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Question #72125

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(1) = 2e^{-1}$ and satisfies the differential equation $(x^2 - 3x) |x-2| f'(x) = (x^2 - 3x) (x-2) (f(x) - x)$ for all $x \in \mathbb{R} \setminus \{0, 2, 3\}$ . Let $S$ be the set of all points in $\mathbb{R}$ where $f(x)$ is not differentiable. If $k = \sum_{x \in S} x$ , then the value of $\int_0^k f(x) dx$ is:

Options

A.
$2 - 2e^{-2}$
B.
$2 + 2e^{-2}$
C.
$4 - 2e^{-2}$
D.
$2 - 4e^{-2}$

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

Implicit to Explicit Conversion|Right-Hand vs Left-Hand Derivatives|The Zero-Factor Cancellation Trap

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

The question tests the 'Zero-Factor Cancellation Trap' by embedding roots x=0 and x=3 into the differential equation.

Students often blindly cancel the algebraic factor and incorrectly assume that points of undefined DE behavior automatically translate to points of non-differentiability for the continuous solution.

This leads to an incorrect set S={0,2,3} and a wrong upper limit k=5 for the integral.

The rigorous approach requires recognizing that continuity at x=0 and x=3 forces the integration constants to match, ensuring smooth transitions, leaving x=2 as the sole point of non-differentiability due to the absolute value term.

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