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Question #41736

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 1 & -1 & 2 \\ 2 & 1 & -1 \\ 4 & -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 1 & 0 & 1 \end{pmatrix}$ . Let $\Pi$ be the plane consisting of all points $(x, y, z)$ such that the matrix equation $BA \begin{pmatrix} u \\ v \\ w \end{pmatrix} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ has at least one solution for $(u, v, w) \in \mathbb{R}^3$ . If $d$ is the perpendicular distance from the point $P(1, -1, 1)$ to the plane $\Pi$ , then the value of $d$ is:

Options

A.
$\frac{\sqrt{70}}{5}$
B.
0
C.
$\frac{2\sqrt{70}}{35}$
D.
$\frac{\sqrt{70}}{7}$

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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

Distance from point to plane|Singular vs Non-Singular Matrices|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of column spaces and matrix transformations, specifically targeting the non-commutativity of matrix multiplication ( $C(BA) \neq C(AB)$ ).

Students often mistakenly assume that since $B$ is invertible, the column space of $BA$ is identical to the column space of $A$ (which is only true for $AB$ ).

If a student falls for this trap and uses the column space of $A$ (or $AB$ ), they will derive the incorrect plane equation $2x + y - z = 0$ .

The point $P(1, -1, 1)$ is intentionally chosen to lie exactly on this incorrect plane, yielding a distance of $0$ (Option B), heavily punishing the commutativity assumption.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Distance from point to plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free