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Question #41698

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_k = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 25, \; x^2 - y^2 \ge k, \; x \ge 0, \; y \ge 0 \}$ . Let $M(k)$ be the maximum value of the objective function $Z = 3x + 4y$ subject to $(x,y) \in S_k$ . Find the sum of all real values of $k$ for which $M(k) = 24$ .

Options

A.
$7$
B.
$-\frac{7378}{625}$
C.
$-\frac{11753}{625}$
D.
$14$

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

Inconsistency Detection|Intersection of Conics|Domain Constraint Intersections

Why This Question Is Tricky

Common Mistakes

This question tests the 'Domain Constraint Intersection' trap in constrained optimization.

Students typically find the boundary intersection points by solving the objective function and the binding constraint simultaneously, yielding two mathematical solutions for the parameter $k$ .

The critical trap is failing to verify if the unconstrained global maximum of the objective function falls inside the feasible region for those $k$ values.

For the extraneous $k$ , the feasible region expands enough to include the unconstrained maximum, making the boundary intersection merely a local maximum.

This forces students to rigorously analyze the domain consistency of the LPP.

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