Finite Mathematics - Sample Tests and Study Hints Part 2

MIDTERM EXAMS Sample exams included in textbook, Appendix D 13 Exam 3 – Version 1 Exam 3 – Version 1 This is a 75 minute test. Part 1. Problems 1 through 13 are worth 5 points each, a total of 65 points. In each of the problems 1 through 4, the augmented matrix shown has been obtained by a sequence of row operations. In each case, decide which of the following statements is true about the associated system of equations. (A) (B) (C) (D) (E)  1.   2.   3.    4.   The system has a unique solution. The system has no solution. The system has an infinite number of solutions with one arbitrary parameter. The system has an infinite number of solutions with two arbitrary parameters. None of the above.   1 0 5  10 0 3 −3  6  0 1 −1  2   1 0 6  4 0 3 −3  6  0 1 −1  −2   1 0 1 0  2 0 1 −1 −4  0  0 2 −2 −8  0   1 0 0 2  2 0 1 −1 3  1   0 1 −1 −3  1  0 0 1 1  −2 5. Find the maximum value of the objective function in the following linear programming problem.  x≥0      y≥0 Maximize 3x + y, subject to 5x + 2y ≤ 90   3x + 4y ≤ 96    y ≤ 30 (A) 96 (B) 51 (C) 54 (D) 30 (E) none of the others For problems 6, 7, and 8, the matrices A, B, and C are defined as      −1 3 5 4 −2 1 1  B= 3 C =  −1 A= 2 −5 6 2 −2 3  −3 2  0 14 Exam 3 – Version 1 6. Find the (2, 1) element of AB. (A) −2 (B) 8 (C) 12 (D) −5 (E) none of the others 7. Find the (1, 2) element of 2B − C. (A) 7 (B) 5 (C) 9 (D) 3 (E) none of the others 8. A matrix D satisfies 3B + D = 2C. Find the (2, 2) element of the matrix D. (A) −1 (B) 1 (C) 2 (D) 3 (E) none of the others 9. Which of the following systems of equations has a unique solution?    2x − 3y = 2 2x − 3y = 2 2x − 3y = 2 (B) (C) (A) −4x + 6y = 4 4x − 6y = 4 4x + 6y = −4  2x − 3y = 2 (E) none of the others (D) −4x + 6y = −4 10. Find the y-coordinate of the intersection of the lines 3x − 4y = 11 and 5x + 2y = 1. (A) −2 (B) 1 (C) 4 (D) 1 2 (E) none of the others 11. Which of the following constraints describes the shaded set shown on the coordinate system below? y 1 −3 −2 −1 0 1 2 x −1 (A) x − 2y ≤ 2 (D) x − 2y ≥ −2 (B) x − 2y ≥ 2 (C) x − 2y ≤ −2 (E) none of the others 12. A set S is described by the system of inequalities   6x + 2y ≤ 10 2x − y ≥ 0  x − 3y ≤ 15 Find the y-coordinate of the corner point of S in the first quadrant. (A) 2 (B) 4 (C) 1 (D) 3 (E) none of the others 15 Exam 3 – Version 1 13. Find the equation of the line whose graph is shown below. y 4 (0, 3) 3 2 (−2, 1) 1 x −4 −3 −2 −1 0 1 −1 (A) x − y = 3 (B) 3x − y = −3 (D) x + y = 3 (E) none of the others (C) x − y = −3 Part 2. Problem 1 is worth 15 points, problems 2 and 3 are worth 10 points each, a total of 35 points. 1. Formulate and solve the following problem. Show all work. Barbara’s Basket’s, Inc., produces two types of decorative straw baskets, Colonial and Southwest styles. One Colonial basket requires 30 pieces of yellow straw and 30 pieces of brown straw, and one Southwest basket requires 50 pieces of yellow straw and 10 pieces of brown straw. Each week the company has 4500 pieces of yellow straw and 1500 pieces of brown straw to be used to produce baskets. There is a commitment to produce at least 30 Southwest baskets. The profit is $4.00 for each Colonial basket and $6.00 for each Southwest basket. How many baskets of each type should be produced each week to maximize profit? 2. Find all solutions of the following system of equations. Show all work.  = 0  2x − 5y x − 3y − z = −1  −x + 2y − z = −1  1 3. Find the inverse of the matrix A =  0 2 0 1 1  1 1 . Show all work. 4 16 Exam 3 – Version 2 Exam 3 – Version 2 This is a 75 minute test. 1. Find an equation for the straight line which goes through (2, 3) and is parallel to the line 9x + 3y = 1. (A) 3x − y = 3 (B) 3x + y = 9 (C) 3x + y = 1 (D) 2x + y = 7 (E) 2x − y = 1 (F) none of these 2. Find the equation which describes a line with y-intercept −2 and slope 34 ? (A) 3x + 4y = 6 (D) 4x − 3y = 6 (B) 3x − 4y = 6 (E) 4x − 3y = 2 (C) 4x + 3y = 6 (F) none of these 3. Find the value of x at the intersection of the line 2x + 3y = 18 and x − y = 4. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 (F) none of these 4. Select the correct statement about the three lines given by the equations: x + y = 2, x = −2, x + 2y = 7. (A) Two of these lines are parallel. (B) Two of these lines have negative slopes, and the slope is undefined for the other line. (C) Two of these lines have negative slopes, and the slope is zero for the other line. (D) Two of these lines have positive slopes, and the slope is undefined for the other line. (E) These lines all go through a single point (F) none of the above. 5. Suppose that the cost of renting in-line skates is related to the number of hours the skates are rented by a linear equation. Also, suppose the cost of a 2-hour rental is $20 and the cost of a 3-hour rental is $25. Find the cost of a 6-hour rental. (A) $40 (B) $42 (C) $44 (D) $46 (E) $48 (F) none of these For each of the augmented matrices in the next three problems, determine which of the following statements is true about the associated system of linear equations: (A) The system has no solution. (B) The system has exactly one solution. (C) The system has exactly two solutions. (D) The system has infinitely many solutions in which one variable can be selected arbitrarily. (E) The system has infinitely many solutions in which two variables can be selected arbitrarily. (F) none of the above.    1 2 0 0  7 6.  0 0 1 −1  2  0 0 0 0  0 17 Exam 3 – Version 2 7. 8.  1  0 0  1  0 0 2 0 0 0 1 0 0 0 0 2 0 0 0 1 0 0 0 1              7 2 1  7 2 0 9. Determine which of the following matrices are in reduced     1 0 −1 3  1 1 B=0 A = 0 1 4 2  1   0 0 0 1 3 0     1 0 0 0  1 1 D= 0 C =  0 1 0 0  1  0 0 0 1  3 0 (A) C only (D) B, C and D only form.       0 −1 0 1 4 0 0 0 1 1 1 0 (B) B and C only (E) A and C only 0 0 0 0 0 1        1 1 3  1 1 3 (C) A, B and C (F) none of these 10. Ralph’s Fill Dirt and Croissant Shop in Spencer makes both grand and petit croissants. Each grand requires 1 ounce of flour and 2 ounces of butter, while each petit requires only 14 ounce of flour and 13 ounce of butter. It is the end of the month and Ralph has only 4 ounces of flour and 6 ounces of butter. Ralph calculates how many of each type of croissant he should make in order to use all his ingredients. Determine the number of grand croissants Ralph calculated. (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problems 11 and 12 refer to the following matrices     2 4 1 −2 A= B= C = (1 2) 1 1 0 1 (F) none of these D=  2 0 3 1  E = (3 −1 ) 11. Determine which of the following is defined. (A) A − DB 12. Find CA. (B) 3B + DC (C) BC + 2D (A) ( 1 6 1 ) (D) ( 4 6 1 ) (D) AE (B) ( 4 3 1 ) (E) undefined (E) BC + E (C) ( 4 6 −1 ) (F) none of these 13. Find the value of x which solves the system of equations: 2x + 4y + 10z = −2 y − 3z = 4 x + 2y + 6z = −2 (F) none of these 18 Exam 3 – Version 2 (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 (F) none of these 14. Find all solutions to the following system of equations: x − y + 3z = 5 2x + y + 3w = 7 −x + y − 3z + 4w = 3 (A) x = 2, y = −3 + 3z, z arbitrary, w = 2 (B) x = 2 + z, y = −3 + 3z, z arbitrary, w = 2 (C) x = 3 − z, y = −3 + 3z, z arbitrary, w = 2 (D) x = 2 − z, y = −3 + 2z, z arbitrary, w = 2 (E) x = 2 − z − 2w, y = −3 + 2z + w, z arbitrary, w arbitrary (F) none of these 15. A certain 3 × 3 matrix A has as its inverse the  1 A−1 =  1 0 Determine the value of y where (A) 1 matrix  1 1 −1 0  . −1 2     x 9 A  y =  4  z −3 (B) 2 (C) 3 (D) 4 (E) 5 (F) none of these 16. An economy has two goods: steel and grain. One needs .25 units of steel to produce 1 unit of steel, and .27 units of grain to produce 1 unit of grain. In addition, one needs .2 units of steel to make 1 unit of grain, and .4 units of grain to make one unit of steel. Let x1 be the number of units of steel produced and x2 be the number of units of grain produced. Find the technology matrix associated with the corresponding Leontief economic model.       .25 .2 .27 .4 .25 .4 (B) (C) (A) .27 .27 .25 .2 .4 .2     .25 .27 .27 .2 (E) (F) none of these (D) .4 .25 .2 .4 17. Let A=  .4 .2 .9 .2  , D=  30 60  , and X=  x1 x2  be the technology matrix, demand vector, and production schedule for a Leontief economic model. Find x1 . (A) 80 (B) 90 (C) 100 (D) 110 (E) 120 (F) none of these 19 Exam 3 – Version 2 18. An enterprising student plans to make and sell novelty T-shirts at the Little 500. A simpler style requires 1 ounce of ink and 20 minutes of labor for each shirt, while a more elaborate style requires 3 ounces of ink and 70 minutes to make each shirt. The student has 12 gallon of ink and can spend at most 20 hours making the T-shirts. The profit on the simpler style is $3 per shirt, and the profit on the more elaborate shirt is $5. Let x represent the number of simpler style shirts and y the number of more elaborate style shirts. Formulate a linear programming problem to maximize the profit. (Recall that there are 128 ounces in a gallon.) Maximize 3x + 5y subject to (A) x + 3y ≤ 128, 20x + 70y ≤ 20, x ≥ 0, y ≥ 0 (B) x + 3y ≤ 128, 20x + 70y ≤ 1200, x ≥ 0, y ≥ 0 (C) 3x + y ≤ 64, 20x + 70y ≤ 1200, x ≥ 0, y ≥ 0 (D) 3x + y ≤ 64, 70x + 20y ≤ 20, x ≥ 0, y ≥ 0 (E) x + 3y ≤ 64, 20x + 70y ≤ 1200, x ≥ 0, y ≥ 0 (F) none of these 19. Determine the feasible set given by the constraints x + y ≥ 2, y −1 −x + y ≥ 1, y x ≥ 0, y≥0 y y y 3 3 3 3 3 2 2 2 2 2 1 1 1 1 0 1 2 x −1 (A) (F) none of these 0 1 2 x −1 0 (B) 1 2 x −1 1 0 (C) 1 (D) 2 x −1 0 1 (E) 20. Find the minimum of x + y on the feasible set shown below. y 3 2 1 x −1 (A) 3 (B) 4 (C) 5 0 (D) 6 1 2 3 4 (E) there is no minimum (F) none of these 2 x FINAL EXAMS Sample exams included in textbook, Appendix D ANSWERS FOR SAMPLE EXAMS 1 AND 3 Sample Exam 1, Version 1 1. 2. 3. 4. C B C E 5. 6. 7. 8. E B B C 9. C 10. C 11. C 12. D 13. 14. 15. 16. A C A A 17. 18. 19. 20. 9. A 10. E 11. D 12. D 13. 14. 15. 16. C True False False 7. C 8. B 9. C 10. A 11. B 12. A C B C B Sample Exam 1, Version 2 1. 2. 3. 4. B A C A 5. 6. 7. 8. A B D A Sample Exam 3, Version 1 Part 1. 1. C 2. B 3. D 4. A 5. C 6. D 13. C Part 2. 1. To maximize profit, Barbara’s should produce 25 Colonial baskets and 75 Southwest baskets each week.    2 −5 0  0 2. First, write this system as an augmented matrix  1 −3 −1  −1 . Then perform −1 2 −1  −1   1 0 5 5   0 1 2 2 . So x + 5z = 5 and y + 2z = 2, and the answer is Gaussian elimination to get 0 0 0 0 x = 5 − 5z, y = 2 − 2z, and z is arbitrary.  1 To find the inverse, reduce the augmented matrix  0 2   3 1 −1 A−1 is the matrix  2 2 −1 . −2 −1 1 3. 0 1 1 1 1 4   1   0   0 0 1 0  0 0 . The inverse 1 Sample Exam 3, Version 2 1. 2. 3. 4. B D E B 5. 6. 7. 8. A E A D 9. B 10. A 11. C 12. D 13. 14. 15. 16. B D E B 17. 18. 19. 20. E E C A