Download April 29, 2001 - Penn Math - University of Pennsylvania

May 3, 2001
THIRD DRAFT OF MATH 151 FINAL EXAM
INSTRUCTIONS. This exam consists of 17 multiple choice questions.
Please circle one answer for each question.
To get credit for a question, you must circle the right answer (and no others), and you
must show supporting work which in our opinion fully derives your answer. Even if you
can do the work in your head, or the information is already on the sheet that you bring to
the exam, you must show it on your exam paper, and explain what you are doing.
There is no partial credit for the questions.
No books, notes, calculators or computers during the exam...except that you may use
one 8 1/2 by 11 inch sheet of paper with information written on both sides.
This exam is conducted under the Code of Academic Integrity of the University of
Pennsylvania.
Please write your name on each sheet, in the space provided.
Good luck!
Topics covered in this exam:
Permutations and combinations (two questions)
Calculation of probabilities
Conditional probabilities and tree diagrams
Probability distributions, expected value, variance, binomial trials,
normal distributions (three questions)
Convergence and divergence of infinite series (two questions)
Taylor series
Maxima, minima and saddle points
Lagrange multipliers
Double integrals
Systems of linear equations
Inverse of a matrix
Markov processes, stable distributions
Linear programming (two variables, done graphically)
Problem 1. How many ways are there to arrange the letters P P E N N N ?
Answer. 6! / (1! 2! 3!) = 60
Choices. 720, 120, 60, 24, 6
Problem 2. There are 5 college students, 4 business students, and 3 engineering
students who are eligible to serve on an undergraduate activities committee.
If the committee is to consist of 2 college students, 2 business students and
2 engineering students, how many possible different committees are there?
Answer. (5C2) (4C2) (3C2) = 180.
Choices. 60, 120, 180, 1080, 1440
Problem 3. Two fair dice, each marked 1,2,3,4,5,6, with each number equally
likely to appear, are thrown. What is the probability that the sum is 5 ?
Answer. 4/36 = 1/9
Choices. 1/36, 1/18, 1/9, 1/6, 1/5
Problem 4. A supermarket has three employees, A, B and C, who package and
weigh produce. They record the correct weight 98%, 97% and 95% of the time,
respectively, and they handle 40%, 40% and 20% of the packaging, respectively.
A customer complains about the incorrect weight on a package he has purchased.
What is the probability that the package was weighed by employee C ?
Answer. 10 / (8 + 12 + 10) = 1/3
Choices. 4/15, 1/3, 2/5, 9/15, 2/3
Problem 5. A continuous random variable X on 0 ≤ x ≤ 1 has probability
density function f(x) = 5x4 . Calculate the expected value of X .
Answer. Expected value = 5/6
Choices. 1/6, 1/3, 2/3, 5/6, 1
Problem 6. At the beginning of a Philadelphia 76'ers basketball game, the
career statistics of the star player, Alan Iverson, indicate that his shooting
percentage from the free throw line is exactly 80%. If during the game Iverson
attempts only 5 free throws, what is the probability that his career shooting
percentage changes?
Answer. 1 – (256/625), hence between 1/2 and 1
Choices. 0, between 0 and 1/2, 1/2, between 1/2 and 1, 1
Problem 7. The scores on a Math 151 final exam are normally distributed with
a mean of 60 and a standard deviation of 20. A score between 50 and 80 is given
a letter grade of B. If there are 200 students in the class, then the expected number
to get B's falls in which of the following ranges?
(Include the table in Appendix E, page A22 of our calculus text.)
Answer. 200 [A(1/2) + A(1)] = 106.56
Choices. 50 - 70, 70 - 90, 90 - 110, 110 - 130, 130 - 150
Problem 8. When the repeating decimal 0.252525... is expressed as the quotient
of two integers in lowest terms, find the sum of the numerator and denominator.
Answer. 124
Choices. 23, 115, 124, 126, 1249
Problem 9. Which of the following infinite series are convergent?
 1/√n
(2)  1 / (4n + 1)
Answer. Only series (2)
Choices. (1), (2), (3), (1) and (2), all
(3)  1 / (4n + 1)
Problem 10. What is the coefficient of x5 in the Taylor series expansion of
f(x) = x3 / (1 + x2) about the origin?
Answer. –1
Choices. –1, –1/2, 0, 1/2, 1
Problem 11. Find the critical point of the function f(x) = x2 + 4xy + 6y2 – 2x+ 4y
and determine if it is a relative maximum, relative minimum or saddle point.
Answer. Relative minimum at (5, –2) .
Choices. Rel min at (5, –2), rel max at (5, –2), saddle at (5, –2), rel min at (–2, 5),
rel max at (–2, 5)
Problem 12. Let x0 and y0 be the coordinates of the point on the line
5x + 4y = 20 that is closest to the origin. (Note that this is equivalent to
minimizing x2 + y2 subject to 5x + 4y = 20.) Find the value of x0 + y0 .
Answer. x0 = 100/41 and y0 = 80/41 , hence x0 + y0 = 180/41 .
Choices. 4, 180/41, 40/9, 9/2, 5
Problem 13. A region in the plane is bounded by the lines
y = x, y = 2x, x = 1 and x = 3 .
Find the value of the double integral of the function xy over this region.
Answer. 30
Choices. 30, 60, 90, 120, 240
Problem 14. Find the value of k which makes the following system of
equations consistent.
3x – y + z = 8
2x + y – 2z = 7
x – 2y + 3z = k
Answer. k = 1
Choices. –15, –1, 0, 1, 15
Problem 15. Consider the 3 by 3 matrix A given by
1
0
0
0 3
1 –4
0 1
Find the sum of the elements in the first row of the matrix A–1 .
Answer. –2
Choices. –7, –2, 1, 2, 7
Problem 16. A regional planning board knows that although the population of
their region remains stable, each year 30% of the people in the eastern portion
move to the west and 40% of the people in the western section move to the east.
Find the fraction of the people in this region who, in the long run, live in the west.
Answer. 3/7
Choices. 3/10, 4/10, 3/7, 4/7, 7/10
Problem 17. Infotron Inc. makes electronic hockey and soccer games. Each
hockey game requires 2 labor-hours of assembly and 2 labor-hours of testing.
Each soccer game requires 3 labor-hours of assembly and 1 labor-hour of testing.
Each day there are 42 labor-hours available for assembly and 26 labor-hours
available for testing. Find the maximum number of games (hockey + soccer)
that Infotron can produce each day.
Answer. 9 hockey games and 8 soccer games per day, hence 17
Choices. 13, 14, 15, 16, 17