Math 152 (MWF)
Spring 2013
Exam 2
Mar 20, 2013
Time Limit: 50 Minutes
Name (Print):
Student number
This exam contains 12 pages (including this cover page) and 4 problems. Check to see if
any pages are missing. Enter all requested information on the top of this page, and put your
initials on the top of every page, in case the pages become separated.
You may not use your books, notes, or any calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• Organize your work, in a reasonably neat
and coherent way, in the space provided. Work
scattered all over the page without a clear ordering will receive very little credit.
• Mysterious or unsupported answers will
not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect
answer supported by substantially correct calculations and explanations might still receive
partial credit.
• If you need more space, use the back of the
pages; clearly indicate when you have done
this.
The maximum grade you can get is 25. You may use
the table to the right to get an idea of the weight of
each question, but please don’t write in it.
Problem
Points
1
8
2
7
3
6
4
4
Total:
25
Score
Math 152 (MWF)
Exam 2 - Page 2 of 12
Mar 20, 2013
The following table may be useful.
θ
0(0◦ )
π/6(30◦ )
π/4(45◦ )
√
√
cos θ
1
3
2
sin θ
0
1
2
2
2
√
2
2
π/3(60◦ )
1
2
√
3
2
π/2(90◦ )
0
1
Math 152 (MWF)
Exam 2 - Page 3 of 12
Mar 20, 2013
I. Short Answer Problems
1. In problems (A)–(B), let z = 2 + i and u = −3 − 3i.
z (A) (1 point) Calculate the value of .
u
(B) (1 point) Express u in polar form (i.e., find r > 0 and θ such that u = reiθ ).
Math 152 (MWF)
Exam 2 - Page 4 of 12
Mar 20, 2013
(C) (1 point) Write w in polar form (i.e., find r > 0 and θ such that w = reiθ ), where
w = 4(cos
3π
3π
5π
5π
+ i sin ) · (cos
+ i sin ).
4
4
4
4
(D) (1 point) Suppose we run the following MATLAB code.
A=ones(4,4)-eye(4);
for j=1:3;
A(j,:)=j*A(j,:);
end;
Specify the output if you now type:
>> A(2,:)
Math 152 (MWF)
Exam 2 - Page 5 of 12
Mar 20, 2013


a1
(E) (1 point) Let A =  a2  be a 3 × 3 matrix with rows a1 , a2 , a3 (where each aj is
a3


a1 + a2
1 × 3). Suppose that det A = 3. Find det  2a2 .
a3 + a2
(F) (1 point) Suppose that T is a linear transformation from R3 to R3 and that u, v
and w are three linearly independent vectors in R3 . Are the vectors T (u), T (v)
and T (w) necessarily linearly independent for R3 ? If yes, explain. If no, give a
counterexample.
Math 152 (MWF)
Exam 2 - Page 6 of 12
Mar 20, 2013
(G) (1 point) Write the matrix representing the linear transformation from R2 to R2
that rotates a vector ~x (counterclockwise) by π/2 radians. (Note that sin(π/2) = 1
and cos(π/2) = 0.)
(H) (1 point) Consider the following circuit with R1 = 1Ω, R2 = 2Ω, R3 = 3Ω, R4 =
4Ω, R5 = 5Ω, E1 = 10V and E2 = 20V .
Write an equation that describes the sum of voltage drops around loop 2 (i.e., the
loop with current i2 ).
Math 152 (MWF)
Exam 2 - Page 7 of 12
Mar 20, 2013
II. Long Answer Problems
2. Consider the matrix


2 4 5
A = 0 1 1 .
1 2 2
(a) (3 points) Find A−1 .
(b) (1 point) Find det(AT ).
(c) (1 point) Let b1 and b2 be unknown 1×3 matrices. Consider the system of (matrix)
equations given by
b1 A = 2 4 5
b2 A = 1 3 3
where A is as above. Express this system as a single matrix equation BA = C.
(d) (2 points) Find b1 and b2 .
Math 152 (MWF)
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Exam 2 - Page 8 of 12
Mar 20, 2013
Math 152 (MWF)
Exam 2 - Page 9 of 12
Mar 20, 2013
3. A is a 3 × 3 matrix depending on a parameter k. After row reductions, the following
matrix results:


k
1
1
U = 0 2 − k 0  .
0
0
−1
The row operations used were (in order): exchange row 1 and row 2; multiply row 2 by
2; subtract row 1 from row 3
(a) (2 points) Find det(A) as a function of the parameter k.
(b) (2 points) Determine all values of k for which A is invertible.
(c) (2 points) Now set k = 1 and find A−1 . (Note: A is different from the given
matrix U !)
Math 152 (MWF)
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Exam 2 - Page 10 of 12
Mar 20, 2013
Math 152 (MWF)
Exam 2 - Page 11 of 12
Mar 20, 2013
4. Let T be the linear transformation acting on vectors in R2 by first rotating the vector by
π/3 (counterclockwise) and then projecting√the resulting vector onto the vertical axis.
Recall that cos(π/3) = 1/2 and sin(π/3) = 3/2.
(a) (2 points) Find the matrix for T .
(b) (2 points) Is T invertible? If so, find the matrix for T −1 . If not, find a non-zero
vector x such that T (x) = 0.
Math 152 (MWF)
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Exam 2 - Page 12 of 12
Mar 20, 2013