SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

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1 Math 1324 Final Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of two equations in two variables. 1) 8x + 9y = -20 3x + 5y = -14 Write an augmented matrix for the system of equations. 2) -2x + 2z = -2 6y + 6z = 84-2x + 7y + 4z = 56 Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z. 3) x + y + z = -1 x - y + 2z = -13 5x + y + z = -17 Use the Gauss-Jordan method to solve the system of equations. 4) x + y + z = 8 x - y + 4z = -5 5x + y + z = 24 5) x + y + z = 9 2x - 3y + 4z = 7 x - 4y + 3z = -2 6) x - y + 3z = -8 x + 5y + z = 40 5x + y + 13z = 10 Perform the indicated operation where possible ) Perform the indicated operation. 8) Let A = 1 3 and B = Find 4A + B. Given the matrices A and B, find the matrix product AB. 9) A = -1 3, B = Find AB Find the inverse, if it exists, of the given matrix. 10)

2 Solve the matrix equation for X. 11) A = , B = , AX = B Find the production matrix for the input-output and demand matrices. 12) A = , D = ) Suppose the following matrix represents the input-output matrix of a primitive economy. How much of each commodity should be produced to produce 70 bushels of yams and 61 pigs? 14) If A is a 3 4 matrix and the matrix sum A + B exists, what can you say about the matrix B? Find the value(s) of the function on the given feasible region. 15) Find the maximum and minimum of z = 9x - 20y. Introduce slack variables as necessary and write the initial simplex tableau for the problem. 16) Maximize z = 4x1 + 2x2 subject to: 2x1 + x2 27 3x1 + 5x2 61 x1 0, x2 0 2

3 Find the pivot in the tableau. 17) Use the indicated entry as the pivot and perform the pivoting once. 18) Use the simplex method to solve the linear programming problem. 19) Maximize z = 2x1 + 5x2 + 3x3 subject to: 2x1 + x2 + 3x3 9 4x1 + 3x2 + 5x3 12 with x1 0, x2 0, x3 0 A bakery makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, 1 dozen eggs, and 2 lb of sugar. A batch of donuts requires 5 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex tableau to maximize profit. 20) The bakery has 700 lb of flour, 800 dozen eggs, 400 lb of sugar. The profit on a batch of sweet rolls is $ and on a batch of donuts is $ ) A political mailing will have several pages on the economy, the military, and the environment. The total number of these pages in the booklet should be less than 100. For the target group that will receive the booklet, market research suggests that there will be a positive impact proportional to 5 times the number of pages on the economy, a positive impact proportional to 2 times the number of pages on the military, and a negative impact proportional to -4 times the number of pages on the environment. The candidate, however, insists that the number of pages on the environment exceed the number on the military by at least 5 and that the number of pages on the economy also exceed the number on the military by at least 5. Find the number of pages that should be devoted to the economy, the military, and the environment. Find the transpose of the matrix. 22)

4 State the dual problem. Use y1, y2, and y3 as the variables. Given: y1 0, y2 0, and y ) Minimize w = 6x1 + 3x2 subject to: 3x1 + 2x2 33 2x1 + 5x2 40 x1 0, x2 0 Use duality to solve the problem. 24) Minimize w = 4y1 + 2y2 subject to: 3y1 + 2y2 60 4y1 + y2 40 y1 0, y2 0 Find the number of subsets of the set. 25) {mom, dad, son, daughter} Let U = {q, r, s, t, u, v, w, x, y, z}; A = {q, s, u, w, y}; B = {q, s, y, z}; and C = {v, w, x, y, z}. List the members of the indicated set, using set braces. 26) A' B Shade the Venn diagram to represent the set. 27) C' (A B) Use a Venn Diagram and the given information to determine the number of elements in the indicated set. 28) n(a) = 65, n(b) = 73, n(c) = 67, n(a B) = 13, n(a C) = 15, n(b C) = 9, n(a B C) = 7, and n(a' B' C') = 131. Find n(u) Use a Venn diagram to answer the question. 29) A survey of 142 college students was done to find out what elective courses they were taking. Let A = the set of those taking art; B = the set of those taking basket weaving; and C = the set of those taking canoeing. The study revealed the following information: n(a) = 45; n(b) = 55; n(c) = 40; n(a B) = 12; n(a C) = 15; n(b C) = 23; n(a B C) = 2. How many students were not taking any of these electives? Find the probability of the given event. 30) A card drawn from a well-shuffled deck of 52 cards is an eight or a 9. 31) A bag contains 8 red marbles, 2 blue marbles, and 1 green marble. A randomly drawn marble is not blue. 4

