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Question #41548

Difficulty: 8/10 | 🟢 Free

The Question

Consider the region $R = \{ (x, y) \in \mathbb{R}^2 : |y| \le e^{-|x|} \sin(\pi x), \ -1 \le x \le 1 \}$ . The area of $R$ is

Options

A.
$\frac{2\pi(e + 1)}{e(\pi^2 + 1)}$
B.
$\frac{4\pi(e + 1)}{e(\pi^2 + 1)}$
C.
$\frac{\pi(e + 1)}{e(\pi^2 + 1)}$
D.
$0$

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

Symmetry Utilization|Modulus Function Symmetry|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Sign Dependency' trap within symmetry utilization.

Students frequently observe the $|y|$ and $|x|$ terms and incorrectly assume 4-quadrant symmetry, calculating the first-quadrant area and multiplying by 4.

However, the presence of the odd function $\sin(\pi x)$ breaks this symmetry.

The critical realization is that an inequality of the form $|y| \le f(x)$ only produces a valid real region when $f(x) \ge 0$ .

For $x < 0$ , $f(x)$ becomes negative, meaning the curve does not exist in the second and third quadrants.

Failing to verify the domain of existence leads directly to the primary distractor (multiplying by 4 instead of 2).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Symmetry Utilization
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free