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Question #40750

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the solution to the differential equation $\frac{dy}{dx} = \frac{y^2 - 2xy}{x^2}$ satisfying $f(1) = 1$ . The general solution can be expressed in the form $1 - \frac{3x}{y} = C x^3$ , where $C$ is the constant of integration.

Let the vectors $\vec{u} = C\hat{i} + \hat{j} + \hat{k}$ , $\vec{v} = \hat{i} + C\hat{j} + \hat{k}$ , and $\vec{w} = \hat{i} + \hat{j} + k\hat{k}$ be coplanar.

If the three lines $L_1: kx + 2y + 1 = 0$ , $L_2: x + ky + 1 = 0$ , and $L_3: 2x + 2y + \lambda = 0$ are concurrent, find the value of $\lambda$ .

Options

A.
-7
B.
-1
C.
1
D.
2

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

Concurrent Lines Condition|Scalar Triple Product Volume Logic|Integration Constant Trap|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link differential equations, vector algebra, and coordinate geometry while navigating two specific traps.

The 'Integration Constant Trap' occurs if the student incorrectly evaluates or entirely omits the constant $C$ from the homogeneous differential equation, leading to an incorrect scalar $k$ .

The 'Cross Multiplication Trap' targets the determinant expansions for both the scalar triple product (coplanarity) and the concurrency condition; a common sign error in the cofactor expansion (e.g., adding instead of subtracting the middle term) will yield $k = -4/5$ or $\lambda = -1$ , both of which are designed to derail the solution path.

Quick Facts About This Question

Question Information

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