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Question #40542

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a real-valued differentiable function defined on the interval $\left[ \frac{1}{2}, \frac{\sqrt{3}}{2} \right]$ satisfying the differential equation $\frac{dy}{dx} = \frac{4x^3 - 4x}{x^4 - 2x^2} e^{-y}$ If $y\left(\frac{1}{\sqrt{2}}\right) = \ln \alpha$ , then the set of all possible values of $\alpha$ is:

Options

A.
$\left( \ln \frac{12}{7}, \infty \right)$
B.
$\left( \ln \frac{16}{7}, \infty \right)$
C.
$\left( \ln \frac{4}{5}, \infty \right)$
D.
$\left( 0, \infty \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Modulus Inequality Logic|Logarithmic Integration Modulus|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

This question tests the 'Domain of Existence' for differential equations involving logarithmic solutions.

The trap lies in treating the constant of integration $C$ as a purely arbitrary real number or only evaluating the domain constraint at the initial condition point.

Students must realize that for the solution $y(x)$ to exist over the entire specified interval, the value of $e^y$ must remain strictly positive globally.

This transforms the integration constant into a constrained parameter, requiring optimization of the modulus function $-\ln(2x^2 - x^4)$ over the interval to find the true lower bound for $\alpha$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Inequality Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free