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

Name  Vaserstein

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, x2+ y2  ≤ 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  x2 + y2 = 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