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Question #40531

Difficulty: 8/10 | 🟢 Free

The Question

Consider the linear programming problem: Minimize $Z = 353x + 57y$ subject to the constraints: $\frac{3x + y - 8}{y - x} \ge 2$ $\frac{x + 3y - 8}{x - y} \ge 2$ $x \ge 0, y \ge 0$ The minimum value of $Z$ is:

Options

A.
820
B.
410
C.
0
D.
\text{Does not exist}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Denominator Restriction|Objective Function Gradient|Non-Negativity Restriction|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the 'Sign Flipping Rule' and 'Denominator Restriction' in rational inequalities within an LPP context.

Students commonly cross-multiply without considering the sign of the variable denominator, incorrectly yielding the constraints $5x - y \ge 8$ and $5y - x \ge 8$ .

This flawed logic produces a false feasible region with a minimum at $(2,2)$ , leading to the trap answer of 820.

However, properly splitting the inequalities into cases based on the sign of $(x-y)$ reveals contradictory conditions (e.g., $x > 2$ and $y < 2$ implying $x > y$ , which contradicts the case assumption $y > x$ ).

Consequently, the actual feasible region is an empty set, meaning no minimum exists.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Denominator Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free