of School Mathematics and Statistics SEMESTER 2 EXAMINATIONS 2007 MATH2040 ENGINEERING MATHEMATICS FAMILY NAME: GIVEN NAMES: STUDENT ID: SIGNATURE:
This Paper contains: 10 pages (including title page) Time allowed: Three hours and 10 minutes
INSTRUCTIONS: Attempt all questions. For Part Is 20 questions, write your answers only in the spaces dotted or blank on this exam paper. (Any working for Part I you do in your answer book should be neatly cancelled.) Marks from your answer book will only be for the 6 questions of Part II. The marks for each part question are shown on the paper. There is a total of 100 marks on the paper.
PLEASE NOTE
Examination candidates may only bring authorised materials into the examination room. If a supervisor nds, during the examination, that you have unauthorised material, in whatever form, in the vicinity of your desk or on your person, whether in the examination room or the toilets or en route to/from the toilets, the matter will be reported to the head of school and disciplinary action will normally be taken against you. This action may result in your being deprived of any credit for this examination or even, in some cases, for the whole unit. This will apply regardless of whether the material has been used at the time it is found. Therefore, any candidate who has brought any unauthorised material whatsoever into the examination room should declare it to the supervisor immediately. Candidates who are uncertain whether any material is authorised should ask the supervisor for clarication.
Semester 2 Examinations June 2007
2. MATH2040
PART I : Short Answer Part 1. The inverse of a unit lower triangular matrix is unit lower triangular, true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[1 mark]
2. The inverse of an invertible banded matrix is always banded with the same bandwidth, true or false? . . . . . . . . . . . . . . . . . . . . . [1 mark]
3. Give a one sentence description of one engineering application in which banded matrices arise. ANSWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2 marks]
4. A nonzero scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue: true or false? . . . . . . . . . . . . . . . . [1 mark]
5. An n 1 matrix u of zeros is never an eigenvector of an n n matrix A: true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
6. Let A be an n n matrix with real entries. If is a complex eigenvalue, then is also an eigenvalue of A: true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
7. An n n matrix A always possesses n linearly independent eigenvectors: true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
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Semester 2 Examinations June 2007
3. MATH2040
8. If a 3 3 matrix A is diagonalizable, then it possesses 3 linearly independent eigenvectors: true or false? . . . . . . . . . . . . . . . . . . . . . [1 mark]
9. A Matlab session gives: A= sym([4,-1,0;-1,3,-1;0,-1,4]); [v,d]=eig(A) v = [ 1, [ -1, [ 1, 1, -1] 2, 0] 1, 1]
d = [ 5, 0, 0] [ 0, 2, 0] [ 0, 0, 4] With A the matrix given in the Matlab above, write down the general solution of the system of d.e.s dy = Ay. dt ANSWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [3 marks]
10. With matrix A dened as in the preceding question, and given that Matlab A\[0;10;0] returns [1;4;1], write down a particular solution to the system of d.e.s 0 dy = Ay + 10 . dt 0 ANSWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [3 marks]
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Semester 2 Examinations June 2007
4. MATH2040
11. Complete the following denition: A square matrix is said to be stable if, for any of its eigenvalues, the real part of the eigenvalue is . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
12. The matrix A =
1 1 is orthogonal since its columns are orthogonal vectors: 1 1 true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
13. The eigenvalues of a symmetric matrix with real entries are always real numbers: true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1 mark]
14. Complete the following denition: An n by n real matrix A is positive denite if it is symmetric and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................. [1 mark]
15. If f (x, y, z) = x2 + y 2 yz, then f (x, y, z) = . . . . . . . . . . . . . . . . . . . . . . . . . . . .
[1 mark]
16. The surface x2 + y 2 yz = 1 at the point (x, y, z) has a normal vector
n = (. . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . .)
