M1 Final Exam
MATH M118 Full name ________________________
Exam 3
Please do each of the following problems. Choose the correct answer. Put all of your answers on the cover sheet. You may not use a calculator, cell phone, headphones, etc.
Also, do NOT try to find answers on another student’s test.
1. Let A and B be the matrices below. What is the (2, 1) entry of AB?
a.
b. −1
c. 2
d. 5
e. −15
f. 19
g. 20
h. 27
i. 50
j. 69
k. 74
l. None of the above
2. Stefan receives a $3000 check. He spends some of this money on ginger snaps and the rest of it on Cincinnati Reds tickets. It is known that ginger snaps cost $4 per box and Reds tickets cost $30 per ticket. Also known is that the number of boxes of ginger snaps …show more content…
was five times the number of Reds tickets. How many boxes of ginger snaps did Stefan buy?
a.
b. 20
c. 30
d. 40
e. 60
f. 150
g. 200
h. 300
i. 500
j. 750
k. 1500
l. None of the above
3. An economy has 3 goods: honey (good 1), roses (good 2), and telenovelas (good 3). The technology matrix A is given below. Also given are (I-A) and (I-A)−1 .
If there is a demand for 100 units of honey, 90 units of roses, and 40 units of telenovelas, how many units of roses need to be produced?
a.
b. 695
c. 670
d. 460
e. 44
f. 4
g. 358
h. 348
i. 32
j. 2
k. 13
l. None of the above
4. The economy of Finitopolis has two goods: rubber balls (good 1) and urns (good 2). The production of each ball requires 0.7 balls and 0.4 urns. The production of each urn requires 0.2 balls and 0.4 urns. If there is a production schedule for 300 rubber balls and 240 urns, what is the demand vector?
a.
b.
c.
d.
e.
f.
g.
h.
i.
j. None of the above
5. Jaime uses her rowing machine every day. When she rows for 15 minutes, she drinks an extra 5 ounces of water that day. When she rows for 25 minutes, she drinks an extra 9 ounces of water that day. It is known that the amount of extra water she consumes in a day is a linear function of her time on the rowing machine on that day. How long does she row if Jaime drinks 20 ounces of extra water?
a.
b. 7
c. 17.5
d. 7.5
e. 52.5
f. 77.5
g. 21
h. 25
i. 60
j. 21.6
k. 35.1
l. None of the above
6. Recall that I is the identity matrix. Find the (2, 2) entry of the inverse of B.
a.
b. 7/6
c. 7
d. 4
e. 2/3
f. 2
g. 1/3
h. 1
i. 0.5
j. −7/6
k. −3
l. −2/3
m. Not this one
n. −2
o. −1/3
p. −0.5
q. B has no inverse
r. None of the above
7. Let G and H be the matrices below. What is the (2,3) entry of G −2H?
a.
b. −3
c. 3
d. −4
e. 4
f. −6
g. 6
h. −11
i. 13
j. −15
k. 15
l. None of the above
8. When a matrix has no inverse, the matrix is said to be singular. The matrix below is singular for one and only one value of k. What is that value of k?
a.
b. −6
c. −7
d. −8
e. −9
f. 0
g. 6
h. 7
i. 8
j. 9
k. Cannot be determined
l. None of the above
9. Caroline and Rohin are very competitive dart players (and good at Markov chains). Every week they play two games of darts with each other, and one of three things happens: Caroline wins both games (state 1), Rohin wins both games (state 2), or they split the games (state 3).
If Caroline wins both games one week, then the following week the probability that Rohin win both games is 0.70, and the probability that they split is two times the probability that Caroline wins both games.
If Rohin wins both games one week, then the following week the probability that Caroline wins both games is 0.55, and the probability that Rohin wins both games is four times the probability that they split the games.
If they split the games one week, the following week they do not split the games, and there is an equal probability that Caroline wins both games and that Rohin wins both games.
