Fall 2015
Math 141:505
Exam 2
Form B
Last Name:
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On my honor, as an Aggie, I have neither given nor received unauthorized aid on
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INSTRUCTIONS
• Part 1: Multiple Choice/True-False (Problems 1-7) Each multiple choice problem is worth 5 points for a total
of 30 points. Each true-false problem is worth 2 points for a total of 10 points. Answers should be written in the
boxes provided. No partial credit will be given.
• Part 2: Work Out (Problems 8-13) The number of points for each problem is indicated next to the problem for
a total of 60 points. Partial credit will be given. All steps must be written clearly and neatly to receive credit.
If you use your calculator for anything beyond an arithmetic calculation, please indicate how at the appropriate
step. Box your final answer.
“An Aggie does not lie, cheat or steal or tolerate those who do.”
c
Yeong-Chyuan
Chung, October 23, 2015
Part 1: Multiple-Choice. Each multiple choice problem is worth 5 points for a total of 30 points. No
partial credit will be given. Answers should be written in the boxes provided.
1. Let U denote the set of all students in a particular college, and let
T = {x ∈ U|x has taken a course in statistics},
C = {x ∈ U|x has taken a course in calculus},
G = {x ∈ U|x has taken a course in geometry}.
Which of the following is the correct description of the set C ∩ (T ∪ G)c ?
(a) The set of all students who have taken a course in calculus, or have not taken courses in either statistics or
geometry.
(b) The set of all students who have taken a course in calculus, but have not taken courses in either statistics or
geometry.
(c) The set of all students who have taken a course in calculus, or have not taken courses in both statistics and
geometry.
(d) None of the above.
b
2. There are ten candidates for three prizes. In how many ways can the prizes be awarded to these candidates if all
three prizes are the same, and no preference is given to any candidate?
(a) 120
(b) 720
(c) 604,800
(d) 3,628,800
a
3. Eight athletes are competing in a race. In how many ways can the gold, silver, and bronze medalists be decided?
(a) 6
(b) 56
(c) 336
(d) 40,320
c
2
c
Yeong-Chyuan
Chung, October 23, 2015
4. Which system of linear inequalities describes the shaded region?
(a)
x+y ≤ 2
9x + 5y ≤ 90
3x + 5y ≤ 60
x ≥ 0, y ≥ 0
(b)
x+y ≤ 2
9x + 5y ≥ 90
3x + 5y ≤ 60
x ≥ 0, y ≥ 0
(c)
x+y ≥ 2
9x + 5y ≤ 90
3x + 5y ≤ 60
x ≥ 0, y ≥ 0
(d)
x+y ≥ 2
9x + 5y ≤ 90
3x + 5y ≥ 60
x ≥ 0, y ≥ 0
c
3
c
Yeong-Chyuan
Chung, October 23, 2015
5. How many distinguishable permutations can be formed from the 11 letters of the word MATHEMATICS?
(a) 39,916,800
(b) 19,958,400
(c) 9,979,200
(d) 4,989,600
d
6. Let A, B, and C be subsets of a universal set U. Suppose that n(A) = 10, n(B) = 8, n(C) = 6, n(B ∪C) = 10, and
n(A ∩ B ∩C) = 2. What is n(A ∪ (B ∩C))?
(a) 12
(b) 18
(c) 30
(d) 32
a
True-False. Each of the following is worth 2 points for a total of 10 points. No partial credit will be given.
Answers should be written in the boxes provided.
7. Consider the set A = {a, b, c}. State whether each of the following statements is true or false.
(i) 0/ ⊂ A
T
(ii) {a} ∈ A
F
(iii) {b} ⊆ A
T
(iv) c ∈ A
T
(v) {b, c, a} = A
T
4
c
Yeong-Chyuan
Chung, October 23, 2015
Part 2: Work Out. The number of points for each problem is indicated next to the problem. Partial credit will
be given. All steps must be written clearly and neatly to receive credit. If you use your calculator for anything
beyond an arithmetic calculation, please indicate how at the appropriate step. Box your final answer.
8. (8pts) An experiment consists of rolling a standard six-sided die and observing whether the number that falls
uppermost is even or odd, and then selecting a card from a standard deck of playing cards and observing its suit
(hearts, diamonds, clubs, or spades). Hearts and diamonds are red, while clubs and spades are black.
(a) Find the sample space, S, for this experiment.
(b) Find the event, E, that the card selected is black.
(c) Find the event, F, that “the number observed is odd or the card selected is red.”
