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Question #41608

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a curve in the $xy$ -plane satisfying the differential equation $\frac{dy}{dx} - y = x$ . Let $P(x, f(x), 0)$ be a point on the curve and $d(x)$ be its perpendicular distance to the plane $\Pi: x + y + \sqrt{2}z - 3 = 0$ . Given that $d(0) = 3$ and the scalar projection of the position vector $\vec{OP}$ at $x=0$ (where $O$ is the origin) onto the vector $\vec{n} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$ is negative, find $f(\ln 2)$ .

Options

A.
$-5 - \ln 2$
B.
$19 - \ln 2$
C.
$5 - \ln 2$
D.
$-19 - \ln 2$

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

General vs Particular Solutions|Collinearity and Coplanarity|Distance from point to plane|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question bridges Class 12 Differential Equations with Class 11 3D Geometry.

The core trap lies in the distance formula from a point to a plane, which strictly requires an absolute value $|ax+by+cz+d|/\sqrt{a^2+b^2+c^2}$ .

Students often drop the modulus, leading directly to the incorrect constant of integration ( $C=10$ ).

The problem forces the evaluation of the modulus by providing two mathematical branches, requiring the student to use the dot product projection sign convention to eliminate the extraneous constant.

This tests the rigorous application of vector signs and prevents 'auto-piloting' through DE initial value conditions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	General vs Particular Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free