Which of the following is the transition matrix for this Markov chain?
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l. None of the above
10. A squirrel can be found in any of 4 states—Alabama(1), Colorado (2), Delaware (3), and Exhaustion (4). It is known that this is a Markov process with the transition matrix given below. On the first observation, the squirrel is in Delaware. What is the probability that it is in the state of Exhaustion on the next two observations?
a.
b. 0.80
c. 0.03
d. 0.04
e. 0.08
f. 0.09
g. 0.16
h. 0.17
i. 0.20
j. 0.35
k. None of the above
11. A squirrel can be found in any of 4 states—Alabama(1), Colorado (2), Delaware (3), and Exhaustion (4). It is known that this is a Markov process with the transition matrix given below. On the first observation, the squirrel is twice as likely to be in Alabama as to be in Delaware, just as likely to be in Delaware as in Colorado, and is not in the state of Exhaustion.
What is the probability that the squirrel is in the state of Exhaustion on the next observation?
a.
b. 0.1400
c. 0.1775
d. 0.2000
e. 0.2060
f. 0.2080
g. 0.2225
h. 0.2500
i. 0.2600
j. 0.2750
k. 0.2800
l. 0.3200
m. Don’t pick me
n. 0.3260
o. 0.4000
p. None of the above
12. Below are three matrices. Which of these are regular
matrices?
a. A, B, C
b. A, B
c. A, C
d. B, C
e. Only A
f. Only B
g. Only C
h. None of these are regular
13. A transition matrix P is given below. Find the stable vector W.
a.
b.
c.
d.
e.
f.
g.
h.
i. There is no stable vector W
j. None of the above
14. Solve the matrix equation below for Y. You may assume that all operations are defined.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l. None of the above
15. Find the slope of the line through (−8, −9) and (5, 10).
a.
b. 3/19
c. 19/3
d. 19/13
e. 13/19
f. 13
g. 1/13
h. −3/19
i. −3
j. −19/3
k. −1/3
l. None of the above
Exam 3
Please do each of the following problems. Choose the correct answer. Put all of your answers on the cover sheet. You may not use a calculator, cell phone, headphones, etc.
Also, do NOT try to find answers on another student’s test.
1. Let A and B be the matrices below. What is the (2, 1) entry of AB?
a.
b. −1
c. 2
d. 5
e. −15
f. 19
g. 20
h. 27
i. 50
j. 69
k. 74
l. None of the above
2. Stefan receives a $3000 check. He spends some of this money on ginger snaps and the rest of it on Cincinnati Reds tickets. It is known that ginger snaps cost $4 per box and Reds tickets cost $30 per ticket. Also known is that the number of boxes of ginger snaps …show more content…
was five times the number of Reds tickets. How many boxes of ginger snaps did Stefan buy?
a.
b. 20
c. 30
d. 40
e. 60
f. 150
g. 200
h. 300
i. 500
j. 750
k. 1500
l. None of the above
3. An economy has 3 goods: honey (good 1), roses (good 2), and telenovelas (good 3). The technology matrix A is given below. Also given are (I-A) and (I-A)−1 .
If there is a demand for 100 units of honey, 90 units of roses, and 40 units of telenovelas, how many units of roses need to be produced?
a.
b. 695
c. 670
d. 460
e. 44
f. 4
g. 358
h. 348
i. 32
j. 2
k. 13
l. None of the above
4. The economy of Finitopolis has two goods: rubber balls (good 1) and urns (good 2). The production of each ball requires 0.7 balls and 0.4 urns. The production of each urn requires 0.2 balls and 0.4 urns. If there is a production schedule for 300 rubber balls and 240 urns, what is the demand vector?
a.
b.
c.
d.
e.
f.
g.
h.
i.
