Midterm 1 Feb 21, 2014 Math 484 25 questions, 2 pts each. Section 001 no version. Section 002 Version A

Name Dr.V

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Answers are in **boldface**

1. 2x - 1 = -1 is

**(A) **a linear equation for x, y, (B) an affine function of x,y, (C) a linear form in x, (D) none of the above.

2. x^{2} + y^{2}= 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 cannot be solved, **(E) **none of the above.

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

(A) is unbounded, (B) is infeasible, ** (C)** has many optimal solutions,

(D) has optimal value 3, (E) none of the above.

4. The optimal value for the linear program a^{2} x + b^{3} -> min, 0 ≤ x ≤ 1 with an unknown x and given numbers a, b is

(A) it is not a linear program , (B) the program is unbounded, **(C) ** b^{3}** ** (D) none of the above.

5. The mathematical program x/y ->max subject to |x| ≤ 1, y > 0 for two unknowns x, y

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

**(D) i**s unbounded, (E) is not a mathematical program.

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,** **(E) none of the above.

7.The answer to the mathematical program 3x+4y -> min , x^{2}+ y^{2} ≤ 25 is

**(**A**)** max= 25 at x = 3, y = 4, (B) max = 30 at x = 4, y, = 3 (C) max = 50 at x = 5, y =5,

**(D)** min =-25 at x = -3, y =-4, ( E) none of the above.

8. The minimal total time for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

3 1 0 1

is ** (A)** 4, (B) 6, (C) 7, (D) 8, (E) none of the above.

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

(A) is not linear, (B) has a solution, **(C) **is a linear equation, (D) has many solutions, (E) none of the above.

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

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

12. x > 0 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, (E) none of the above.

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 |
2 |
-3 |

(A) is infeasible, (B) is 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 -3 as the optimal value, (E) has only one feasible solution.

16. The standard tableau

x1 x2 x3 1

0 0 0 0 -> min

**(A) **is optimal, (B) has a bad column, (C) has a bad row, (D) is not standard, (E) none of the above.

17. The standard tableau

1

0 = x1

0 = x2

-1 -> min

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

18. Pivoting the tableau

x y

2* 3 = z

produces the tableau

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

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

C) x y D) z y E) x y

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

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

A) is infeasible. B) is unbounded. C) **h**as no optimal solutions.

D) has infinitely many optimal solutions.** E) **non of above** [added after test]**

20.The equation x^{4}+ y^{3} ^{ }= 0 for x, y

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

21.The system x+ 2y = 3, 4x + 8y = 12 for x,y

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

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

**A)** 4x + 5y ≥ 6, B) x ≥ 0, C) y ≥ 1, D) x ≤ 0, ** **E) **no**ne of the above.

23. For any number x, 0 = 0 implies that x ≥ 0 or x ≤ 0. ** A) **True B) False

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

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