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Question #41155

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real values of $x$ satisfying the equation $\sin^{-1}(|x^2 - 2x|) + \cos^{-1}(|x| - 1) = \frac{\pi}{2}$ . Find the sum of the elements of $S$ .

Options

A.
$\frac{1 + \sqrt{5}}{2}$
B.
$2 + \sqrt{5}$
C.
1
D.
$\frac{3 + \sqrt{5}}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Principal Value Branch Boundaries|Modulus Function Breakdown|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reconcile modulus sign dependencies with the restricted principal value branches of Inverse Trigonometric Functions (ITFs).

The 'Lethal Collision' occurs because the algebraic equation $|x^2 - 2x| = |x| - 1$ yields multiple roots (such as $x = \frac{3+\sqrt{5}}{2}$ ) if the modulus is split without checking the ITF domain constraints.

Students frequently equate the arguments without enforcing the condition that the arguments must lie in $[-1, 1]$ , leading to extraneous solutions that satisfy the algebraic identity but violate the domain of $\sin^{-1}$ and $\cos^{-1}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Inverse Trigonometric Functions - Class 12
Concept	Principal Value Branch Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free