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Question #40735

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be the line given by the symmetric equations $X = Y = Z = x$ , where $x$ is a real parameter. Let $P(x)$ be a point on $L$ and $d(x)$ be the perpendicular distance from $P(x)$ to the plane $X + Y - Z = 0$ . Let $f(x) = \sqrt{3} d(x)$ . A continuous function $y(x)$ satisfies the differential equation $x \frac{dy}{dx} + f(x) y = 3x^3$ for all $x \in \mathbb{R} \setminus \{0\}$ . If $y(-1) = -3$ , then the value of $y(1)$ is:

Options

A.
$3 - \frac{12}{e}$
B.
$3 - \frac{18}{e^2}$
C.
$-15$
D.
$3 + \frac{12}{e}$

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

Parametric Form Manipulation|Distance from point to plane|Integrating Factor (IF) Sign Sensitivity

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of 3D coordinate geometry and piecewise differential equations, specifically targeting the 'Integrating Factor Sign Sensitivity' trap.

Students must first correctly extract the absolute value function from the point-to-plane distance formula.

The core trap lies in the differential equation: dividing by $x$ yields a signum function $\text{sgn}(x)$ as the coefficient of $y$ .

A common mistake is to blindly ignore the absolute value, assuming $\frac{|x|}{x} = 1$ globally.

This leads to a single continuous branch $y = 3x^2 - 6x + 6 + K e^{-x}$ , yielding an incorrect constant $K = -18/e$ and the trap answer $3 - 18/e^2$ .

Rigorous boundary matching at $x=0$ is required to find the correct constants of integration for both domains.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Parametric Form Manipulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free