Before You Begin

Question #41262

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_k(\theta) = \frac{1}{\sqrt{2}} \begin{pmatrix} e^{2 i k \theta} & e^{i k \theta} \\ -e^{i k \theta} & 1 \end{pmatrix}$ for $k \in \mathbb{N}$ and $\theta \in [0, \pi]$ . Let $M(\theta) = A_1(\theta) A_2(\theta) \dots A_{10}(\theta)$ . If $\det(M(\theta)) + \det(M^{-1}(\theta)) = \sqrt{3}$ , then the sum of all possible values of $\theta$ is:

Options

A.
$55\pi$
B.
$\frac{55\pi}{2}$
C.
$110\pi$
D.
$5\pi$

Check Your Answer

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

Inversion and Determinant Product|Orthogonal Matrix Conditions|Euler's Form and Rotations

Why This Question Is Tricky

Common Mistakes

The student must recognize that the determinant of the product matrix $M(\theta)$ simplifies to a single complex phase $e^{110i\theta}$ .

The core trap lies in the trigonometric equation $\cos(110\theta) = \sqrt{3}/2$ .

Students often find the principal solutions $\theta = \pm \pi/660$ but fail to account for the modulo $2\pi$ periodicity over the highly stretched domain $110\theta \in [0, 110\pi]$ .

This leads to missing the vast majority of the 110 valid roots.

Furthermore, recognizing the symmetry of the roots about $\pi/2$ is crucial to bypass manually summing the arithmetic progressions, testing both algebraic precision and geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Inversion and Determinant Product
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free