Midterm 1  Feb 21, 2014 Math 484  25 questions, 2 pts each.   Section 001 no version. Section 002 Version A

Name  Dr.V

No electronics is allowed.   Write both answers and details.

1. 2x -  1 = -1   is

(A) a linear equation for x, y, (B) an affine function of x,y,   (C) a linear form in x, (D) none of the above.

2.   x2 + y2= 0  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+2y -> max subject to x+y=1

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

(D) has optimal value 3,   (E) none of the above.

4.  The optimal value for the linear program   a2 x + b3 -> min, 0 ≤ x ≤ 1  with an unknown  x and given numbers  a, b is

(A)  it is not a linear program , (B)   the program is unbounded, (C)  b3   (D) none of the above.

5. The mathematical program x/y ->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) is not a mathematical program.

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, (E) none of the above.

7.The answer to the mathematical program 3x+4y -> min , x2+ y2    25 is

(A) max= 25  at x = 3,  y = 4,  (B) max = 30  at x = 4, y, = 3 (C) max = 50 at x = 5, y =5,

(D) min =-25 at x = -3, y =-4, ( E) none of the above.

8. The minimal total time for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

3 1 0 1

is    (A) 4,  (B) 6,  (C) 7, (D) 8, (E) none of the above.

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

(A) is not linear, (B) has a solution, (C) is a linear equation, (D) has many solutions,  (E) none of the above.

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

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

12. x > 0 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, (E)  none of the above.

14. 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) 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 -3 as the optimal value, (E) has only one feasible solution.

16. The standard tableau

x1 x2 x3 1

0   0   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

0 = 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) z    y                    B)  x         y

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

C) x  y             D) z    y             E) x    y

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

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

A) is infeasible. B) is unbounded. C) has no optimal solutions.

D) has infinitely many optimal solutions. E) non of above [added after test]

20.The equation  x4+ y3  = 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 + 8y = 12  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) none of the above.

23. For any number  x,    0 = 0 implies that  x ≥ 0 or x ≤ 0.   A) True B) False

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

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