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Question #40885

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{v}$ be a vector in $3$ -dimensional space with magnitude $2$ . The lengths of the projections of $\vec{v}$ on the $x, y,$ and $z$ axes are $a, b,$ and $c$ respectively, where $a, b,$ and $c$ are distinct positive real numbers. Let $M$ be a $3 \times 3$ matrix defined as $M = \begin{pmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ bc & ca & ab \end{pmatrix}$ . If $| \text{adj}(\text{adj}(M)) | = (a-b)^4(b-c)^4(c-a)^4$ , find the volume of the tetrahedron whose vertices are the origin $O(0,0,0)$ and the points $P(a,b,c)$ , $Q(b,c,a)$ , and $R(c,a,b)$ .

Options

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

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

Projection vs. Coordinate|Adjoint-Determinant Relation|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D coordinate geometry and linear algebra, specifically targeting the 'Factor Theorem' trap in determinants.

Students often attempt to brute-force the expansion of the $3 \times 3$ matrix $M$ instead of using column operations to reveal the standard cyclic determinant $(a-b)(b-c)(c-a)(ab+bc+ca)$ .

Furthermore, the problem forces students to link the abstract algebraic result to the geometric constraint of projection lengths ( $a^2+b^2+c^2=4$ ).

Failing to recognize that the adjoint-determinant relation $|adj(adj(M))| = |M|^4$ isolates the symmetric sum $ab+bc+ca=1$ prevents the evaluation of the tetrahedron's volume, effectively punishing students who treat the matrix as a purely abstract entity without geometric context.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free