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First midterm exam: Math 232
October 9th, 11:30am–12:20am
Instructions/Remarks:
- Read all instructions.
- The questions are on the 6 numbered single-sided pages following this one. Count them now.
- Please include all details and rough work. You may use the back of the question pages and there
is extra paper should you require it.
- The exam is out of a total of 50 marks. The value of each question is indicated below.
- You have 50 minutes to complete this examination.
- Good luck!
Marks:
- /15 2) /15 3) /10 4) /
Total: /50 Grade:
Name: ID:
Signature:
Question 1 [15pts] Consider the system of linear equations
x 1 − x 2 + 2x 3 = 1
−x 1 + 2x 2 − (1 + p)x 3 = p − 1
−x 1 + 3x 2 − 2 px 3 = 2p − 1
a. Write the system in the form Ax = b
b. For what values of p does the system have a unique solution?
c. For what values of p does the system have no solution?
d. For what values of p does the system have infinitely many solutions?
e. Solve the system for p = 1.
Justify your answers and show all your work.
Question 2 [15 pts]
This question relates to Figure 1 above which describes traffic flow on some downtown streets.
a. What must the flow through the traffic light at point y be?
b. Write down a system of equations of the form Ax = b to determine the flow through the points
x 1 , x 2 , and x 3.
c. Solve the system and explain your answer in regard to Figure 1.
(Continue work here)
Question 4 [10 pts] The transformation T (x) is defined by:
T
x 1
x 2
x 3
x 4
x 5
[
3 x 3 − x 5
x 1 − x 2 − 4 x 3 + 2x 4
]
What is the matrix A such that T (x) = Ax?
Is the vector
v =
[
]
in the range of T? Explain.
