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Question #41246

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 3 \end{pmatrix}$ and let $\lambda_1 < \lambda_2 < \lambda_3$ be the roots of its characteristic equation. Consider two planes $P_1$ and $P_2$ given in parametric form as: $P_1: \vec{r} = \hat{i} + s(-\lambda_1 \hat{i} + \lambda_2 \hat{j}) + t(\lambda_2 \hat{j} + \lambda_3 \hat{k})$ $P_2: \vec{r} = \hat{j} + p(\lambda_3 \hat{i} - \lambda_1 \hat{j}) + q(\lambda_3 \hat{j} + \lambda_2 \lambda_3 \hat{k})$ where $s, t, p, q \in \mathbb{R}$ . The angle between the planes $P_1$ and $P_2$ is:

Options

A.
$\cos^{-1}\left( \frac{36}{49} \right)$
B.
$\pi - \cos^{-1}\left( \frac{36}{49} \right)$
C.
$\cos^{-1}\left( \frac{18}{49} \right)$
D.
$\pi - \cos^{-1}\left( \frac{18}{49} \right)$

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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

Characteristic Equation|Angle between two planes|Parametric Form Manipulation

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of linear algebra and 3D geometry.

Students must first find the eigenvalues of a matrix to construct the parametric equations of two planes.

The primary trap lies in the definition of the angle between two planes.

Students often compute the dot product of the normal vectors (which evaluates to -108) and directly use it to find the angle, forgetting that the angle between planes is conventionally defined as the acute angle between their normal vectors, requiring an absolute value $|\vec{n}_1 \cdot \vec{n}_2|$ .

Failing to apply this yields the obtuse angle supplement, leading directly to the trap option.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Characteristic Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free