Midterm 1 Feb 15, 2013  Math 484.2  25 questions, 2 pts each.

Name________Vaserstein__________________

1. 2x-3y ≥  5  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.

2.   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 -> max subject to x,y ≥ 0

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

(D) has optimal value 2,   (E) is not linear.

4. The function  x/y  of x,y,z is

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

(C) an  equation,  (D) not defined for some  x,y,z, (E) not a function.

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) is 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 bounded, (E) is not a mathematical program.

7.The answer to the mathematical program 2x+ y -> max, |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 =1, y =0.

8. The minimal total cost for the job assignment problem

3 2 1 4

3 3 2 2

4 5 3 6

3 0 0 1

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

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

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

10. x ≤ 0 provided that x > 1.   A) True B) False

11.If x > 0 or y > 0 then xy > 0.   A) True B) False

12. x > 0 only if x  ≥ 0.   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 0 0 0 -> max

(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 -> max, 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 x2 + y4= 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 + y = 9 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) y ≤ 1

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

24. A standard tableau without zero entries cannot have both a bad row and a bad column.   A) True   B)False

25. Any linear program given by a feasible tableau with a bad column is unbounded.  A) True  B) False