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Question #40423

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B, C$ be the angles of a triangle. Consider the matrix $M = \begin{pmatrix} -1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{pmatrix}$ . Let $\lambda_1, \lambda_2, \lambda_3$ be the eigenvalues of $M$ . If $X = \begin{pmatrix} \csc A \\ \csc B \\ \csc C \end{pmatrix}$ , find the value of $X^T \text{adj}(M) X + \lambda_1 + \lambda_2 + \lambda_3$ .

Options

A.
6
B.
-6
C.
0
D.
12

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

Conditional Identity Constraints|Characteristic Equation|Singular vs. Non-Singular Threshold|Minors and Cofactors Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the synergy between conditional trigonometric identities and matrix properties, specifically targeting the cofactor sign convention trap.

Students must use the constraint $A+B+C=\pi$ to evaluate the cofactors of the singular matrix $M$ .

A critical error occurs when calculating off-diagonal cofactors (e.g., $C_{12}$ ); students frequently forget the negative sign associated with the checkerboard pattern of the cofactor matrix, incorrectly computing $C_{12} = -\sin A \sin B$ instead of $\sin A \sin B$ .

This mistake cascades into the quadratic form $X^T \text{adj}(M) X$ , yielding a false value of $-3$ instead of $9$ , which directly leads to the trap option $-6$ .

Furthermore, the problem elegantly bypasses the need to explicitly solve the characteristic equation by utilizing the fundamental trace property for the sum of eigenvalues.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Conditional Identity Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free