Solution:
Let E and O indicate even and odd respectively. Let H, D, C, S indicate hearts, diamonds, clubs, and spades
respectively.
(a) S = {EH, ED, EC, ES, OH, OD, OC, OS}.
(b) E = {EC, ES, OC, OS}.
(c) F = {OH, OD, OC, OS, EH, ED}.
5
c
Yeong-Chyuan
Chung, October 23, 2015
9. (8pts) In a survey of 600 college students, the following data was obtained:
• 312 students watch baseball games.
• 234 students watch soccer games.
• 76 students watch all three games.
• 240 students watch football games and baseball games.
• 114 students watch football games and soccer games.
• 110 students watch soccer games only.
• 210 students do not watch football games.
In the Venn diagram below, U represents the set of all students surveyed, A represents the set of students who
watch football games, B represents the set of students who watch baseball games, and C represents the set of
students who watch soccer games.
Fill in the number of elements in each region of the diagram.
6
c
Yeong-Chyuan
Chung, October 23, 2015
10. (11pts) Consider the following word problem.
“An exercise equipment manufacturer manufactures two models of steppers used for aerobic exercises. Manufacturing each luxury model requires 10 lb of plastic and 10 min of labor. Manufacturing each standard model
requires 16 lb of plastic and 8 min of labor. The profit for each luxury model is $40, and the profit for each
standard model is $30. If 6000 lb of plastic and 60 labor-hours are available for the production of the steppers
per day, how many steppers of each model should they produce each day to maximize profit?”
Clearly define the appropriate variables, and set up (but do not solve) a linear programming problem that would
be used to answer the question.
Solution:
Let x be the number of luxury steppers manufactured.
Let y be the number of standard steppers manufactured.
Maximize P = 40x + 30y
subject to 10x + 16y ≤ 6000
10x + 8y ≤ 3600
x ≥ 0, y ≥ 0
7
c
Yeong-Chyuan
Chung, October 23, 2015
11. (13pts) Consider the following linear programming problem.
Minimize C = −2x + 5y
subject to 5y ≥ 2x + 5
y ≥ −2x + 4
1
19
y ≤ x+
5
2
x ≥ 0, y ≥ 0
The figure below shows the lines y = 52 x + 1, y = −2x + 4, and y = 15 x + 19
2.
Indicate the feasible set, label the corner points of the feasible set, and then find the optimal solution(s).
Solution:
Corner
C = −2x + 5y
(0, 4)
20
(0, 9.5)
47.5
(1.25, 1.5)
5
(42.5, 18)
5
There are infinitely many optimal solutions given by the line segment joining the points (1.25, 1.5) and (42.5, 18),
and the optimal value of C is 5.
8
c
Yeong-Chyuan
Chung, October 23, 2015
12. (10pts) A container has 7 blue marbles and 8 red marbles. A sample of 6 marbles is selected.
(a) How many ways are there to select such a sample if we want either exactly 4 blue marbles or exactly 4 red
marbles?
(b) How many ways are there to select such a sample if we want at least one blue marble and at least one red
marble?
Solution:
(a) We want to have 4 blue and 2 red, or 4 red and 2 blue. The number of such samples is C(7, 4) · C(8, 2) +
C(8, 4) ·C(7, 2) = 2450.
(b) We can consider the following cases: (1) 1 blue and 5 red, (2) 2 blue and 4 red, (3) 3 blue and 3 red, (4) 4
blue and 2 red, (5) 5 blue and 1 red.
Then the number of desired samples is C(7, 1) · C(8, 5) + C(7, 2) · C(8, 4) + C(7, 3) · C(8, 3) + C(7, 4) ·
C(8, 2) +C(7, 5) ·C(8, 1) = 4970.
Alternatively, we can consider the complement, which consists of samples with either no blue marbles or
no red marbles. The number of such samples is C(8, 6) +C(7, 6) = 35. Then we subtract that from the total
number of samples so we get C(15, 6) − 35 = 4970.
13. (10pts) A password is to be made from a string of six characters chosen from the lowercase letters of the English
alphabet, the uppercase letters of the English alphabet, and the numbers 0 through 9. How many passwords are
possible if the first two characters must be lowercase letters, the third character must be a number, the last two
characters must be uppercase letters, and no repetition of characters is allowed?
Solution:
26 · 25 · 10 · 57 · 26 · 25 = 240, 825, 000.
Problem Part 1
8
9
10
11
Score
9
12
13
Total