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The University of British Columbia Final Examination - April 20, 2009 Mathematics 152 All Sections
Closed book examination. No calculators. Time: 2.5 hours
Last Name First Signature
Student Number
Section :
Instructor :
Special Instructions:
No books, notes, or calculators are allowed. Show all your work, little or no credit will be given for a numerical answer without the correct accompanying work. If you need more space than the space provided, use the back of the previous page.
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishon- est practises shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other mem- ory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
- Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
part A 30
B1 5
B2 5
B3 5
B4 5
B5 5
B6 5
Total 60
Page 1 of 12 pages
Part A - Short Answer Questions, 1 mark each
A1: Evaluate (1, 2 , −1) × (4, − 2 , 1).
A2: A linear system of three equations in four unknowns has
(a) always a unique solution. (b) either a unique solution or no solutions. (c) either a unique solution or an infinite number of solutions. (d) either no solutions or an infinite number of solutions.
A3: An electrical network with 2 voltage sources, 3 current sources, 5 resistors (all given) arranged in 6 elementary loops can be described using the loop current technique described in the notes and computer labs as a linear system with the following number of unknowns:
(a) 6 (b) 8 (c) 9 (d) 11
A4: An electrical network with 2 capacitors, 3 inductors, 5 resistors (all given) arranged in 6 elementary loops can be described as a system of the following number of differential equations:
(a) 5 (b) 6 (c) 8 (d) 16
A5: Find the determinant of the matrix
A6: For what values of λ does the matrix [ 3 + λ 2 2 3 + λ
]
not have an inverse?
A13: What is the matrix representation of the 2D projection onto the x axis?
A14: The variables x and y are defined by the MATLAB commands
x = [1 0 2]; y = [3 2 1];
which of the following MATLAB commands will result in an error message?
(a) dot(x,y) (b) cross(x,y) (c) x.y (d) xy
A15: The variable A is defined by the MATLAB command
A=[1 2 3; 4 5 6];
What is the result of the command A(2,1)?
A16: A matrix A is entered into MATLAB. The eigenanalysis of A is performed using the command [T D] = eig(A) which gives the following results:
T = 0.7071 0. 0.7071 0. D = 2.0000 0 0 3.
Using these results, determine A^5 [1, 1]T^ :
(a) [32, 32]T (b) [32, 243]T (c) [1, 1]T (d) [243, 243]T
A17: The linear transformation T : R^3 → R^3 is given by
T (x, y, z) = (2x + 2y, 3 x + 3z, x + y + z).
Write the matrix representation of T.
A18: Write down the matrix A that will result from the following lines of MATLAB code:
A=zeros(3,3); c=[1 2 3]; A(3,:) = -c; for i=1: A(i,i+1) = 1; end
For questions A19-A23 below, u and z are the complex numbers given below:
u = i + 1 z = 2 − i
A19: Evaluate u + 2z. Your answer should be in the form a + ib where a and b are real numbers.
A20: Evaluate |z|.
A21: What is the polar representation of u?
A22: Evaluate uz. Put your answer in the form a + ib where a and b are real numbers.
A23: Evaluate u/z. Put your answer in the form a + ib where a and b are real numbers.
A24: If a = (1, − 1 , 1) and b = (2, 3 , 4) find projab.
Part B - Long Answer Questions, 5 marks each
B1: Three friends, Hiro, Wan and Bob together have $16. Hiro has twice as much money as Bob and Wan has $1 more than Hiro.
(a) [2 marks] Let x = (x 1 , x 2 , x 3 )T^ be the vector of unknowns, where x 1 is the amount of money that Hiro has, x 2 the amount that Wan has, x 3 the amount that Bob has. Describe the information above as a linear system in the form
Ax = b
(write A and b with specific values). (b) [1] Write the system you found above in augmented matrix form. (c) [2] Solve the system above using Gaussian elimination on the augmented matrix. How much money do each of the three friends have?
B2: Let P be the plane defined by x + 2y + 3z = 1
(a) [2 marks] Consider the intersection of P with a second plane Q defined by
2 x − y − z = 0
This intersection is geometrically a line. Find a parametric description of this line. (b) [3] Find the point on P closest to the point (2, 0 , 0).
B4: Consider the differential equation system
dy 1 dt
= ay 1 − 3 y 2 dy 2 dt
= − 3 y 1 + ay 2
where a is a real parameter.
(a) [1 point] Let y(t) =
[
y 1 (t) y 2 (t)
]
Write the matrix A so that the system above is written
dy dt
= Ay.
(b) [2] Find the eigenvalues and eigenvectors of A when a = 1. (c) [1] Write the general solution to the differential equation system when a = 1. (d) [1] For what values of a (if any) do all solutions of the system satisfy y 1 (t) → 0 and y 2 (t) → 0 as t → ∞?
B5: Consider the differential equation system
dx dt
= Ax
where A has eigenvalues λ 1 = −1 + i and λ 2 = − 1 − i with corresponding eigenvectors
k 1 = [1 + i, 1 − i]T^ , and k 2 = [1 − i, 1 + i]T
(a) [2 marks] Write the general solution of the DE system. (b) [3] Find the solution (written in terms of real functions of t) that satisfies initial conditions x(0) = [1, 2]T^.
