Date: Tuesday May 8 .
Time: 11:00AM -1:00 PM
Place: CHEM 102 ( Corner of 34th and Spruce)
Note: For the Final Exam calculators are NOT permitted. Otherwise our customary
cheat sheet rule is in effect. There will be multiple choice questions in which one is required
to choose the right answer AND to justify it with relevant calculations. Failure
to do either must result in no credit.
I will not have regular hours on Tuesday morning May 1 but wil have extra
office hours on Wednesday May 2 and Thursday May 3. 10:00 AM -1:00 PM
both days.
Also on Monday May 7 , 10:00 AM -1:00 PM. **
MATH 151 - PRACTICE FINAL EXAM
Suggestions:.
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 review session.
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.
ANS 120
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?
ANS 3/10
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?
ANS 1/9
Problem 4. A continuous random variable X on 0 <= X 2 <=1 has
probabililty density function f(x) = 3 * x^2 . Compute the variance of X .
ANS 3/80
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.
ANS Roughly 340
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 ?
ANS 1
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)?
ANS 4
Problem 8. Approximate e^(1/10) by using the first four terms of the power
series for e^x .
ANS 1+1/10 + 1/ 200 + 1/6000 = 1.1051666
Problem 9. Find the sum of the infinite series Sum((7 * (- 3/4)^n),n=1..infinity) .
ANS -3
Problem 10. Which of the following three infinite series converge(s)?
a. Sum( (6 ln n) / n, n=1..infinity)
b. Sum( n / (n^3 + 1), n=1..infinity)
c. Sum( n^(1/999), n=0..infinity)
ANS b. only
Problem 11. Consider the function f(x, y) = x^2 + 2*x*y - y^2 + 2x - 6y + 7
Find its critical points and determine their type.
ANS Saddle point at ( 1 , -2)
Problem 12. Sketch the level curves of the function f(x, y) = x^2 - y^2 .
ANS Family of Hyperbolas together with the lines y =x and y =-x
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 x^2 + y^2 = 1 ?
ANS ( 3/5 , 4/5 )
Problem 14. Let R be the triangle with vertices (0, 0) , (1, 0) and
(1, 2) . Compute the double integral of z = 30*x^2*y over R
ANS 12
Problem 15. Find all solutions to the following system of linear
equations.
{ x + 3y =1 , x +2 y = 3 , - 4x + y + z = -2 )
ANS ( 7 , -2 , 28 )
Problem 16. Let A denote the 3 by 3 matrix:
Row 1 = { 1 , 2 , 0 }
Row 2 = { 0 , 1 , -1 }
Row 3 = { 0 , 2 , -1 }
Find the sum of the entries in the first row of the inverse matrix A ^ (-1) .
ANS 1
Problem 17. A simple economy consists of two sectors, agriculture and
tourism.
The input-output matrix is
Row 1 = { 0.2 0.2 }
Row 2 = { 0.3 0.3 }
How many units should be produced by each sector to meet the consumer
demand of 20 units of agriculture and 10 units of tourism?
ANS ( 32 , 28 )
Problem 18. Find the stable distribution of the Markov chain whose
transition matrix is
Row 1 = { 0.9 0.2 }
Row 2 = { 0.1 0.8 }
ANS ( 2/3 , 1/3 )
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?
ANS 0 (One uses the absorbing stochastic matrix)
Problem 20. Given the objective function 3x + 4y , 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?
ANS ( 0 , 4) at which point the objective function has value 16.