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Question #40858

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(4, -2, -2)$ , $B(-2, 4, -2)$ , and $C(-2, -2, 4)$ be three vertices of a tetrahedron $ABCD$ . The centroid $G$ of the tetrahedron lies on the plane $x + y + z = 3$ . The distance of $G$ from the point $(1, 1, 1)$ is strictly greater than $\frac{1}{\sqrt{2}}$ and strictly less than $\sqrt{\frac{3}{2}}$ . If the fourth vertex $D(x, y, z)$ has integer coordinates, find the number of possible positions for $D$ .

Options

A.
12
B.
18
C.
24
D.
30

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Centroid of a Tetrahedron|Locus via Distance Constraints|Distance Formula in 3D|Graphical Feasible Region

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map 3D distance constraints onto a discrete locus of points, specifically targeting the 'Graphical Feasible Region' boundary trap.

By forcing the centroid into a restricted space, the problem reduces to a Diophantine inequality $4 < u^2+uv+v^2 < 12$ .

The primary trap lies in the strict inequalities: students frequently ignore the strict bounds and include the boundary values $K=4$ and $K=12$ , which incorrectly adds 12 extra solutions (leading to the trap answer 30).

Furthermore, the coordinate shift $u=x-4$ elegantly tests translation of axes in 3D coordinate geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Centroid of a Tetrahedron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free