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Question #41697

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real values of $\lambda$ for which the linear programming problem:

Maximize $Z = 5x + 2y + 3z$

subject to the equality constraints: $4y - 2z = \lambda^2 - 2\lambda$ $-2x + 3z = 1 - 3\lambda$ $3x - 9y = 2$ $x, y, z \ge 0$

has a non-empty feasible region. The sum of all elements in $S$ is:

Options

A.
2
B.
4
C.
-4
D.
0

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

Inconsistency Detection|Singular vs Non-Singular Matrices|Skew-Symmetric Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to detect inconsistency in a system of linear constraints within an optimization context.

The coefficient matrix is a diagonally scaled skew-symmetric matrix of odd order, meaning its determinant is inherently zero.

Students who mechanically attempt to find corner points or apply Cramer's rule will be trapped in tedious algebra.

The elegant 'Advanced' approach requires recognizing the singularity, computing the left null space, and enforcing the consistency condition $\mathbf{v}^T \mathbf{b} = 0$ .

This directly yields a quadratic in $\lambda$ , bypassing the need to explicitly solve the inequalities.

Quick Facts About This Question

Question Information

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