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Question #40846

Difficulty: 8/10 | 🟢 Free

The Question

A point $P(x,y,z)$ moves in 3D space such that its $L_\infty$ distance from the origin, defined as $\max(|x|, |y|, |z|)$ , is at most 3. If the coordinates of $P$ also satisfy the inequality $\frac{x y z}{(x-1)(y-2)(z-1)} \ge 0$ what is the volume of the region representing the locus of $P$ ?

Options

A.
124
B.
104
C.
100
D.
92

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

Octant Sign Convention|Locus Problems via Transformation|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate 3D geometric constraints with algebraic inequalities, specifically targeting the 'Octant Sign Convention' and the 'Wavy Curve' method.

The lethal trap lies in assuming spatial symmetry or ignoring the roots at the origin ( $x=0, y=0, z=0$ ) which represent the octant boundaries.

Students who calculate the volume in the first octant and multiply by 8 (assuming symmetry) will incorrectly obtain 104.

Students who ignore the numerator variables (failing to map the wavy curve critical points to the octant boundaries) will incorrectly calculate the lengths and obtain 100.

The correct approach requires applying the wavy curve independently to each variable across its full domain, effectively summing the valid sub-volumes across all eight octants.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Octant Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free