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Question #41792

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all points $(x,y)$ in the Cartesian plane for which the function $f(x,y) = \sec^{-1}\left( \frac{x^2+y^2}{2} \right) + \cos^{-1}\left( \frac{|x|+|y|}{2} \right)$ is defined. Let $A$ be the maximum possible area of a rectangle, with sides parallel to the coordinate axes, that is completely contained within the region $S$ . Find the value of $A$ .

Options

A.
$3 - 2\sqrt{2}$
B.
$\frac{3}{2} - \sqrt{2}$
C.
$6 - 4\sqrt{2}$
D.
$2 - \sqrt{2}$

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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

Word Problem Modeling|Geometry Constraint Modeling|Modulus Function Symmetry|Domain Constraint Intersections

Why This Question Is Tricky

Common Mistakes

This question employs the 'Lethal Collision' architecture by intersecting a modulus domain constraint (a square) with an inverse trigonometric domain constraint (a circle).

The primary trap lies in the visual and geometric interpretation of the feasible region $S$ .

Students will correctly identify $S$ as the space between the circle and the square, but will intuitively assume that any inscribed rectangle must be confined to a single quadrant (yielding the trap answer $\frac{3}{2} - \sqrt{2}$ ).

The rigorous AOD approach requires testing boundary conditions across the axes, revealing that a rectangle crossing the y-axis (or x-axis) can exploit the 'concavity' of the circular exclusion zone, exactly doubling the maximum possible area.

Quick Facts About This Question

Question Information

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