For the linear programming problem (LPP),
Maximize \(Z = 4x + y\)
\(x + y \le 5\)
\(3x + y \le 9\)
\(x, y \ge 0\).
Which of the following are NOT true?
(A) The given LPP has unbounded feasible region.
(B) The corner points of the feasible region are (0, 0), (0, 5), (3, 2) and (3, 0).
(C) The optimal value of the objective function is 12.
(D) The given LPP has a unique optimal solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
The feasible region lies in the first quadrant below both lines. The lines \(x + y = 5\) and \(3x + y = 9\) meet where
\( \displaystyle (3x + y) - (x + y) = 9 - 5 \)
\( \displaystyle 2x = 4 \)
\( \displaystyle x = 2,\quad y = 3 \)
The line \(3x + y = 9\) cuts the x-axis at (3, 0), and \(x + y = 5\) cuts the y-axis at (0, 5). Values of Z = 4x + y at the corner points:
\( \displaystyle Z(0, 0) = 0 \)
\( \displaystyle Z(3, 0) = 12 \)
\( \displaystyle Z(2, 3) = 11 \)
\( \displaystyle Z(0, 5) = 5 \)
So the feasible region is the bounded quadrilateral with corners (0, 0), (3, 0), (2, 3) and (0, 5), and the maximum value of Z is 12, at (3, 0) only.
The question asks for the statements that are NOT true.
Checking the statements
(A) The given LPP has unbounded feasible region. — Incorrect. The region is a closed quadrilateral, so it is bounded.
(B) The corner points of the feasible region are (0, 0), (0, 5), (3, 2) and (3, 0). — Incorrect. The corner is (2, 3), not (3, 2); (3, 2) is not even feasible, since \(3(3) + 2 = 11 \gt 9\).
(C) The optimal value of the objective function is 12. — Correct. The maximum of Z is 12, at (3, 0).
(D) The given LPP has a unique optimal solution. — Correct. Only the corner (3, 0) gives 12; the next value is 11, so the optimum is unique.
The statements that are not true are (A) and (B).