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Question #41759

Difficulty: 8/10 | 🟢 Free

The Question

Let the feasible region $S$ be defined by the inequality $|x| + |y| \le 4$ . A matrix $M$ is defined as $M = \begin{pmatrix} |x| - 2 & |y| - 2 \\ |y| - 2 & |x| - 2 \end{pmatrix}$ If $Z = \det(M)$ , find the maximum value of the objective function $Z$ for $(x,y) \in S$ .

Options

A.
0
B.
4
C.
8
D.
16

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Boundary Value Theorem|Determinant as Geometric Area/Volume|Modulus Inequality Logic|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

This question targets the fundamental flaw in blindly applying the Fundamental Theorem of Linear Programming (vertex testing) to non-linear objective functions.

The determinant creates a quadratic surface $Z = (|x|-2)^2 - (|y|-2)^2$ .

Students relying on standard LPP algorithms will test the vertices $(\pm 4, 0)$ and $(0, \pm 4)$ , or even the boundaries, all of which deceptively yield $Z=0$ .

The trap is bypassed only by recognizing that the objective function has non-differentiable critical points (ridges) along the axes due to the modulus functions.

The true maximum lies strictly inside the feasible region at $(0, \pm 2)$ , testing the student's ability to decouple the variables and optimize via critical point analysis rather than boundary value theorems.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Boundary Value Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free