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Question #41247

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(\theta) = \begin{pmatrix} \sin 2\theta & \cos 2\theta & \sin \theta \\ \cos 2\theta & -\sin 2\theta & \cos \theta \\ \sin \theta & \cos \theta & \sin\theta + \cos\theta \end{pmatrix}$ . Let $S$ be the set of all $\theta \in [0, 2\pi]$ such that the matrix $M(\theta)$ has at least one integer eigenvalue. Find the sum of all elements in $S$ .

Options

A.
$\frac{11\pi}{2}$
B.
$5\pi$
C.
$\frac{9\pi}{2}$
D.
$6\pi$

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

Characteristic Equation|Parametric form pitfalls|Compound Angle Expansion

Why This Question Is Tricky

Common Mistakes

The question tests the 'Parametric Form Pitfall' within the context of characteristic equations.

Students must expand the determinant of a trigonometric matrix to find a cubic characteristic equation $\lambda^3 - k\lambda^2 - 2\lambda + k = 0$ , where $k = \sin\theta + \cos\theta$ .

The primary trap lies in assuming all integer eigenvalues $\lambda$ yield valid solutions for $\theta$ .

Students must recognize that the compound angle restricts the parameter $k \in [-\sqrt{2}, \sqrt{2}]$ , forcing them to reject extraneous integer roots like $\lambda = \pm 3$ .

Furthermore, calculating the sum of the valid $\theta$ values requires careful navigation of the periodic boundary conditions of inverse trigonometric functions in the interval $[0, 2\pi]$ .

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