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Question #41637

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two lines in $\mathbb{R}^3$ passing through the origin, having direction ratios $(1, 2, 2)$ and $(2, 3, 6)$ respectively. Let $P_1$ and $P_2$ be the $3 \times 3$ matrices representing the orthogonal projections onto $L_1$ and $L_2$ respectively. If $M = P_1 + P_2 - \lambda I$ , where $I$ is the $3 \times 3$ identity matrix, then the set of all real values of $\lambda$ for which $M$ is a singular matrix is:

Options

A.
$\left\{0, \frac{1}{21}, \frac{41}{21}\right\}$
B.
$\left\{0, \frac{20}{21}, \frac{22}{21}\right\}$
C.
$\left\{1, \frac{1}{21}, \frac{41}{21}\right\}$
D.
$\left\{0, \frac{41}{441}, \frac{400}{441}\right\}$

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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

Parametric form pitfalls|Orthogonal projection of a line|Singular vs Non-Singular Matrices|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

This question exploits a common trap where students attempt to explicitly construct the $3 \times 3$ projection matrices using the parametric direction ratios and then solve the cubic characteristic equation $\det(P_1 + P_2 - \lambda I) = 0$ .

This brute-force approach leads to massive arithmetic fatigue and a high probability of error.

The conceptual bypass requires recognizing that an orthogonal projection onto a line yields a rank-1 matrix, making the sum of two such matrices at most rank-2 (guaranteeing 0 as an eigenvalue).

Furthermore, utilizing trace properties and the geometric invariant that the non-zero eigenvalues of $P_1 + P_2$ are $1 \pm \cos \theta$ completely circumvents the matrix algebra, rewarding deep conceptual understanding of linear transformations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free