5 Use the given table to find the probability of the indicated event. Round your answer to the nearest thousandth. 32) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. toppings freshman sophomore junior senior cheese meat veggie A randomly selected student prefers a cheese topping. 33) A single die is rolled one time. Find the probability of rolling a number greater than 2 or less than 6. Find the odds in favor of the indicated event. 34) Rolling a 4 with a fair die. 35) Below is a table of data from a survey given to 1600 teenagers asking them to estimate what percentage of their classmates are using drugs. Find the probability that a randomly selected girl thinks that 50% or more of her classmates are using drugs. Round your answer to the nearest hundredth. None 1% - 24% 25% - 49% 50% - 74% 75% or more Boys Girls ) If two fair dice are rolled, find the probability of a sum of 6 given that the roll is a double. Assume that two marbles are drawn without replacement from a box with 1 blue, 3 white, 2 green, and 2 red marbles. Find the probability of the indicated result. 37) The second marble is red, given that the first marble is white. Use the given table to find the indicated probability. 38) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. toppings freshman sophomore junior senior cheese meat veggie P(favorite topping is meat student is junior)? Round the answer to the nearest thousandth. Express the answer as a percentage. 39) 40% of the workers at Motor Works are female, while 46% of the workers at City Bank are female. If a worker is selected at random, what is the probability that the worker will be from Motor Works, given that the worker is female? Round your answer to the nearest tenth, if necessary. 5

6 Provide an appropriate response. 40) Can an event E for a sample space S contain an outcome that is not in S? Evaluate the expression. 41) 10P3 42) 9 2 Use the multiplication principle to solve the problem. 43) A shirt company has 4 designs each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company? 44) How many ways can 6 people be chosen and arranged in a straight line if there are 8 people to choose from? 45) Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red cards and three black cards? 46) A bag contains 4 blue, 4 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include exactly two red marbles? A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability. 47) 2 cherry, 1 lemon A die is rolled five times and the number of fours that come up is tallied. Find the probability of getting the given result. 48) Exactly one four 49) Suppose a charitable organization decides to raise money by raffling a trip worth $500. If 3000 tickets are sold at $1.00 each, find the expected value of winning for a person who buys 1 ticket. Find the equilibrium vector for the transition matrix. 50) Find the requested long-range probabilities based on the transition matrix or data given. 51) At a liberal arts college, students are classified as humanities or science majors. There is a probability of 0.1 that a humanities major will change to science and of 0.5 that a science major will change to humanities. Find the long-range prediction for the proportion of students in each major. Find the median. 52) 46, 22, 6, 1, 26, 13, 28, 32, 34, 31 6

7 Find the mean. 53) The normal monthly precipitation (in inches) for August is listed for 20 different U.S. cities. Find the mean of the data. Round to the nearest hundredth Find the standard deviation. 54) 270, 250, 191, 283, 123, 228, 291, 141, 244 Find the area under the normal curve. 55) Find the percent of the area under a normal curve between the mean and 0.35 deviations from the mean. 56) Find the percent of the area under the standard normal curve between z = and z = Assume the distribution is normal. Use the area of the normal curve to answer the question. Round to the nearest whole percent. 57) The mean monthly income of trainees at a local mill is $1100 with a standard deviation of $150. Find the probability that a trainee earns less than $900 a month. A company installs 5000 light bulbs, each with an average life of 500 hours, standard deviation of 100 hours, and distribution approximated by a normal curve. Find the approximate number of bulbs that can be expected to last the specified period of time. 58) Between 290 hours and 500 hours A die is rolled five times and the number of twos that come up is tallied. Find the probability of getting the indicated result. 59) Two comes up two times. Suppose 500 coins are tossed. Using the normal curve approximation to the binomial distribution, find the probability of the indicated results. 60) Exactly 250 heads 7

8 Answer Key Testname: MATH 1324 FRS13 1) (2,-4) 2) ) (-4, 5, -2) 4) (4, 5, -1) 5) -7z , 2z , z 6) No solution 7) ) ) ) ) ) ) X = ) B is a 3 4 matrix 15) 45, ) x1 x2 x3 x4 z ) 2 in row 1, column 2 18) 19) Maximum is 20 when x1 = 0, x2 = 4, x3 = 0 8

9 Answer Key Testname: MATH 1324 FRS13 20) x1 x2 x3 x4 x5 x ) 44, 25, 30 22) ) Maximize z = 33y1 + 40y2 subject to: 3y1 + 2y2 6 24) 64 25) 16 26) {q, r, s, t, v, x, y, z} 27) 2y1 + 5y2 3 28) ) 50 30) ) ) ) 1 34) 1 to 5 35).22 36) ) ) ) 46.5% 40) No 41) ) 36 43) 56 9

10 Answer Key Testname: MATH 1324 FRS13 44) 20,160 45) 845,000 ways 46) ) ) ) -$.83 50) [ ] 51) [ ] 52) 27 53) 2.94 in. 54) ) 13.68% 56) 87.7% 57) 9% 58) ) )