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Question #40901

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(1, 0, 0)$ , $B(0, 1, 0)$ , and $C(0, 0, 1)$ be three points in $\mathbb{R}^3$ . A variable point $P(x, y, z)$ moves on the line segment joining the origin $O(0,0,0)$ and the point $M(2, 2, 2)$ . Let $V$ be the volume of the tetrahedron $PABC$ , and $d$ be the distance of $P$ from the origin. If the locus of $P$ consists of all points on the segment $OM$ satisfying the equation $[6V] = [d^2]$ where $[ \cdot ]$ denotes the greatest integer function, then the total length of the locus of $P$ is:

Options

A.
$1 + \sqrt{2} - \frac{2}{\sqrt{3}}$
B.
$1 + \sqrt{2} - \frac{1}{\sqrt{3}}$
C.
$\sqrt{2} + \frac{1}{\sqrt{3}}$
D.
$\sqrt{3} + \sqrt{2} - 1$

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

Locus via Distance Constraints|Determinant as Geometric Area/Volume|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle locus problems constrained by discrete step functions (Greatest Integer Function) rather than continuous algebraic equations.

The primary trap is twofold: first, students often fail to rigorously partition the domain into intervals where the floor functions evaluate to the same integer $k$ , instead attempting to solve $|3t-1| = 3t^2$ directly.

Second, even if the parameter intervals for $t$ are found correctly, students frequently forget that the point $P(t,t,t)$ moves in 3D space, meaning the actual geometric length of the locus is scaled by a factor of $\sqrt{3}$ relative to the parameter $t$ .

This effectively filters out candidates who lack geometric visualization of parameterized space curves.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Locus via Distance Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free