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Question #41787

Difficulty: 8/10 | 🟢 Free

The Question

Consider the linear programming problem: Maximize $Z = 5x + 7y$ subject to the constraints that $z = x + iy$ ( $x, y \in \mathbb{R}$ ) satisfies the following conditions: (i) $|z - 2| \le |z - 6|$ (ii) $|z - 2i| \le |z - 6i|$ (iii) $|z| \ge |z - 6 - 6i|$ and $(x,y)$ lies in the domain of the real-valued function $f(x,y) = \sin^{-1}\left(\frac{x^2+y^2}{18}\right)$ .

Find the maximum value of $Z$ .

Options

A.
48
B.
38
C.
36
D.
34

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

Inconsistency Detection|Geometry of Complex Inequalities|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize complex number loci with linear programming and domain analysis.

The primary trap lies in the separation of the constraints: the complex inequalities define a standard triangular feasible region with vertices $(2,4)$ , $(4,2)$ , and $(4,4)$ .

A student rushing to apply the corner-point method will evaluate the objective function at these vertices, yielding a maximum of 48 at $(4,4)$ .

However, the subtle domain restriction of the inverse sine function imposes a circular constraint $x^2+y^2 \le 18$ .

The entire triangular region lies strictly outside this circle, except for a single degenerate point $(3,3)$ on the boundary.

This forces the feasible region to collapse to a singleton, making the true maximum 36.

This penalizes algorithmic application of LPP without verifying the global intersection of all constraints.

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