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Question #41683

Difficulty: 8/10 | 🟢 Free

The Question

Consider the linear programming problem: Minimize $Z = 3x + y$ subject to the constraints: $y - 3|x - 4| \ge 3$ $3x + y \ge 12$ $x + y \le 13$ $2x - y \le 10$ $x \ge 0, y \ge 0$ Let $S$ be the set of all points $(x,y)$ in the feasible region at which $Z$ attains its minimum value. If $S$ is a line segment of length $L$ , find the value of $L^2$ .

Options

A.
$90$
B.
$160$
C.
$\frac{45}{2}$
D.
$100$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Multiple Optimal Solutions|Redundant Constraints|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle modulus sign dependencies in linear programming and identify redundant constraints.

The trap occurs when students misidentify $3x + y \ge 12$ as the active constraint for the minimum, or when they ignore the piecewise domain $x \le 4$ of the modulus branch $3x + y \ge 15$ .

In both trap scenarios, they calculate the length of a 'phantom' segment extending to the x-axis (e.g., from $(1,12)$ to $(5,0)$ ), incorrectly yielding $L^2 = 160$ .

The redundant constraint $2x - y \le 10$ is specifically designed to intersect the phantom branch exactly at $(5,0)$ , reinforcing the false path.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Multiple Optimal Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free