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Question #40516

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: (0, \infty) \to \mathbb{R}$ be a continuous function satisfying the differential equation $x f'(x) + f(x) = 15x^2 \frac{|x-2|}{x-2}$ for all $x \in (0, \infty) \setminus \{2\}$ . If $f(1) = 27$ , then the value of $f(8)$ is

Options

A.
312
B.
314
C.
316
D.
324

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

Sign Flipping Rule|Integrating f(x) + xf'(x) forms|Integration Constant Trap|General vs Particular Solutions

Why This Question Is Tricky

Common Mistakes

The question tests the 'Integration Constant Trap' within a piecewise continuous differential equation.

Students easily identify the exact differential $\frac{d}{dx}(xf(x))$ , but the modular condition forces a sign flip at $x=2$ .

The trap occurs when students assume a single integration constant $C$ for the entire domain or incorrectly transfer $C_2$ to the $x > 2$ domain.

To find the correct particular solution, they must enforce continuity at the boundary $x=2$ , solving for a new constant $C_1$ that bridges the discontinuity.

Distractor 312 catches a common arithmetic error during the boundary evaluation, while 324 catches the false assumption that $C_1 = C_2$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Sign Flipping Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free