Midterm 1 Oct 3, 2014 Math 484.003 25 questions, 2 pts each.

Name Dr.V

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1. 2x - 1 ≠ -1 is

(A) a linear constraint for x, (B) an affine function of x (C) a linear form in x, **(D)** a constraint on x.

2. x + y ≤ 0 is

**(A)** a linear constraint for x, y, (B) a linear equation for x in standard form,

(C) an affine function of x, y, (D) an equation which can be solved,

3. The linear program x+2y -> max subject to x+y = 1

**(A) **is unbounded, (B) is infeasible, (C) has many optimal solutions, (D) has optimal value 2.

4. The optimal value for the linear program a^{2}x + b -> max, 0 ≤ x ≤ 1 with an unknown x and given numbers a, b is

(A) it is not a linear program , (B) is unbounded, **(C)** a^{2} + b,** **(D) b.

5. The mathematical program x/y -> max subject to |x| ≤ 1, y > 1 for two unknowns x, y

(A) is a linear program, (B) is infeasible, (C) has an optimal solution,(D) is unbounded.

6. The mathematical program x -> max, x < 1

**(A) **is not a linear program, (B) is infeasible, (**C) h**as an optimal solution,

(D) is unbounded.

7. The answer to the mathematical program x^{2}+ y^{2} -> min , x+ y^{ } ≥ 8 is

**(**A**)** max= 25 at x = 3, y = 4, (B) min = 0 at x = 0, y = 0 (C) max = 50 at x = 5, y =5,

**(D)** min =32 at x = 4, y = 4.

8. The minimal total time for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

0 1 2 1

is ** (A)** 4, (B) 5, (C) 6, (D) 7.

9. For each number t, the equation (sin(t) +2)x= cos(t) for unknown x

(A) is not linear,** (B)** has a solution, (C)** **is unbounded, (D) has many solutions.

10. 1 ≤ 0 provided that 0 = 1. **A) **True, B) False.

11. x > 0 and y > 0 if xy > 0. A)** **True, **B) **False

12. x > 0 only if x ≥ 2. ** **A) True , **B)** False.

13. The linear program given by a standard tableau with the matrix

-1 |
-2 |
0 |
-3 |
0 |

-4 |
0 |
-2 |
0 |
1 |

0 |
1 |
0 |
2 |
-3 |

(A) is infeasible, (B) is unbounded, **(C) **has an optimal solution,

(D) has only 2 linear constraints.

14. The linear program given by a standard tableau with the matrix

-1 |
-2 |
0 |
3 |
0 |

-4 |
0 |
2 |
0 |
1 |

0 |
-1 |
0 |
-1 |
-3 |

(A) is infeasible, **(B) i**s unbounded, (C) has an optimal solution,

(D) requires at least two pivot steps to solve by simplex method.

15. The linear program given by a standard tableau with the matrix

-1 |
-2 |
0 |
-3 |
0 |

-4 |
0 |
2 |
0 |
1 |

0 |
-1 |
0 |
2 |
-3 |

(A**)** is infeasible, (B) is unbounded, **(C) **has an optimal solution,

(D) has -4 as the optimal value.

16. The standard tableau

x1 x2 x3 1

0 0 0 -1 -> min

**(A) **is optimal, (B) has a bad column, (C) has a bad row, (D) is not standard.

17. The standard tableau

1

-1 = x1

0 = x2

-1 -> min

(A) is optimal. (B) has a bad column, **(C) h**as a bad row, (D) is not standard.

18. Pivoting the tableau

x y

2* 3 = z

produces the tableau

**A) **z y B) x y C) x y

1/2 -3/2 = x 2 -3/2 = z 1/2 3 = z

19. The linear program x+y -> max, x≤ 2, y ≤ - 1 with 2 unknowns x, y

A) is infeasible. B) is unbounded. **C**) **h**as an optimal solution.

D) has infinitely many optimal solutions.** **

20. The number of choices for the pivot entry in the standard tableau

1 2 3 -1 0

0 -1 1 -3 3

1 -1 2 -3 4

which are consistent with the simplex method is (A) 0, (B) 1, (**C**) 2, (D) 3.

21.The system x+ 2y = 3, 4x + 8y = 10 for x,y

**A)** has no solutions. B) has exactly one solution. C**) **has infinitely many solutions.

22. For any column x of numbers, 0 = 0 implies that x ≥ 0 or -x ≥ 0. A) True **B)** False

23. The number of choices for the pivot entry in the standard tableau

1 - 2 3 -1 0

0 -1 1 -3 0

1 -1 2 -3 4

which are consistent with the simplex method is (A) 0, (B) 1, (C ) 2, (D) 3.

24. For any number x, if x^{2}^{ }= 0 then x ≥ 0 and x ≤ 0. ** ** **A) **True B) False

25. The set of constraints x + y > 2, 2x + 3y > 4 implies that

A) 4x + 5y ≥ 9, B) x ≥ 0, C) y ≥ 1, D) x ≤ 0.