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Question #41167

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sin^{-1}(\sin x) + \cos^{-1}(\cos x) + \tan^{-1}(\tan x)$ for $x \in (0, \pi) \setminus \left\{\frac{\pi}{2}\right\}$ . Let $L(x)$ be the linear approximation of $f(x)$ at $x = \frac{29\pi}{60}$ . If $E$ is the absolute error when $L(x)$ is used to approximate the value of $f\left(\frac{31\pi}{60}\right)$ , then $E$ equals:

Options

A.
$\frac{31\pi}{30}$
B.
$\frac{29\pi}{30}$
C.
$\frac{\pi}{30}$
D.
$\pi$

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

Composition Inverse Identity Traps|The Quadrant Adjustment Logic|Approximation via Differentials

Why This Question Is Tricky

Common Mistakes

This question tests the 'Identity Trap' in inverse trigonometric functions combined with linear approximations.

Students often incorrectly assume $f^{-1}(f(x)) = x$ universally, which would lead to $f(x) = 3x$ everywhere and an error of 0.

Furthermore, applying a differential approximation across a threshold ( $x = \pi/2$ ) where the function undergoes a jump discontinuity (due to quadrant adjustments like $\pi - x$ and $x - \pi$ ) compounds the error.

The linear model $L(x)$ assumes a continuous smooth slope and fails to capture the sign flips and domain restrictions, resulting in a massive deviation from the true value.

Quick Facts About This Question

Question Information

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