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Question #40397

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a differentiable function satisfying the differential inequality $\frac{dy}{dx} - \left( \frac{\cos 2x}{1 + \sin 2x} \right) y < -\cos\left(x - \frac{\pi}{4}\right)$ for $x \in \left[ \frac{5\pi}{6}, \pi \right]$ . If $y(\pi) = 0$ , then which of the following inequalities is necessarily true?

Options

A.
$y\left(\frac{5\pi}{6}\right) > \frac{\pi(1 - \sqrt{3})}{12\sqrt{2}}$
B.
$y\left(\frac{5\pi}{6}\right) < \frac{\pi(1 - \sqrt{3})}{12\sqrt{2}}$
C.
$y\left(\frac{5\pi}{6}\right) > \frac{\pi(\sqrt{3} - 1)}{12\sqrt{2}}$
D.
$y\left(\frac{5\pi}{6}\right) < \frac{\pi(\sqrt{3} - 1)}{12\sqrt{2}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Compound Angle Expansion|Homogeneous Function Homogeneity|Integrating Factor (IF) Sign Sensitivity

Why This Question Is Tricky

Common Mistakes

The question tests the sign sensitivity of the Integrating Factor (IF) in linear differential inequalities, combined with compound angle reductions.

Students must first reduce the trig-term to a homogeneous form.

The critical trap lies in the integration $\int P(x)dx = -\ln|\sin x + \cos x|$ .

In the domain $[5\pi/6, \pi]$ , the argument is negative, so the modulus introduces a negative sign, making IF $= \frac{-1}{\sin x + \cos x}$ .

Students who blindly write IF $= \frac{1}{\sin x + \cos x}$ will multiply the inequality by a negative quantity without reversing it, leading to the exact opposite inequality bound (Option B).

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