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Question #70184

Difficulty: 8/10 | 🟢 Free

The Question

Consider the family of curves given by the differential equation $\frac{dy}{dx} = \frac{a^2+\sin^2 x}{y - y^3}$ , where $a > 0$ . Let $y=f(x)$ be the orthogonal trajectory to this family. If $f(0) = y_0$ (where $y_0 \notin \{-1, 0, 1\}$ ), find the value of $\lim_{x \to \pi} \frac{|1 - (f(x))^{-2}|}{|1 - y_0^{-2}|}$ .

Options

A.
$e^{\frac{2\pi}{a\sqrt{a^2+1}}}$
B.
1
C.
$e^{\frac{\pi}{a\sqrt{a^2+1}}}$
D.
$e^{\frac{2\pi}{a^2+1}}$

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

Orthogonal Trajectories|Partial Fractions Decomposition|ASTC Quadrant Sign Convention|Symmetry and Periodicity

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This problem fuses orthogonal trajectories, partial fraction decomposition, and a lethal periodicity trap.

The integral $\int \frac{dx}{a^2+\sin^2 x}$ evaluates to an inverse tangent function with a restricted domain $(-\pi/2, \pi/2)$ .

Students who blindly apply the naive antiderivative without the step function $\lfloor \frac{x+\pi/2}{\pi} \rfloor$ will incorrectly evaluate the definite integral from $0$ to $\pi$ as $0$ , leading to the trap answer $1$ .

Recognizing the asymptotes of $\tan x$ and enforcing continuity is mandatory for the correct exponential ratio.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Orthogonal Trajectories
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free