Midterm 1 Feb 20, 2015 Math 484.002  25 questions, 2 pts each.

Name  Dr.V

No electronics is allowed.   Write both answers and details.

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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.

2.   x + y     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,

3. Every linear program  without constraints

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

4. The optimal value for the linear program   a2x + b  -> min, 0 ≤  x  ≤ 1  with an unknown  x and given numbers  a, b is

(A)  it is not a linear program , (B)   is unbounded, (C) a2 + b,   (D) b.

5. The mathematical program x/y -> min subject to |x| ≤ 1, y > 1 for two unknowns x, y

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

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

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

(D) is unbounded.

7. The answer to the mathematical program  x2+ y2  -> min , x+ y     - 8 is

(A) max= 25  at x = 3,  y = 4,  (B) min = 0  at x = 0, y = 0 (C) max = 50 at x =- 5, y =-5,

(D) min =32 at x = -4, y = -4.

8. The minimal total time for the job assignment problem

6  2  1  4

3  0  2  1

2  5  3  6

0  1  2  1

is    (A) 4,  (B) 5,  (C) 6, (D) 7.

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

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

Solution. When sin(t) ≠ -1, we have one solution. Otherwise, cos(t) = 0 and every x is a solution.

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

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

12. x > 2 only if x  ≥ 2.   A) True , B) False.

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

 -1 -2 0 -3 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.

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

 -1 -2 0 -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.

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

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

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

(D) has -4 as the optimal value.

16. The standard tableau

x1 x2 x3 1

0   2   3   4 -> min

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

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.

18. Pivoting the tableau

x     y

2*  3 = z

4   5 = u

produces the tableau

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

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

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

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

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

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)  2, (D) 3.

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.

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

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

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.

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

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

A) 4x + 5y ≥  9,  B) x ≥ 0,   C) y ≥ 1,   D) x ≤ 0.