Midterm 1 Oct 2, 2015.  Math 484.003 25 questions, 2 pts each.

Name  Dr.V

No electronics is allowed.   Write both answers and details.

If you do not like given answers, write this here and choose E on scantron.

Answers are in boldface Details are not given.

1.x ≤  0   is

(A) a linear constraint for x, (B) an affine function of x  (C) a linear form in x, (D) an equation.

A.

2.   x + y   - 1  is

(A) a linear constraint for x, y,  (B) a linear equation for x, y  in standard form,

(C) an affine function of x, y,  (D) an equation which can be solved,

C.

3. Every linear program  with more constraints than the number of unknowns

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

E.

4. For any numbers a, b, the mathematical  program   a2x + b  -> min, 0 <   x  <  1  with an unknown  x

(A)  it is not a linear program , (B)   is unbounded, (C) has no optimal solutions   (D)has no feasible solutions.

A.

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.

D.

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

(A) is not a linear program, (B) is infeasible, (C) has an optimal solution,

(D) is unbounded.

D.

7. The answer to the mathematical program  x2 + y2   -> min , 2x+ y2   = 6  is

(A) max= 25  at x = 3,  y = 4,  (B) min = 5  at x = 1.  y = 2, (C) min = 4 at x =0, y =-2,  (D) min =3 at x = 3, y = 1.

B.

8. The minimal total time for the job assignment problem

6  2  1  4

3  0  2  1

2  5  3  6

1  1  2  1

is    (A) 4,  (B) 5,  (C) 6, (D) 7.

A.

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

(A) is not linear, (B) has a solution, (C) is unbounded, (D) has many solutions.

B.

10. 1 =  0 provided that 0 = 2.   A) True,  B) False.

A.

11. x > 0 or  y > 0 only if  xy > 0.   A) True,  B) False

B.

12. x > 2 provided that x  ≥ 2.   A) True , B) False.

B.

13. The linear program given by a  standard tableau with the matrix

 -1 -2 0 -3 0 -4 0 -2 0 0 0 1 0 2 -3

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

(D) has only 2 linear constraints.

C.

14. The linear program given by a  standard tableau with the matrix

 -1 -2 -1 3 0 -4 0 2 0 1 0 -1 -1 -1 -3

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

(D) requires at least two pivot steps to solve by simplex method.

B.

15. The linear program given by a  standard tableau with the matrix

 -1 -2 0 -3 0 -4 0 2 0 1 -1 1 0 2 -3

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

(D) has -4 as the optimal value.

C.

16. The standard tableau

x1 x2 x3 1

0    0  0  0  = x4

0   2   3   4 -> min

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

A.

17. The standard tableau

1

-1 = x1

-2 = x2

-3 -> min

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

C.

18. Pivoting the tableau

x     y

2   3* = z

4    3 = u

produces the tableau

A) z    y                    B)  x       z                      C) x   y                           D) x    z

-2/3  1/3 = x                 2/3     1/3 = y                  1/2  3 = z                    -2/3  1/3 = y

2     1   = u                   2        1   = u                  -2   -1  = u                      2     1  = u

D.

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

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

D) has infinitely many optimal solutions.

C.

20. The number of choices for the pivot entry in the standard tableau with the matrix

1   2   -3    -1       0   0

0  -1    -1    -3     -3  3

-1  -1   -2   -3     -4   3

which are consistent with the simplex method is  (A) 0, (B) 1, (C)  2, (D) 3.

A.

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.

C.

22.  For any  column   x  of numbers,    x ≥ 0 or -x ≥ 0.   A) True  B) False

B.

23. The number of choices for the pivot entry in the standard tableau with the matrix

1  - 2    3    -1    0

0  -1    1      3    0

-1  -1   -2    -3   4

which are consistent with the simplex method is  (A) 0, (B) 1, (C )  2, (D) 3.

A.

24.  For any number  x,  if    x2 = 0 then     x ≥ 0 or x ≤ 0.     A) True   B) False

A.

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

A) 4x + 5y ≥  8,  B) x > 0,   C) y > 1,   D) x < 0.

A.