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Question #40400

Difficulty: 8/10 | 🟢 Free

The Question

Let $M(\theta) = \begin{pmatrix} 4\cos^3\theta - 3\cos\theta & 3\sin\theta - 4\sin^3\theta \\ 3\sin\theta - 4\sin^3\theta & 3\cos\theta - 4\cos^3\theta \end{pmatrix}$ and $N(\theta) = \begin{pmatrix} 2\cos^2\theta - 1 & 2\sin\theta\cos\theta \\ 2\sin\theta\cos\theta & 1 - 2\cos^2\theta \end{pmatrix}$ .

Let $S$ be the set of all $\theta \in [0, 2\pi]$ satisfying $(M(\theta) N(\theta))^3 = \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{pmatrix}$ Find the sum of all elements in $S$ .

Options

A.
$\frac{8\pi}{3}$
B.
$\frac{10\pi}{3}$
C.
$\frac{44\pi}{3}$
D.
$\frac{2\pi}{9}$

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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

Double and Triple Angle Ratios|Orthogonal Matrix Conditions|Matrix Multiplication Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the recognition of multiple-angle trigonometric identities embedded within matrices and the properties of orthogonal matrices.

The primary trap lies in the non-commutativity of matrix multiplication: students often incorrectly assume $P(x)P(y) = R(x+y)$ or $R(y-x)$ by misapplying scalar angle addition formulas or reversing the multiplication order.

If they use $R(y-x)$ , they solve $R(-3\theta) = R(2\pi/3)$ , leading to the incorrect sum $10\pi/3$ .

Additionally, the equation $(R(\theta))^3 = M$ requires finding all periodic solutions in $[0, 2\pi]$ using De Moivre's concept; missing the $2k\pi$ period yields the partial answer $2\pi/9$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Double and Triple Angle Ratios
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free