MAT 119 Place name label here
WILLIAMS
Test #1
Scratch Paper Is Provided by the Testing Center.
Do Not Use Your Own Paper!
Read the instructions carefully:
- There are 5 pages and 8 numbered questions worth a total of 100 points.
- Read all the questions carefully. Proctors may not explain a problem to you.
- You must show your work or explain a process as appropriate for each problem to receive full credit!!
- Answers must be complete, organized, and exact unless directed otherwise. Always indicate how a calculator was used (i.e., sketch a graph and explain your steps, etc. . . . ). When possible,
your answer.
No calculator that does symbolic algebra may be used.
Casio FX2 calculators or ones with
QWERTY keyboards are not allowed!
Honor Statement:
By signing below you confirm that you have neither given nor received any unauthorized assistance on this exam. This includes any use of a graphing calculator beyond those uses specifically authorized by the Mathematics Department and your instructor. Furthermore, you agree not to discuss this exam with anyone until the exam testing period is over. In addition, your calculator's memory and menus may be checked at any time and cleared by any testing center proctor or Mathematics Department instructor.
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1.(15 points) Solve the following system of equations using a matrix method of your choice. Include the row operations you use (e.g. R2 = r2 + r1) and show all your steps.
2.(15 points) A physics laboratory can be used by 38 students at one time. The lab has 16 work stations, some set up for 2 students each and the rest set up for 3 students each. How many work stations of each type are in the lab? Use any method you choose but SHOW ALL WORK.
3.(10 points) Solve the following system of equations:
-27x + 10y + 4z = 0
22x – 8y – 3z = -1
-8x + 3y + z = 2
…given that the inverse of is .
4.(15 points) For each matrix expression below, determine if the operation can be carried out.
If so, evaluate the expression BY HAND to find the resulting matrix. SHOW WORK.
If a solution does not exist, write “not possible,” and EXPLAIN WHY NOT (be specific!!).
A = B = C =
(a)C + 2AB
(b)CA - 3B
(c)CB
(d) A-1
(e) B-1
5.(10 points) a) Graph the system of linear inequalities. (Read the inequality
symbols carefully, and show accordingly!)
b) Find all corner points mathematically, and list each one on the
graph.
c) Tell whether the graph is bounded or unbounded.
6.(20 points) An independent shoe manufacturer offers 2 types of shoes: casual and tennis. The primary materials used to make the shoes are leather—40 units available, and rubber—16 units available. The manufacturer wants to know how many pairs of each type of shoe should be made in order to maximize profit. The manufacturer has gathered the information below:
Leather neededRubber neededProfit / pair
per pairper pair
Casual5 units1 units$20
Tennis2 units2 units$16
(a)Define your variables and write the objective function. (4 points)
(b)Write down all the constraints. (5 points)
(c)Graph the feasible region, find and list all the corner points. (6 points)
(d)Use your answer from part (c) to maximize your objective function from part (a). How many pairs of casual shoes should be produced? How many pairs of tennis shoes? What is the maximum profit? (5 points)
7.(5 points) (a) Define a pair of 2 x 2 matrices, A and B, such that AB = BA. Neither matrix may have a row of all zeros. Validate your answer.
(b) Define a pair of 2 x 2 matrices, A and B, such that AB BA. Neither
matrix may have a row of all zeros. Validate your answer.
8.(10 points) Each augmented matrix given below represents a system of equations in either x and y or in x, y and z. Determine which matrices have solutions and which have no solution.
If there is a unique solution, write the solution.
If there are an infinite number of solutions, list two of them.
If there is no solution, state why.
a)b)
c)d)
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