Midterm 1 Oct 3, 2013 Math 484.002 25 questions, 3 pts each.

Name Vaserstein

No electronics is allowed. Return this page and the scantron.

Answers are in **boldface**

1. 2x - 1 is

(A) a linear equation for x, y, **(B) **an affine function of x,y,z, (C) a linear form in x,

(D) a linear constraint for x,y, (E) none of the above.

2. xy 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+3y -> 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 function (x+z)/y of x,y,z is

(A) an affine function of x,y,z, (B) a linear constraint for x,y,z,

(C) an equation, ** (D) **not defined for some x,y,z, (E) none of the above.

5. The mathematical program x/y ->min 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 a linear program, (B) is infeasible, (C) has an optimal solution,

(D) is unbounded,** (E) n**one of the above.

7.The answer to the mathematical program 3x+4y -> max , 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 cost for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

3 0 1 1

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

9. For each number t, the equation tx= 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.If x > 0 and y > 0 then xy > 0. **A) **True B) False

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

13. The linear program given by the standard tableau

x1 x2 x3 x4 1

1 -2 0 3 0 =x5

-4 0 2 0 1 =x6

0 1 0 2 -3 -> min

(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 the standard tableau

x1 x2 x3 x4 1

1 2 0 3 0 =x5

4 0 -2 0 1 =x6

0 -1 0 2 -3 -> min

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

(D) requires at least one pivot step to solve by simplex method.

15. The linear program given by the standard tableau

x1 x2 x3 x4 1

1 2 0 3 0 =x5

0 0 0 0 -1 =x6

0 -1 0 2 -3 -> min

**(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

-1 = x1

0 =x2

-1 -> min

(A) is optimal. (B) has a bad column, **(C) **has 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 an optimal solution.

D) has infinitely many optimal solutions. E) has -3 as the optimal solution.

20.The equation x^{4}+ y^{4} ^{ }= 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 + 7y = 0 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 ≥ 10, B) x ≥ 0, C) y ≥ 1, D) x ≤ 0, **E) **none of the above.

23. Unless said otherwise, a number in this course means a real number. ** A) **True B) False

24. A standard tableau with a bad row cannot be feasible **A) **True B)False

25. Any linear program given by a standard tableau with a bad column is unbounded.

A) True **B)** False