Practice Review for Final for Math 1324

MATH 1324 Review for Final

Below is a list of the types of questions you should expect to see on the final exam.

Sets and counting:

Be able to look at a shaded Venn diagram and decide which of several expressions it represents. Know the notation for union, intersection, and complement. Be able to use a Venn diagram to solve a word problem. Be able to use the multiplication principle, permutations, and combinations in counting problems. Know about a deck of cards and a pair of dice. Know what is meant by a "code word".

Probability:

Know the basic principles of probability.

Know when Bayes’ theorem applies to conditional probabilities and when it doesn't.

Be able to find the expected value of a game.

Be able to use the z-tables to find a probability given a normal distribution.

Be able to find the mean value when given a frequency table.

Be able to use a normal curve approximation for a binomial distribution. The instructions for this problem will tell to use the approximation. Know how to deal with the words "at least" and "at most".

Finance:

Be able to use the TVM solver to work with compound interest. Be able to decide if there is to be one payment or many.

Be able to decide if the question asks for a present value or a future value. Be able to find the total interest on a loan. Know the difference between compound and simple interest, and how to find the number of compounding periods per year.

Matrices:

Be able to multiply two matrices without using your calculator. There will be at least one variable in the matrices.

Be able to use row operations to reduce a matrix and find the solution to a system of equations.

Minimization and Maximization:

Be able to minimize or maximize an objective function given the feasible region. Be able to write the linear programming problem that models a word problem. Be able to give the initial simplex tableau for a linear programming problem.

Be able to determine the correct pivot element for a simplex tableau. Be able to find the solution to a minimization problem solved by the dual method.

Practice Review for Final for Math 1324

Please review all old tests and any course packets for the final also. These review questions will not be exact examples of problems on the final exam!

1. Solve the following system of equations using row operations:

(a) x + 2y = 3

2x + 4y = 7

(b) x + y + z = 1

2x + y = -2

3y + z = 2

2x + 3y = -3

3x + 5y + 3z = 1

2. Use the given matrices to compute the following operations, if possible.

A = B = C =

(a) What is the size of A? of B?

(b) Find a12 and b32

(d) Find AB. Find BA.

3. If the feasible region for a system of linear inequalities has corner points of (0, 8), (4, 3) and

(5, 0), find both the maximum and minimum value of the objective function z = 6x + 4y.

4. It takes 3 hours to build a planter box and 1 hour to paint it. It takes 4 hours to build a step-stool and 2 hours to paint it. My grandpa has 12 hours available to paint and he has 8 hours available for building. He wants to sell the planter box for $15 each and each step-stool for $20 each. Set up the linear programming problem that would maximize the income my grandpa would make. Do not work out.

5. The EB Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring and 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $40 and on an SST ring is $30?

6. Each day Ron needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill 1 contains 4 units of A and 3 of B. Pill 2 contains 1 unit of A, 2 of B, and 4 of C. Pill 3 contains 10 units of A, 1 of B, and 5 of C. If pill 1 costs 6 cents, pill 2 costs 9 cents, and pill 3 costs 1 cent, how many of each pill must Ron take to minimize his cost?

7. Set up the initial simplex tableau for the following problem:

Maximize P = 5x + 9y

Subject to: 3x + 4y 48

2x + 3y 60

x 0, y 0

8. Given the following initial simplex tableau, find the first pivot:

9. Read the solution from the following simplex tableau:

10. Read the solution from the following dual tableau:

You were told to minimize P = 18y1 + 20y2 + 2y3

11. (a) Andra borrowed $ 10,000 to purchase a new car at an annual interest rate of 11%. She is to pay it back in equal monthly payments over a 5 year period. How much total will be paid over the period of the loan?

(b) Find the amount of interest earned if $3954 is deposited at 8% interest compounded semiannually for 12 years.

(d) Kristin wants to start an IRA that will have $ 460,000 in it when she retires in 15 years. How much should she invest semiannually in her IRA to do this if the interest is 15% compounded semiannually?

(e) At the end of every 3 months, Teresa deposits $100 into an account that pays 6% compounded quarterly. After 4 years, she puts the accumulated amount into a certificate of deposit paying 7.5% compounded semiannually for 1 year. When this certificate matures, how much will Teresa have accumulated?

12. Answer the following true or false:

(a) 6 {5, 6, 10}

(b) 5 {a, 2, b, 5}

(d) {2, 4, 6}

13. If A = {3, 4, 7, 9, 12} and B = {2, 3, 7, 8, 12}, find A B and A B.

14. Shade a venn diagram that would represent (A C ‘ ) B.

15, Fill in a venn diagram for the following problem:

Of 100 Aggies surveyed concerning where they prefer to go on Friday nights:

35 like Harry’s and Shadow Canyon

69 like The Chicken

59 like more than one of the three places.

