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Question #41183

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x) = \begin{pmatrix} \frac{1}{\pi}\sin^{-1}(\sin x) & \frac{1}{\pi}\cos^{-1}(\sin x) & 0 \\ -\frac{1}{\pi}\cos^{-1}(\sin x) & \frac{1}{\pi}\sin^{-1}(\sin x) & 0 \\ 0 & 0 & 1 \end{pmatrix}$ . If $S$ is the sum of all solutions to the equation $|\text{adj}(2A(x))| = 25$ in the interval $x \in [0, 10\pi]$ , find the value of $S$ .

Options

A.
$45\pi$
B.
$50\pi$
C.
$55\pi$
D.
$65\pi$

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

Inverse Function Periodicity|Determinant Scaling Property|Orthogonal Matrix Conditions

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of determinant scaling properties and the periodicity of inverse trigonometric functions.

A common trap is miscalculating the determinant of the adjoint matrix by incorrectly scaling the constant (e.g., assuming $|\text{adj}(2A)| = 4|A|^2$ instead of $64|A|^2$ for a $3 \times 3$ matrix).

Furthermore, students often incorrectly assume $\sin^{-1}(\sin x) = x$ globally, which would lead to an algebraic equation with non-periodic roots.

By forcing the evaluation of $\sin^{-1}(\sin x) = -\pi/4$ , the problem requires students to correctly identify the piecewise periodic solutions and sum them over a large interval, filtering out those who rely on rote algebraic simplification without domain analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Inverse Function Periodicity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free