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Question #41651

Difficulty: 8/10 | 🟢 Free

The Question

Let $ABCD$ be a tetrahedron such that its two opposite edges $AB$ and $CD$ lie on the lines $L_1$ and $L_2$ respectively, where: $L_1: x+y+z-1 = 0 = x-y+z-1$ $L_2: x+y-z+2 = 0 = -x+y+z+2$ The vertices $A$ and $B$ are at a distance of $1$ unit from the origin, while the vertices $C$ and $D$ are at a distance of $\sqrt{6}$ units from the origin. Let $G(\alpha, \beta, \gamma)$ be the centroid of the tetrahedron $ABCD$ . If $G$ lies in the $k^{\text{th}}$ octant (using the standard I-VIII convention), then the value of $k^2 + 8(\alpha + \beta + \gamma)$ is:

Options

A.
12
B.
13
C.
0
D.
60

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

Symmetric vs Non-symmetric form|Centroid of a Tetrahedron|Octant Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to extract parametric coordinates from lines given in non-symmetric (intersection of planes) form and compute the centroid of a 3D figure.

The primary trap lies in the 'Octant Sign Convention' and algebraic sign retention.

Students frequently misidentify the octant of $(+,-,+)$ as II or VIII due to memorization errors, leading to incorrect $k$ values (yielding 0 or 60).

Additionally, a common calculation trap occurs when solving for $L_2$ : a sign error yields $y=2$ instead of $y=-2$ , placing the centroid in Octant I and leading to the trap answer 13.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Symmetric vs Non-symmetric form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free