The optimal value of the objective function of the LPP, Minimize \(Z = 3x-2y\) subject to constraints \(x+y \le 10\), \(3x+5y \ge 15\), \(x \ge 0, y \ge 0\), is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −20
On a bounded feasible region, the minimum of Z occurs at a corner point.
Corner points from the boundary lines:
\(x + y = 10\) meets the axes at (10, 0) and (0, 10).
\(3x + 5y = 15\) meets the axes at (5, 0) and (0, 3).
The feasible region lies between these two lines in the first quadrant. Its corners are (5, 0), (10, 0), (0, 10) and (0, 3).
\(Z(5, 0) = 15\)
\(Z(10, 0) = 30\)
\(Z(0, 10) = -20\)
\(Z(0, 3) = -6\)
The smallest corner value is \(-20\), at (0, 10).
Why the other options are incorrect
Option (1) 30 — Incorrect. This is the maximum of Z, at (10, 0).
Option (3) −6 — Incorrect. This is Z at (0, 3), which is not the smallest corner value.
Option (4) 15 — Incorrect. This is Z at (5, 0).
Note: NTA dropped this question. For the benefit of students and to give them additional practice, the question has been suitably modified so that it has one clear answer.