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Question #72098

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the continuous solution to the differential equation $x \frac{dy}{dx} - y = (x^2 + y^2) \frac{\pi}{8} [x]$ for $x \in (0, 3)$ , satisfying $\lim_{x \to 0^+} \frac{f(x)}{x} = 0$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . If $k = f\left(\frac{5}{2}\right)$ , then the value of $8 \int_{0}^{k} \{x\} dx$ is: (where $\{x\}$ denotes the fractional part of $x$ )

Options

A.
8
B.
9
C.
10
D.
12

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

Implicit to Explicit Conversion|Variable Transformation Substitution|Greatest Integer Function Traps

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to handle implicit differential equations requiring variable transformation ( $v = y/x$ ) alongside piecewise functions.

The primary trap lies in the integration of the Greatest Integer Function $[x]$ and Fractional Part Function $\{x\}$ .

Students frequently integrate $[x]$ blindly as $x$ or $x - 0.5$ , or integrate $\{x\}$ as $x^2/2$ , failing to partition the domain at integer boundaries.

This violates the Fundamental Theorem of Calculus across jump discontinuities.

The recursive structure ensures that any error in handling the piecewise integration in Part A propagates, leading to an incorrect upper limit for the fractional part integral in Part B.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Implicit to Explicit Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free