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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2009
Module Title: Engineering Mathematics 311
Module Code: MATH
School: Building & Civil
Programme Title: B. Eng Degree (Honours) in Structural Engineering-Y
Programme Code: CSTRU_8_Y
External Examiner(s): Dr. P. Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Answer 4 questions. These questions carry equal marks.
Duration: 2 Hours
Sitting: Autumn 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) The matrices below have the same eigenvalues but different eigenvectors. Find the eigenvalues of one matrix and the eigenvectors of both matrices.
A B^ =
Deduce whether or not that the eigenvectors of A are linearly independent and mutually orthogonal. Find the general solution of the system of simultaneous differential equations d dt =
x (^) Ax (15 marks)
(b) Two masses are attached to two springs and the displacements of these masses x 1 and x 2 are found by solving the system of differential equations
2 1 2
1 1 2 3x 16 x 11 x
3x 19 x 8x ″= −
By assuming periodic solutions of the form xi =Ri cos(ωt-α) find the general solution of this set of simultaneous differential equations. (10 marks)
- (a) A prism has vertices O(0,0,0), A(1,0,0), B(0,1,0), C(0,0,3), D(1,0,3) and E(0,1,3). (i) Evaluate the surface integral
S
∫∫(2y+2z)dS
over the surfaces OBEC and ABED. (ii) Evaluate the triple integral 2 V
∫∫∫2zy dV
where V is the prism above. (15 marks) (b) (i) If a= 2xz i+ 2yz j +2z 2 k and n ˆ is a unit vector normal to the surface of the volume described by V: x^2 +y^2 ≤ 9 0 ≤z≤2. evaluate the surface integral
S
∫∫ a n. dSˆ^.
(ii) For the volume described in part (b) (i) evaluate the triple integral
V
∫∫∫ (div ) dV a^.^ (10 marks)
- (a) Using spherical polar coordinates (r,θ, φ) find the moment of inertia of the sphere of unit density described by x^2 +y^2 +z 2 ≤ 4 about the z-axis. The Jacobian is given by J=r 2 sinφ. (8 marks)
(b) Find the Fourier Series of the periodic function
f(t)= (^) 2-t if 1t if 0^ ≤ ≤t t≤ ≤^12
f(t+2)=f(t)
2 2
2 2
Note: tcos(nπt)dt= tsin(nπt)^ +cos(nπt) nπ n π tsin(nπt)dt=- tcos(nπt)^ +sin(nπt) nπ n π
(10 marks)
(c) By using a method of your own choice find the general solution of the system of differntial equatons dx = 5x+y dt dy = - 4x+y dt (7 marks)
