Midterm 1 Oct 3, 2013 Math 484.001 25 questions, 3 pts each.

Name Vaserstein

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

The answers are in **boldface**

1. 2x-3y > 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. 0 ≥ 3 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 ≥ 0, x+y=1

(A) is unbounded, (B) is infeasible, (C) has many optimal solutions,(D) has optimal value 2, **(E)** none of the above.

4. The function x - 2 of x,y,z is

**(A)** an affine function of x,y,z, (B) is a linear form in x,y,z,(C) is an equation, (D) is not defined for some x,y,z,(E) none of the above.

5. The mathematical program xy ->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)** is unbounded, (E) none of the above.

6. The mathematical program x -> min, x ≤ 1

**(A)** is a linear program, (B) is infeasible, (C) has an optimal solution,(D) is bounded, (E) none of the above.

7.The answer to the mathematical program x+y -> min, x^{2}+ y^{2} ≤ 1 is

(A) x= 2 at x = y = 1, (B) x =y, (C) max = 1 at x = 1, y = 0,

**(D)** min = -√2 at x = -1/√2, y =-1/√2 , ( E) none of the above.

8. The minimal total cost for the job assignment problem

1 2 3 4

3 3 2 2

4 5 3 6

0 2 2 2

is (A) 5, (B) 6, **(C)** 7, (D) 8, (E) none of the above.

9. For each number t, the equation tx = sin(t) +1 for unknown x

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

10. x> 1 provided that x ≥ 2. ** A) **True, B) False

11.If x > 0 and y > 0 then x + y ≥ 0. ** A)** True B) False

12. x >1 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

-4 0 -2 0 1 =x6

0 1 0 0 -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 1 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.

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

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

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

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

D) has infinitely many optimal solutions,

20.The equation x^{2} + y^{2} = 0 for x, y

A) has no solutions, **B)** has exactly one solution, C) has infinitely many solutions, D) none of the above.

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, D) none of the above.

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

**A)** 4x + 5y ≥ 0, 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 rational number. A) True** B) **False

24. Any linear program given by a standard tableau with a bad column is unbounded. A) True A construction of projective modules on the cuspidal cubic.False

25. No linear equation has exactly 2 solutions.** A) **True B)False