Midterm 1 Feb 15, 2013 Math 484.1 25 questions, 2 pts each.

Name_________**Vaserstein**___________________

**Correct 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) a linear program.

**The students with (A) or (D) answer will get 3 bonus points; students who wrote that both are correct will get **

**up to 10 bonus points (in e!). The total bonus points will be entered at Angel as e! on Sunday.**

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) nonsense.

3. The linear program 2x+3y -> min 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) is not linear.

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

(A) not 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) is not a function.

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) is not a mathematical program.

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

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

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

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

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

**(D) **min = -2 at x = 0, y = -1, ( E) max = 2 at x = 0, y = 1.

8. The minimal total cost for the job assignment problem

1 2 3 4

3 3 2 2

4 5 3 6

0 1 1 1

is (A) 5, (B) 6, **(C)** 7, (D) 8, (E) 6.5.

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

(A) is not linear, (**B) **has a solution, (C) is not a constraint.

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

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

12. x > 2 only if x > 1. ** 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) has only 4 unknowns.

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., (E) cannot be solved 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 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 -1 0 0 -> min

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

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) x y

-1/2 -3/2 = z

B) x y

2 -3/2 = z

C) x y

1/2 3 = z

D) z y

1/2 3/2 =x

**E)** z y

-1/2 3/2 = x

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

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

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

20.The equation x^{2} + 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 > 7 B) x ≥ 0 C) y ≥ 1 D) x ≤ 0 E) y ≤ 1

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

24. Any linear program given by a standard tableau with a bad row is infeasible. **A) **True B) False

25. A standard tableau with nonzero entries (all of them) cannot have both a bad row and a bad column. ** A) **True B)False