j. None of the above
5. Jaime uses her rowing machine every day. When she rows for 15 minutes, she drinks an extra 5 ounces of water that day. When she rows for 25 minutes, she drinks an extra 9 ounces of water that day. It is known that the amount of extra water she consumes in a day is a linear function of her time on the rowing machine on that day. How long does she row if Jaime drinks 20 ounces of extra water?
a.
b. 7
c. 17.5
d. 7.5
e. 52.5
f. 77.5
g. 21
h. 25
i. 60
j. 21.6
k. 35.1
l. None of the above
6. Recall that I is the identity matrix. Find the (2, 2) entry of the inverse of B.
a.
b. 7/6
c. 7
d. 4
e. 2/3
f. 2
g. 1/3
h. 1
i. 0.5
j. −7/6
k. −3
l. −2/3
m. Not this one
n. −2
o. −1/3
p. −0.5
q. B has no inverse
r. None of the above
7. Let G and H be the matrices below. What is the (2,3) entry of G −2H?
a.
b. −3
c. 3
d. −4
e. 4
f. −6
g. 6
h. −11
i. 13
j. −15
k. 15
l. None of the above
8. When a matrix has no inverse, the matrix is said to be singular. The matrix below is singular for one and only one value of k. What is that value of k?
a.
b. −6
c. −7
d. −8
e. −9
f. 0
g. 6
h. 7
i. 8
j. 9
k. Cannot be determined
l. None of the above
9. Caroline and Rohin are very competitive dart players (and good at Markov chains). Every week they play two games of darts with each other, and one of three things happens: Caroline wins both games (state 1), Rohin wins both games (state 2), or they split the games (state 3).
If Caroline wins both games one week, then the following week the probability that Rohin win both games is 0.70, and the probability that they split is two times the probability that Caroline wins both games.
If Rohin wins both games one week, then the following week the probability that Caroline wins both games is 0.55, and the probability that Rohin wins both games is four times the probability that they split the games.
If they split the games one week, the following week they do not split the games, and there is an equal probability that Caroline wins both games and that Rohin wins both games.
Which of the following is the transition matrix for this Markov chain?
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l. None of the above
10. A squirrel can be found in any of 4 states—Alabama(1), Colorado (2), Delaware (3), and Exhaustion (4). It is known that this is a Markov process with the transition matrix given below. On the first observation, the squirrel is in Delaware. What is the probability that it is in the state of Exhaustion on the next two observations?
a.
b. 0.80
c. 0.03
d. 0.04
e. 0.08
f. 0.09
g. 0.16
h. 0.17
i. 0.20
j. 0.35
k. None of the above
11. A squirrel can be found in any of 4 states—Alabama(1), Colorado (2), Delaware (3), and Exhaustion (4). It is known that this is a Markov process with the transition matrix given below. On the first observation, the squirrel is twice as likely to be in Alabama as to be in Delaware, just as likely to be in Delaware as in Colorado, and is not in the state of Exhaustion.
What is the probability that the squirrel is in the state of Exhaustion on the next observation?
a.
b. 0.1400
c. 0.1775
d. 0.2000
e. 0.2060
f. 0.2080
g. 0.2225
h. 0.2500
i. 0.2600
j. 0.2750
k. 0.2800
l. 0.3200
m. Don’t pick me
n. 0.3260
o. 0.4000
p. None of the above
12. Below are three matrices. Which of these are regular
matrices?
a. A, B, C
b. A, B
c. A, C
d. B, C
e. Only A
f. Only B
g. Only C
h. None of these are regular
13. A transition matrix P is given below. Find the stable vector W.
a.
b.
c.
d.
e.
f.
g.
h.
i. There is no stable vector W
j. None of the above
14. Solve the matrix equation below for Y. You may assume that all operations are defined.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l. None of the above
15. Find the slope of the line through (−8, −9) and (5, 10).
a.
b. 3/19
c. 19/3
d. 19/13
e. 13/19
f. 13
g. 1/13
h. −3/19
i. −3
j. −19/3
k. −1/3
l. None of the above