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Question #66251

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: [-\pi, \pi] \to \mathbb{R}$ be a continuous function that satisfies the differential equation $\frac{dy}{dx} + y = \frac{\sin x - 2\cos x + 2}{2y}$ for all $x \in (-\pi, \pi) \setminus \{0\}$ . It is given that $f(\pi) = \sqrt{2}$ and $f(-\pi) = \sqrt{2}$ . Let $k$ be the number of points in the interval $(-\pi, \pi)$ where $f(x)$ is not differentiable, and let $I = \int_{-\pi}^{\pi} f(x) \, dx$ . Find the value of $10k + \frac{I^2}{4}$ .

Options

A.
18
B.
8
C.
10
D.
12

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

Differentiability vs Continuity|Implicit to Explicit Conversion|Bernoulli Equation Transformation

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the 'Implicit to Explicit Conversion' trap within a Bernoulli differential equation.

Students must apply the transformation $v = y^2$ to solve the DE, yielding an implicit solution $y^2 = 1 - \cos x$ .

The critical trap lies in transitioning to the explicit function $y = f(x)$ .

Because the DE is undefined at the singular solution $y=0$ (which occurs at $x=0$ ), the domain splits into two branches.

The boundary conditions force $f(x)$ to be positive on both sides, resulting in $f(x) = \sqrt{2}|\sin(x/2)|$ .

Students often incorrectly assume that solutions to DEs are everywhere differentiable, missing that the piecewise junction at $x=0$ forms a cusp, leading to a failure in differentiability despite maintaining continuity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Differentiability vs Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free