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Question #70162

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ symmetric orthogonal matrix. Let $\vec{y}(x)$ be a differentiable vector-valued function in $\mathbb{R}^3$ satisfying the differential equation: $x A \frac{d\vec{y}}{dx} + A \vec{y} = x^2 (\vec{y}^T A \vec{y}) A \vec{y}, \quad x > 0$ If $\vec{y}(1)^T A \vec{y}(1) = \frac{1}{c}$ for some constant $c$ , which of the following represents $w(x) = \vec{y}^T A \vec{y}$ ?

Options

A.
$\frac{1}{x^2 (c - 2 \ln x)}$
B.
$\frac{1}{x^2 (c + 2 \ln x)}$
C.
$\frac{1}{x (c - 2 \ln x)}$
D.
$\frac{x^2}{c - 2 \ln x}$

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

Bernoulli Equation Transformation|Orthogonal Matrix Conditions|Exact Differential Recognition

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

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

Difficulty Boost : +4

Common Mistakes

The student is first trapped by the matrix-vector formulation, often attempting to decompose the symmetric orthogonal matrix A into components, leading to an intractable non-linear coupled system.

The first shortcut is multiplying by A and then taking the dot product with the transformed vector to collapse the system into a scalar ODE.

The second 'collision' occurs here: students may fail to recognize the exact differential rearrangement d(x²w) = 2x³w²dx or the underlying Bernoulli form.

Instead, they might attempt to find a complex bivariate integrating factor for the non-exact differential form, leading to algebraic dead ends.

This tests the ability to elevate local algorithmic solving into global structural recognition.

Quick Facts About This Question

Question Information

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