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Question #59476

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying the differential equation $x f'(x) - f(x) = \mu x^2 \ln(x^2)$ for all $x \neq 0$ , where $\mu$ is a positive constant. If $f(1) = p$ and $f(-1) = q$ , then which of the following relations is necessarily true?

Options

A.
$p + q = -4\mu$
B.
$p - q = 4\mu$
C.
$p + q = 4\mu$
D.
$p - q = -4\mu$

Check Your Answer

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

Right-Hand vs Left-Hand Derivatives|General vs Particular Solutions|Logarithmic Integration Modulus

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

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

Difficulty Boost : +4

Common Mistakes

This question tests the 'Logarithmic Integration Modulus' trap and the transition from a general to a particular solution using calculus constraints.

Students often incorrectly integrate $\ln(x^2)$ as $2\ln x$ without the modulus, failing to define the function properly for $x < 0$ .

Furthermore, integrating across a discontinuity at $x=0$ generates two independent constants, $C_1$ and $C_2$ .

The critical step is applying the differentiability constraint at $x=0$ (RHD = LHD) to prove $C_1 = C_2$ , thereby linking the piecewise domains and allowing the elimination of the constant to find the algebraic relation between $p$ and $q$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Right-Hand vs Left-Hand Derivatives
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free