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Question #41646

Difficulty: 8/10 | 🟢 Free

The Question

Let $M = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & -1 & 2 \end{pmatrix}$ and let $\lambda_1 < \lambda_2 < \lambda_3$ be the eigenvalues of $M$ . A plane $\Pi$ is defined by the equation $\lambda_1 x + \lambda_2 y + \lambda_3 z = 12$ . Let $A = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$ . A line $L$ passes through the point $P(3, 3, 1)$ and has a direction vector $\vec{v} = AB\vec{u}$ , where $\vec{u} = \hat{i} + \hat{j} - \hat{k}$ . If $S$ is the set of all points of intersection of the line $L$ and the plane $\Pi$ , let $d$ be the shortest distance from the origin to the set $S$ . Find the value of $d^2$ .

Options

A.
$\frac{32}{3}$
B.
$19$
C.
$\frac{72}{7}$
D.
$14$

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

Line lying in a plane|Characteristic Equation|Matrix Multiplication Non-Commutativity|Elementary Row Operations

Why This Question Is Tricky

Common Mistakes

This question tests the containment of a line in a plane and the non-commutativity of matrix multiplication representing geometric transformations.

The primary trap lies in calculating the direction vector $\vec{v}$ .

If a student incorrectly assumes matrix multiplication is commutative or applies the transformations in the wrong order ( $BA\vec{u}$ instead of $AB\vec{u}$ ), they will obtain $\vec{v}_{wrong} = \langle 1, -1, -1 \rangle$ .

This incorrect vector is not orthogonal to the plane's normal ( $\vec{n} \cdot \vec{v}_{wrong} = -4 \neq 0$ ), leading the student to falsely conclude that the line intersects the plane at exactly one point, $P(3,3,1)$ .

Consequently, they would calculate the distance to this single point, arriving at the trap answer of 19.

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