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Question #41641

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 0 & 2 & 0 \\ 4 & -2 & 4 \\ 2 & -3 & 2 \end{pmatrix}$ . A plane $P$ has the equation $ax + by + cz = d$ . The system of linear equations $A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is known to be consistent. If the plane $P$ passes through the points $(1, 1, 1)$ and $(2, 0, 1)$ , find the square of the perpendicular distance from the point $(10, 10, -8)$ to the plane $P$ .

Options

A.
144
B.
225
C.
256
D.
324

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

Distance from point to plane|System of Linear Equations (Cramer/Inverse)|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the consistency of linear systems and its geometric interpretation.

The trap lies in the matrix A being nilpotent (A^3 = 0), which implies det(A) = 0.

Students who blindly attempt to use Cramer's rule or invert A to find the normal vector (a, b, c) will encounter a singular matrix and might assume the problem is unsolvable or make a division-by-zero error.

The correct approach requires recognizing that for the system to be consistent, the vector (a, b, c) must lie in the column space of A, which is equivalent to being orthogonal to its left null space.

This provides the crucial constraint 2a - b + 2c = 0 needed to uniquely determine the plane's normal vector alongside the geometric points provided.

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