axis. the line y= 43 x and the circle x 2 +y^2 =25. (11 marks)
(b) (i) Evaluate the line integral
∫ C 6y^2 dx+6x^2 dy
where C is the perimeter of the region in the first quadrant enclosed by the line y=2x and the parabola y=x^2.
(ii) If V is the volume with a constant cross section described in part (b) (i) and the height is described by 0 ≤ z ≤ 6 evaluate the triple integral
I zz= ∫∫∫
V
xyzdV (9 marks)
- (a) A variate can only assume values between x=0 and x=2. The probability density function is given by p(x)=k(2x-x^2 ). Find the value of k. Show that the mean value, the median value and the modal value of this distribution are equal. ( marks)
(b) The diameters of washers produced are assumed to be Normally Distributed. If 96% of diameters are less than 50.07 mm and 99.7% are less than 50.11 mm, find the mean value and standard deviation (correct to two places of decimal) of the diameters produced. If diameters outside of the range 50±0.15 mm are deemed to be faulty find the average defective rate. (8 marks)
(c) Ten samples of fifty items are taken at random from the output of a machine and the number of defective items in each sample were counted and were recorded : 2, 2, 1, 3, 2, 1, 2, 1, 0, 1 Calculate the mean number of defective items per sample.. Calculate the probability that a batch of 100 of these items contains three or more defectives items? Use both 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 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
∫ UdV=^ UV−∫VdU
