October 7, 2014
MATH 2L03
Term Test 2
Dr. Jessie Yang
Duration: 45 minutes
SURNAME:
Given Name:
Student ID:
THIS EXAMINATION PAPER INCLUDES 6 PAGES AND 5 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE PAPER IS COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR.
INSTRUCTIONS: No aids except the standard Casio fx991 calculator are permitted.
Question Points Your Score
Q1
2
Q2
4
Q3
6
Q4
2
Q5
6
TOTAL
20
page 1 of 6
MATH 2L03 Term Test 2
[Q1] [2 points] There is no partial credit. Your mark will be based solely on what is
written in the “Solution” box provided; use the rest of the space for scratch work.
Let
A=
1.5 −2.35 5.6
44.2
0
12.2


1.4 7.8
10
20
30
, B =  5.4 0  , C =
.
−10 −20 −30
5.6 6.6
1. (1 point) Find (2.1A − 2.3C)T .

Solution:

−19.85 115.82
 −50.935 46

−57.24 94.62
2. (1 point) Find AB.
Solution:
20.77 48.66
130.2 425.28
MATH 2L03 Term Test 2
[Q2] [4 points] You may get partial credits on your works. However, for the full credit
you need to write clearly your solution in the “Solution” box.
According to the nutritional information on a package of cereal, each 1-ounce serving
of Cheerios contains 3 grams protein and 24 grams carbohydrates. Each half-cup serving
of enriched skim milk contains 4 grams protein and 6 grams carbohydrates.
I am planning a meal of cereal and milk and I want it to supply 26 grams of protein
and 78 grams of carbohydrates.
1. (2 points) Find the matrix equation which models to solve how to prepare my meal.
Solution:
Let x be the number of servings of Cheerios (each serving is for 1-ounce)
and let y be the number of servings of milk (each serving is for half-cup).
Then the matrix equation we are looking for is as follows:
3 4
x
26
=
.
24 6
y
78
2. (2 points) Describe how to prepare my meal.
Solution:
x
y
=
3 4
24 6
−1 26
78
=
1
− 13
4
13
2
39
1
− 26
26
78
=
Therefore, Use 2 servings of Cheerios and 5 servings of milk.
2
5
.
MATH 2L03 Term Test 2
[Q3] [6 points] You may get partial credits on your works. However, for the full credit
you need to write clearly your solution in the “Solution” box.
You invested a total of $6, 000 in the three funds at the beginning of 2011, including an
equal amount in the company U and the company V. Your total year-to-date loss amounted
to $360. We want to find how much you invested in each of the three funds.
1. (2 points) Find the matrix equation AX = B which models to solve the problem.
Solution:
Let x, y, z be the investments to the company U, V, W respectively. Then
A, X, B are as follows:


 


1
1 1
x
6000
−1 0  , X =  y  , B =  0  .
A= 1
0.06 0.05 0.07
z
360
2. (2 points) Find the inverse of A which you found above.

Solution:
A−1

7/3
2/3 −100/3
7/3 −1/3 −100/3  .
=
−11/3 −1/3
200/3
3. (2 points) Find how much you invested in each of the three funds.

Solution:

2000
X = A−1 B =  2000  .
2000
Thus you invested $2000 in each.
MATH 2L03 Term Test 2
[Q4] [2 points] There is no partial credit. Your mark will be based solely on what is
written in the “Solution” box provided; use the rest of the space for scratch work.
Reduce the payoff matrices by dominance.
1. (1 point)


1 −1 −5
 4
0
2 

P1 = 
 3 −3 10  .
3 −5 −4
Solution:
0
2. (1 point)


2 −4
9
 1
1
0 

P2 = 
 −1 −2 −3  .
1
1
1
Solution:
1
.
MATH 2L03 Term Test 2
[Q5] [6 points] You may get partial credits on your works. However, for the full credit
you need to write clearly your solution in the “Solution” box.
Let
P =
−2 −1
−1 −3
.
1. (2 points) Find the optimal row strategy.
Let R =
Solution:
x 1−x
be the optimal rowstrategy.
We
the two pure
consider
1
0
strategies for the column player: C1 =
, C2 =
. For C1 we obtain
0
1
e1 = RP C1 = −x − 1, and for C2 we obtain e2 = RP C2 = 2x − 3. Therefore the
optimal row strategy is
R = 2/3 1/3 .
2. (2 points) Find the optimal column strategy.
Solution:
y
Let C =
be the optimal column strategy. We consider the two
1−y
pure strategies for the row player: R1 = 1 0 , R2 = 0 1 . For R1 we
obtain e1 = R1 P C = −y − 1, and for R2 we obtain e2 = R2 P C = 2y − 3.
Therefore the optimal column strategy is
2/3
C=
.
1/3
3. (2 points) Find the expected value of the game in the event that each player uses
his/her optimal strategy.
The expected value of the game is
Solution:
RP C =
2/3 1/3
P
2/3
1/3
= (−5/3).