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Question #41738

Difficulty: 8/10 | 🟢 Free

The Question

Consider the system of linear equations $AX = B$ , where $A = \begin{pmatrix} 2 & 1 & -1 \\ 1 & -1 & 2 \\ \alpha & 1 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 3 \\ 4 \\ \beta \end{pmatrix}$ The equations represent three planes in $\mathbb{R}^3$ . It is given that the line of intersection of the first two planes is strictly parallel to the third plane (i.e., it does not lie in the third plane). Let $C = \text{adj}(A)$ . Which of the following is a possible value for the column vector $CB$ ?

Options

A.
$\begin{pmatrix} 2 \\ -10 \\ -6 \end{pmatrix}$
B.
$\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$
C.
$\begin{pmatrix} 2 \\ 10 \\ 6 \end{pmatrix}$
D.
$\begin{pmatrix} -2 \\ 10 \\ -6 \end{pmatrix}$

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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

Line lying in a plane|Collinearity and Coplanarity|Adjoint-Determinant Relation

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of systems of linear equations and the Adjoint-Determinant relation.

A common trap is assuming that $|A|=0$ is the only condition needed for parallel planes, or confusing the condition for a line lying in a plane ( $(\text{adj} A)B = 0$ , infinite solutions) with the line being strictly parallel ( $(\text{adj} A)B \neq 0$ , no solution).

Students falling for the trap will calculate $CB$ and incorrectly equate it to the zero vector, choosing the null vector option.

The problem forces them to recognize that strict parallelism requires inconsistency, hence a non-zero vector proportional to the adjoint's column space.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line lying in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free