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Question #41851

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, \dots, 10\}$ . A homogeneous function $f(x, y) = a x^2 + b xy + c y^2$ is constructed by choosing the parameters $a, b, c$ independently and uniformly at random from the set $S$ with replacement. Consider the differential equation: $x^2 \frac{dy}{dx} = xy + f(x, y)$ Let $E$ be the event that the particular solution $y(x)$ satisfying the boundary condition $y(1) = 0$ also satisfies $y(e) = e$ . What is the probability of the event $E$ ?

Options

A.
0
B.
$\frac{47}{250}$
C.
$\frac{38}{125}$
D.
$\frac{1}{1000}$

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

Multiplication Principle Scope|Homogeneous Function Homogeneity|Arbitrary Constants vs Parameters|Empty and Universal Relation

Why This Question Is Tricky

Common Mistakes

This question creates a lethal collision between Differential Equations and Probability by testing the Multiplication Principle's scope against an empty relation.

Students often mechanically separate variables and assume that a valid combination of parameters $(a, b, c)$ must exist to satisfy the boundary conditions, leading them to fall for combinatorics traps (like the 304/188 error patterns derived from misapplied derangements or unconstrained integer partitions).

The core trap lies in failing to check the existence criteria of the solution set: because the parameters are strictly positive integers, the integrand is strictly bounded below 1, making the definite integral strictly less than 1.

The boundary condition forces a relation that contradicts the parameter space, collapsing the cardinality of the solution set to zero.

This filters out candidates who do not verify the domain and range constraints of arbitrary constants in initial value problems.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Multiplication Principle Scope
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free