Midterm 1  Sep 30, 2016.  Math 484.004  25 questions, 2 pts each.  

Name  Dr. V

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Answers are in boldface Details are not given.


1. 0 ≠   1   is

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

E.


2.   x  =  y   - 1  is

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

(C) a  linear form in  x, y,  (D) an equation which cannot be solved,

B.


3. Every linear program  with  10  constraints  for 2 unknowns  

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

E.


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

(A)    is   a linear program , (B)   is unbounded, (C) has no optimal solutions   (D )has no feasible solutions.

A.


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.

D.


6. 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.


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

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

E.


8, If x> 2 and y > 3 then x+y≥ 6. A) True,  B) False.

B.


9. The bound  x < 9 is weaker than  x < 9.5.  A) True,  B) False.

B.


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

A.


11. x > 0 and  y > 0   if  xy ≥ 0.   A) True,  B) False

B.



12. x≥ 2 given that x  =  2.   A) True , B) False.

A.


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

-1

-2

-3

-4

0

-4

0

-2

0

1

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

0

1

-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

1

-4

0

2

0

1

-1

1

1

2

3

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

(D) has 2  as the optimal value.

C.


16. The standard tableau

x1   x2   x3   1  

-1    1   -1   -1   = x4

-1    -1  -1   -1   -> min

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

B.


17. For any numbers  a, b  and c,  the standard tableau

x1  x2   1

-1     b   c -> min

(A) is optimal. (B) has a bad column, (C) has a bad row, (D) is terminal.

B. The tableau is independent of  a.


18. Pivoting the standard tableau with matrix

2*   3 

4     3  

produces a tableau which

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

D.


19. Pivoting the standard tableau with matrix

  -2*  -3 

  -4    3  

produces a tableau which

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

B.


20. 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.


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

-1   2   -3    -1      0  1

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)  4, (D) 5.

E.


22. 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   0

1   1    -2    -3     4   3

which are consistent with the simplex method is  (A) 0, (B) 2, (C)  4, (D) 6.

C.


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

0  -1   -2    -3    4

which are consistent with the simplex method.is  (A) 0, (B) 2, (C )  3, (D) 5.

A.


24. The linear program given by  the row tableau   with  x1,  x3, x4, x5  ≥ 0

x1            2 x1       - x3         1         

-1

2

0

-3

= -x4

2

-1

-2

0

= x5

0

-1

0

2

-> min


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

(D) has only 2 linear constraints.

C.


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

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

A.