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Question #74979

Difficulty: 8/10 | 🟢 Free

The Question

Let $a, b, c$ be distinct positive real numbers. Consider the matrices $A = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}$ , $B = \begin{pmatrix} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{pmatrix}$ , and $C = \begin{pmatrix} a^2 & b^2 & c^2 \\ b^2 & c^2 & a^2 \\ c^2 & a^2 & b^2 \end{pmatrix}$ . If $l, m, n$ are the direction cosines of a line such that $A B C \begin{pmatrix} l \\ m \\ n \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ , then which of the following is true?

Options

A.
a^3 + b^3 + c^3 = 3abc
B.
a+b+c = 0
C.
al + bm + cn = 0
D.
\text{No such line exists}

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

Direction Cosines constraint|Determinant Product Rule|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

The problem tests the intersection of matrix singularity and 3D geometry constraints.

Students often mechanically solve $M\vec{v}=\vec{0}$ by assuming the matrix must be singular to yield a relation between variables, or they blindly manipulate the algebraic equations to find a condition like $a^3+b^3+c^3=3abc$ .

The trap lies in failing to evaluate the non-singularity of the product matrix $ABC$ for distinct positive reals.

Because $\det(ABC) \neq 0$ , the only solution is the trivial zero vector.

However, direction cosines must satisfy $l^2+m^2+n^2=1$ , making the zero vector invalid and leading to the conclusion that no such line can exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Direction Cosines constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free