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Question #41652

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a solution to the differential equation $x \ln x \frac{dy}{dx} + y = 2 \ln x, \quad x > 1$ A line $L$ has direction ratios proportional to $(2, y(e), e y'(e))$ . It is given that $L$ intersects the $y$ -axis and lies entirely in the plane $P_1: x + 2y + z = 10$ . If $L$ intersects the plane $P_2: 3x - y + 2z = 17$ at the point $Q(\alpha, \beta, \gamma)$ , then the value of $\alpha + \beta + \gamma$ is:

Options

A.
9
B.
27
C.
-6
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

Symmetric vs Non-symmetric form|Linear Differential Equation Structure|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Integration Constant Trap' within a multi-domain synthesis of Differential Equations and 3D Geometry.

Students frequently assume the constant of integration $C=0$ when asked for 'a solution' to define geometric properties.

Here, the direction ratios of the line are parameterized by $C$ .

The trap is sprung if the student fails to apply the boundary condition—that the line lies entirely within a specific plane—to solve for $C$ .

Furthermore, the fixed point of the symmetric form must be deduced by intersecting the given plane with the specified axis, punishing rote memorization of symmetric forms without spatial reasoning.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Symmetric vs Non-symmetric form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free