MHT CET · Maths · Linear Programming

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by

A \(2 x+3 y \geq 6,-x+2 y \geq 2,3 x+6 y \leq 18, \ x-3 y\) \(\geq 3, x \geq 0, y \geq 0\)
B \(2 x+3 y \geq 6,-x+2 y \leq 2, x-3 y \leq 3 \ x+2 y\) \(\geq 18, x \geq 0, y \geq 0\)
C \(2 x+3 y \leq 6,-x+2 y \geq 2,3 x+6 y \leq 18 \ x-3 y\) \(\leq 3, x \geq 0, y \geq 0\)
D \(2 x+3 y \geq 6,3 x+6 y \leq 18, x-3 y \leq 3 \ -x+2 y\) \(\leq 2, x \geq 0, y \geq 0\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2 x+3 y \geq 6,3 x+6 y \leq 18, x-3 y \leq 3 \ -x+2 y\) \(\leq 2, x \geq 0, y \geq 0\)

Step-by-step Solution

Detailed explanation

Shaded region lies on origin side of \(3 x+6 y=18, x-3 y=3,-x+2 y=2\) and on non-origin side of \(2 x+3 y=6\).
\(\begin{aligned}
\therefore \quad & 2 x+3 y \geq 6,3 x+6 y \leq 18, x-3 y \leq 3, \\
& -x+2 y \leq 2, x \geq 0, y \geq 0
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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