30 like Harry’s or Shadow Canyon, but not the Chicken

15 like only Harry’s and The Chicken

30 like all three places

15 like Harry’s, but not the Chicken

16. (a) Supposed that Robert has 7 shirts, 4 pairs of jeans and 3 pairs of shoes. How many different outfits does Robert have assuming he wears one of each?

(b) How many different arrangements are there of the letters in the word “Howdy”? How many arrangements are there in the word “waterfall”?

(d) A committee of 4 is to be chosen from a group of 5 women and 6 men. How many ways can a committee be chosen if there is at least two women? Exactly two women? No women?

17. (a) Suppose that a single card is drawn from a standard 52 card deck. Find the probability that the card is a red five.

(b) Suppose that one card is drawn from a standard 52 card deck. Find the probability that the card is a red card or a five.

(d) Suppose that the reliability of a test for hepatitis is specified as follows: Of people with hepatitis, 95% have a positive reaction and 5% have a negative reaction; of people free of hepatitis, 90% have a negative reaction and 10% have a positive reaction. From a large population of which .005 of the people have hepatitis, a person is selected at random and given the test. If the test is positive, what is the probability that the person actually has hepatitis?

(e) Box A contains 5 red balls and 1 pink ball; box B contains 2 red balls and 3 pink balls. A box is chosen, and a ball is selected from it. The probability of choosing box A is 3/8. If the selected ball is pink, what is the probability that it came from box A? Find the probability that the ball came from box B, given that it is red. Find the probability that the ball is pink, given that it came from box A.

(f) It is known that 83% of the students taking a certain academic course will pass the course with a grade of C or better. If in a particular class, there are 28 students, what is the probability that at least 24 students will pass?

18. If 4 balls are drawn from a bag containing 5 red, 4 blue and 3 yellow balls, what is the expected number of yellow balls in the sample?

19. Suppose you pay $5 to play a game that offers to pay you $20 if you draw 2 cards from a standard deck of 52 cards and get all hearts. What is the expected payoff? Is it a fair game?

20. Find the mean, median, range, mode, and standard deviation of the following:

(a) 15, 13, 20, 19, 8, 22, 10, 10

(b) x 10 20 30 40

frequency 3 5 2 4

21. The probability that a certain baseball team will win a given game is 0.42. If the team plays 25 games, find the expected number of wins and the standard deviation.

22. It’s found that a random variable X is normally distributed with a mean of 30 and a standard deviation of 4.

(a) Find the probability that the data will be between 24 and 36.

(b) Find the value for X that has 10% of the area less than that value (X).

23. If the odds of something happening is 2 to 3, what is the probability of it happening?

24. If the probability of an event is 0.35, what are the odds for the event.

25. We pay $10 to play a card game. We draw one card from a deck. If it is the queen of spades, we get $30 total back. If it is any other black card, we get $5 back to us. Anything else, and we lose. What is the expected value of this game?

26. Suppose we play a new game. We pay $10 to play, and then roll a pair of dice. If we roll 7 or 11, we win, while a 2, 3, or 12 means we lose. Otherwise, we must roll again, and if the second roll comes up 7 or 11, we still win, but otherwise we lose. If when we win we get $30, then find the expected value of the game.

Answers to 1324 Final Review

1) (A) No solution (B) (-1,0,2) (C) infinitely many solutions of the form (9z-18, -6z+11,z)

2) (A) 2x2, 3x2 (B) x, 0 (C) cannot be done, , (D) cannot be done,

3) max is 36 at (4,3), min is 30 at (5,0)

4) Let x= # of planters, y=# of step stools. MAX: Z = 15x + 20y subject to

5) 12 VIP and 12 SST

6) 0 of pill 1, 0 of pill 2, and 12 of pill 3.

8) the 3 in R2C2

9) max p = 80, x2 = 20, s2 = 16, x1 = s1 = 0

10) min p = 29, y1 = 3/2, y2 = 1/10, y3=0

11) (A) $13,045.20 (B) $6181.30 (C) $454.83 (D) 4448.77 (E) 1930.25

12) (A) True (B) False (C) True (C) True

13) {2, 3, 4, 7, 8, 9, 12} , {3, 7, 12}

14)

15)

16) a) 84 b) 120, 90720 c) 36 d) 215, 150, 15

17) a) 1/26 b) 7/13, c) 0.4, 0.6 d) 0.04556 e) 1/7, 4/9, 1/6 f) 0.4705

18) I yellow ball

19) Lose $3.82, no

20) a) 14.625, 14, 14, 10, 5.2355 b) 25, 20, 30, 20, 11.6024

21) expected number = 10.5 standard deviation = 2.468

22) a) 0.8664 b) 24.874

23) 2/5

24) 7:13

25) Lose $7.02

26) Win $1.11