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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 2 Examinations 2008/
Module Title : Engineering Mathematics 202
Module Code: MATH
School: Mechanical & Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2 Bachelor of Engineering (Honours) in Biomedical Engineering-Stage 2
Programme Code: EMECH_8_Y EBIOM_8_Y
External Examiner(s): Dr.P.Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Select any four questions. The questions carry equal marks.
Duration: 2 Hours
Sitting: Summer 2009
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
- (a) Find the eigenvalues and the corresponding eigenvectors of the matrices
A=
B =
Show that the eigenvectors of A are linearly independent but not mutually orthogonal. (16 marks) (b) The displacements x 1 and x 2 of two masses attached to two springs are found by solving the set of simultaneous differential equations below. By assuming periodic solutions of the form xi =Ri cos(ωt-αi ) find the general solution of this set of differential equations
1 1 2 2 1 2
2x 6x 3x 3x 8x 9x
″ = − −^ (9 marks)
- (a) Find the Inverse Laplace Transform of the expression
-s -2s -3s 2 2 2 2 2 2
2e 4e 4e (^) ... s (s +1) s (s +1) s (s +1)
− + + (12 marks)
(b) By using Laplace Transform solve the differential equations
(i)
2 2
d y (^) +4 dy+3y=12U(t-2) y (0)=y(0)= dt dt
(ii)
2 2
d y (^) 4y 32sin2t y (0) y(0) 0 dt
- (a) Consider the triangular region with vertices (-1,0), (1,0) and (0,2).
(i) If C is the perimeter of this region evaluate the line integral
2 2 C
Ñ∫12y dx +12x dy.
- (a) A triangular plate is of varying thickness and the mass per unit area at any point
(x,y) on the plate is given by ρ=(10+1.2x)kgm-2^. If the plate lies in the triangular region with vertices (0,0), (1,0) and (0,1) find the mass of the plate by summing vertically and by summing horizontally. (10 marks)
(b) By assuming an exponential solution find the general solution of the system of differential equations dx =x+2y+z dt dy =2x+y+z dt dz =x+y+2z dt (10 marks)
(c) Find the Inverse Laplace Transform of the expression below and sketch the function obtained -2s 2 2
(^4) - 2 +4e s s s
(5 marks)
DERIVATIVES AND INTEGRALS
f(x) a=constant f ′ (x) x n nx n−^1 e^ ax a eax sinx cosx cosx -sinx
f(x) a=constant ∫ f(x)dx
x n xn+ if n - n+
x
1 lnx
eax 1 a
a eax sinx -cosx cosx sinx
Note : 2sinAcosB=sin(A+B)+sin(A-B) 2cosAcosB=cos(A+B)+cos(A-B) 2sinAsinB=cos(A-B)-cos(A+B) sin(-A)=-sinA cos(-A)=cosA
Z-TRANSFORMS
f(t) F(z) U(n)= z 1
z − a^ N z a
z − n (z 1)^2
z − n 2 (z 1)^3
z(z 1) −
a n f(n)
a
F z
nf(n) -zF(z)
f(n+1) zF(z)-zf(0)
f(n+2) (^) z 2 F(z)−z^2 f(0)−zf(1)
