Download Tuesday April 24, 2001

Tuesday April 24, 2001
MATH 151 - PRACTICE FINAL EXAM
INSTRUCTIONS.
Work out all the problems on this practice exam.
Give detailed explanations, using words as well as equations.
Use a separate sheet for each problem.
Leave space to include notes from our review session.
Bring the practice exam with you to the review session.
We'll go over the problems at that time.
NOTE. These problems are taken from last semester's Math 151 final exam,
with the multiple choices omitted, and with occasional slight changes
in wording, and do not necessarily correspond to questions on this
semester's exam.
Problem 1. Six school children are the finalists in a spelling contest.
In how many possible ways can the winners of the first, second and third prizes
(one of each) be chosen among them.
Problem 2. Ten equally qualified applicants, six men and four women, apply for three lab
technician positions. Unable to justify choosing any of the applicants over the others, the
personnel director decides to select three at random. What is the probability that one man
and two women will be chosen?
Problem 3. The chances of having a left-handed child are 4 in 10 if both parents are left-handed,
2 in 10 if one parent is left-handed, and only 1 in 10 if neither parent is left-handed. Suppose a
left-handed child is chosen from a population in which 20% of the adults are left-handed.
What is the probability that the child's parents are both left-handed?
Problem 4. A continuous random variable X on 0 ≤ x ≤ 1 has probabililty density function
f(x) = 3x2 . Compute the variance of X .
Problem 5. An IQ test is scaled to give a mean score of 100 with a standard deviation of 20, and
it is assumed that the scores are normally distributed. Children having IQs of less than 80 or
more than 145 are deemed to need special attention. Given a population of 2000 children, how
many of them can be expected to need these additional services? For this problem, please
consult the table in Appendix E, page A22 of our calculus text.
Problem 6. A random variable X takes non-negative integer values 0, 1, 2, 3, ... with
probabilities Pr(X = k) = (1/e) (1/k!) . What is the expected value of X ?
Problem 7. A young couple plans to continue having children until they have their first girl,
and then to stop. Suppose the probability that a child is a girl is 1/2 and that the outcome of
each birth is an independent event. What is the expected family size (including the parents and
their children)?
Problem 8. Approximate e–1/10 by using the first four terms of the power series for ex .
Problem 9. Find the sum of the infinite series
n=2 ∞
7 (–3/4)n .
Problem 10. Which of the following three infinite series converge?
n=1 ∞ (6 ln n) / n
n=2 ∞
n / (n3 + 1)
n=0 ∞ n1/999
Problem 11. Consider the function f(x, y) = x2 + 2xy – y2 + 2x – 6y + 7 .
Find its critical points and determine their type.
Problem 12. Sketch the level curves of the function f(x, y) = x2 – y2 .
(Note: in last semester's Math 151 final exam, a sketch of the level curves was given,
and the question was to choose the corresponding function.)
Problem 13. For which values of (x, y) does the function f(x, y) = 3 – 3x – 4y have a
minimum when subject to the constraint x2 + y2 = 1 ?
Problem 14. Let R be the triangle with vertices (0, 0) , (1, 0) and (1, 2) . Compute the
integral
∫∫R
30x2y dx dy .
Problem 15. Find all solutions to the following system of linear equations.
x + 3y
= 1
x + 2y
= 3
–4x + y + z = –2 .
Problem 16. Let A denote the 3 by 3 matrix
1
0
0
2
1
2
0
–1
–1 .
Find the sum of the entries in the first row of the inverse matrix A–1 .
Problem 17. A simple economy consists of two sectors, agriculture and tourism.
The input-output matrix is
.2 .2
.3 .3
.
How many units should be produced by each sector to meet the consumer demand of
20 units of agriculture and 10 units of tourism?
Problem 18. Find the stable distribution of the Markov chain whose transition matrix is
.9
.1
.2
.8
,
Problem 19. A simple board game has four fields A, B, C and D . Once you end up on
field A you have won and the game is over. One you end up on field B you have lost and
the game is over. From fields C and D you can move to other fields by flipping a fair coin.
If you are on field C and you throw a head, then you move to field A , otherwise to field D.
If you are on field D and you throw a head, then you move to field C , otherwise to field B.
Suppose that you start in field D . What is the probability that you win?
Problem 20. Given the objective function 3x + 3y , subject to the constraints:
x ≥ 0
y ≥ 0
2x + y ≤ 5
x+y ≤ 4.
For which point (x, y) does the objective function (subject to these constraints) reach a maximum?