MA 114 Final Exam Practice
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
x - y = 7 . Which of the following statements is true?
2x - 2y = k
A) If k = 14, the system has no solution.
B) If k = 14, the system has infinitely many solutions.
C) If k ≠ 14, the system has exactly one solution.
D) none of these
1) Consider the system:
1)
2)
2)
-4 1 2
7 -1 -4
xy 4
Let A =
and B = 3 2 4
1
1
0
2
2
1 1 6
In order for A and B to be inverses, x and y must be
A) x = -1, y =-1.
B) x = 1, y = -1.
C) x = -1, y = 1.
D) x = 1, y = 1.
E) none of these
3) Consider the simplex tableau that is the final one in a problem to maximize x + 2y + 3z.
x y z u v M
2 1 0 -3 8 0 2
3 0 1 5 2 0 5
1 0 0 6 5 1 19
What are the values of x, y and z when the maximum value of x + 2y + 3z occurs?
A) x = 2, y = 5, z = 0.
B) x = 19, y = 2, z = 5.
C) x = 0, y = 2, z = 5.
D) x = 2, y = 1, z = 0.
E) none of these
Decide whether the situation involves permutations or combinations.
4) An arrangement of 10 people for a picture.
A) Permutation
B) Combination
5) A committee of 5 delegates chosen from a class of 26 students to bring a petition to the
administration.
A) Permutation
B) Combination
1
3)
4)
5)
Fifty percent of students enrolled in an astronomy class have previously taken physics. Thirty percent of these
students received an A for the astronomy class, whereas twenty percent of the other students received an A for
astronomy.
6) Find the probability that a student selected at random previously took a physics course and did
6)
not receive an A in the astronomy course.
A) 0.10
B) 0.35
C) 0.40
D) 0.15
E) none of these
Find the probability.
7) A sample of 4 different calculators is randomly selected from a group containing 13 that are
defective and 32 that have no defects. What is the probability that at least one of the calculators
is defective?
A) 0.130
B) 0.759
C) 0.241
D) 0.744
′
7)
′
′
Suppose n(U) = 200, n(A) = 80, n(B) = 65, n(C) = 90, n(A ∩ B) = 48, n(A ∩ C) = 62, n(B ∩ C) = 45, and n(A ∩ B ∩ C ) = 80.
8) Find n(A ∩ B ∩ C).
8)
A) 40
B) 20
C) 120
D) impossible to determine
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5 2 -2
1 -2 0
and
2 1 -1
-1 4 1 are inverses of each other.
-3 -1 2
1 -1 1
x - 2y
= 5
Use this fact to solve the system -x + 4y + z = 1 .
x - y +z = 2
9) The matrices
9)
An experiment with outcomes s1, s2, s3, s4, and s5 is described by the probability table below.
Let E = {s1, s4} and F = {s1, s2, s3, s5}.
Outcome Probability
s1
0.05
s2
0.10
s3
0.20
s4
0.50
s5
0.15
10) Compute Pr(E).
10)
11) Compute Pr(E ∪ F).
11)
Solve the problem.
12) At a certain party, 72 people are invited. Only half of those invited come. Of those who
come, twice as many are women as are men. How many men and how many women
come? Let x be the number of women and y be the number of men.
2
12)
13) Big Round Cheese Company has on hand 45 pounds of Cheddar and 49 pounds of Brie
each day. It prepares two Christmas packagesthe "Holiday" box, which has 5 pounds of
Cheddar and 2 pounds of Brie, and the "Noel" box, which contains 2 pounds of Cheddar
and 7 pounds of Brie. Profit on each Holiday assortment is $6, profit on each Noel
assortment is $8. The initial and final tableaux are as follows:
x y
u
v M
7 - 2
1 0
0 7
31
31
x y u v M
2
5
0 1 0 5
5 2 1 0 0 45
31
31
2 7 0 1 0 49
26
28
-6 -8 0 0 1 0
0 0
1 82
31
31
13)
(initial)
(final)
Here, x = the number of Holiday boxes per day and y = the number of Noel boxes per
day.
(a) Give the optimal production schedule and the resulting profit.
(b) How much excess Cheddar and excess Brie remain each day if this plan is followed?
14) How many six-letter words can be formed from the letters in "STABLE" if the first two
letters must be vowels and repetition is allowed?
14)
15) An experiment consists of arranging a white ball (W), a black ball (B), and a red ball (R)
in a row.
(a) What is the sample space S for this experiment?
(b) Describe the event E = "the white ball is in the middle," as a subset of the sample
space.
(c) Describe the event F = "the red ball is next to the black ball," as a subset of the
sample space.
(d) Describe the event E ∪ F.
15)
16) Compute the probability of obtaining three face cards when five cards are dealt from a
standard 52-card deck.
16)
17) There are three children in a family. Are the events "there are more boys than girls" and
"the first child is a girl"
(a) mutually exclusive?
(b) independent?
17)
Write the initial simplex tableau for the linear programming problem.
18) Minimize 3x + 4y subject to the following constraints.
x + y ≤ 10
2x + 3y ≥ 1
x ≥ 0, y ≤ 0
3
18)
Answer Key
Testname: FINAL_P
1) B
2) D
3) C
4) A
5) B
6) B
7) B
8) B
9) x = 23, y = 9, z = -12
10) 0.55
11) 1
12) x = 24 = number of women; y = 12 = number of men
13) (a) x = 7, y = 5, M = $82
(b) none
14) 22 ∙ 64 = 5184
15) (a) {(W, B, R), (W, R, B), (B, W, R), (B, R, W), (R, B, W), (R, W, B)}
(b) {(B, W, R), (R, W, B)}
(c) {(W, B, R), (W, R, B), (B, R, W), (R, B, W)}
(d) {(B, W, R), (R, W, B), (W, B, R), (W, R, B), (B, R, W), (R, B, W)} = S
16)
12 ∙ 40
3
2 = 55 ≈ 0.066
52
833
5
17) (a) no
(b) no
18) x y
1 1
-2 -3
3 4
u
1
0
0
v
0
1
0
M
0 10
0 -1
1 0
4