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Question #41182

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x)$ . The sum of all solutions to the equation $f(x) = \frac{x}{4}$ in the interval $x \in [0, 20\pi]$ can be expressed as $\frac{p\pi}{q}$ , where $p$ and $q$ are coprime positive integers. Find the value of $p+q$ .

Options

A.
487
B.
512
C.
361
D.
424

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

Inverse Function Periodicity|The Quadrant Adjustment Logic|Codomain vs Range Distinction|Surjective Mapping (Onto)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the codomain and the actual range of a composite inverse trigonometric function.

The primary trap lies in the quadrant adjustment logic: students often mechanically evaluate the piecewise definitions across the entire given interval $[0, 20\pi]$ , wasting immense time or making arithmetic errors.

By forcing the student to first establish the global range of $f(x)$ as $[0, \pi]$ , the problem mandates bounding the RHS ( $x/4 \le \pi$ ), instantly collapsing the search space to $[0, 4\pi]$ .

Furthermore, it exploits the common misconception that $\sin^{-1}(\sin x) + \cos^{-1}(\cos x)$ might simplify to a constant or a simple linear function, punishing those who ignore the sign of the derivative in different quadrants.

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