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Question #40414

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = |\sin(\sin^{-1}x + \cos^{-1}(x-1))| + |\cos(\sin^{-1}x + \cos^{-1}(x-1))|$ . If $M$ is the maximum value of $f(x)$ over its domain of definition, then $M$ equals:

Options

A.
$\sqrt{2}$
B.
$\frac{\sqrt{3}+1}{2}$
C.
1
D.
$\frac{\sqrt{6}}{2}$

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

Compound Angle Expansion|Triangle Inequality Mind Games|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of domain constraints with trigonometric inequalities.

A common trap is blindly applying the triangle inequality $|\sin\theta| + |\cos\theta| \le \sqrt{2}$ and assuming the maximum $\sqrt{2}$ is attainable.

However, the modulus inequality acts as a filter: the domain of the inverse trigonometric functions restricts the argument $\theta$ to $[5\pi/6, \pi]$ .

Since the critical angle $3\pi/4$ (where the unconstrained maximum occurs) lies outside this restricted range, students who ignore the simultaneous validity of the domain and the compound angle expansion will incorrectly choose $\sqrt{2}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Compound Angle Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free