MHT CET · Maths · Linear Programming

The function to be maximized is given by \(\mathrm{Z}=3 x+2 y\). The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by

A \(3 x+8 y \leq 24,4 x+5 y \leq 20,5 x+3 y \geq 15\), \(x \geq 0, y \geq 0\)
B \(3 x+8 y \geq 24,4 x+5 y \geq 20,5 x+3 y \leq 15\), \(x \geq 0, y \geq 0\)
C \(3 x+8 y \leq 24,4 x+5 y \geq 20,5 x+3 y \geq 15\), \(x \geq 0, y \geq 0\)
D \(3 x+8 y \geq 24,4 x+5 y \leq 20,5 x+3 y \leq 15\), \(x \geq 0, y \geq 0\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(3 x+8 y \geq 24,4 x+5 y \leq 20,5 x+3 y \leq 15\), \(x \geq 0, y \geq 0\)

Step-by-step Solution

Detailed explanation

Shaded region lies origin side of \(4 x+5 y \leq 20\), \(5 x+3 y \leq 15\) and on non-origin side of \(3 x+8 y \geq 24\).
\(\begin{array}{ll}
\therefore \quad 3 x+8 y & \geq 24,4 x+5 y \leq 20,5 x+3 y \leq 15 \\
& x \geq 0, y \geq 0
\end{array}\)

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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