The equation of the plane tangent to the surface at the point (x, y, z) = (1, 1, 1) is (. . . . . . . . . . . .)(x 1) + (. . . . . . . . . . . .)(y 1) + (. . . . . . . . . . . .)(z 1) = . . . . . . . . . . . . [2 marks]
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Semester 2 Examinations June 2007
5. MATH2040
17. Let (x, y, z) = f (, ) and (u, v) = g(s, t). f (g( is a valid composition: true or false? . . . . . . . . . . . . p)) g(f ( is a valid composition: true or false? . . . . . . . . . . . . p)) [1 mark]
18. If (u, v) = f (s, t), and s and t are functions of (x, y, z), then the Chain Rule for this is situation (u, v) ( ) ( ) = (x, y, z) ( ) ( ) where (u, v) = (s, t) [2 marks]
19. Let (x, y, z) = f (u, v) for (u, v) in the region R. The area of this surface is given by the double integral ........................ du dv [1 mark]
R
20. Which of the following identities are true in general for suitably dimensioned and dierentiable scalar function f and vector eld g. (a) (b) (c) (d) f = 0 : true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f = 0 : true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g = 0 : true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
f g = 0 : true or false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2 marks]
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Semester 2 Examinations June 2007
6. MATH2040
PART II : Long Answer Part 1. Consider the dierential equation, for an unforced damped linear oscillator: 1 257 y + y + y=0 2 16 or, equivalently, with [y1 ; y2 ] = y = [y; y ], y = Ay with A = 0 1 1 257 16 2 .
Using Matlab one nds that the eigenvalues of A are 1 4i, and that 4 expm(tA) = exp(t/4)
1 1 sin(4t) cos(4t) + 16 sin(4t) 4 257 1 64 sin(4t) cos(4t) 16 sin(4t)
.
Now suppose that the system is started with y(0) = [1; 0], i.e. y(0) = 1, y (0) = 0. (a) Write down the formula for y(t) solving this initial value problem. [4 marks] (b) How many times does y(t) = y1 (t) pass through 0 when t varies from 0 up to 2? [4 marks] (c) On the phase plane plot which follows, sketch, with attention to the main qualitative features, but without too much concern about detailed numerics, the general shape of the trajectory of the solution of parts (a), (b) in the (y1 , y2 )-plane. The trajectory is given for 0 < t < /2 and you are asked to continue this so that you show it over 0 < t < 2. [4 marks]
QUESTION 1(c) CONTINUES OVER THE PAGE
Semester 2 Examinations June 2007 1(c) (Continued)
7. MATH2040
2. A vibrating system is described by the linear system of 2nd-order dierential equations: 4 1 0 d2 y + Ay = 0 where A = 1 3 1 . 2 dt 0 1 4 (a) What are the normal frequencies (resonant frequencies) for this system? Briey explain. (Hint. See the Matlab output given in Part I Question 9.) [6 marks] (b) Solve the initial-value problem for the above dierential equation with initial values: y(0) = 0 , dy (0) = [1, 0, 1]T . dt (Use the Matlab output again.) [6 marks]
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Semester 2 Examinations June 2007
8. MATH2040
3. Let f (t) be the 2-periodic extension of the function with f (t) = sin(t) for < t 0, (a) Sketch the function f (t) for 2 t 2. (b) Is the function f even, odd or neither? f (t) = sin(t) for 0 < t . [2 marks] [2 marks]
(c) Determine the Fourier series of the function f (t) specied at the beginning of this question. You may use any relevant information in the following Matlab outputs. syms n t pretty(int(sin(t)*cos(n*t),t,sym(0),sym(pi))) (cos(pi n) + 1) (n + 1)(n 1)
pretty(int(sin(t)*cos(t),t,sym(0),sym(pi))) % for n=1 0 pretty(int(sin(t)*sin(n*t),t,sym(0),sym(pi))) sin(pi n) (n + 1)(n 1) [4 marks] (d) Using your results from part (c) on the Fourier series of f (t), nd the Fourier series for a particular solution of the dierential equation, d2 y + y = f (t) . dt2 [4 marks]
4. Let f (x, y) = (x2 + 3y 2 ) exp(3x2 y 2 ) (a) In the Matlab code below f is the function, g is its gradient, h is its hessian. gs and hs are the products of strictly positive expressions with g and h respectively. Thus gs is zero i g is. The matlab code nds the critical points of f : list these in your exam booklet. QUESTION 4(a) CONTINUE...