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Question #40568

Difficulty: 8/10 | 🟢 Free

The Question

Let $k$ be the number of distinct ways to seat 5 people around a circular table such that two specific individuals are never adjacent. Consider the homogeneous function $H(x, y) = \sqrt{y^2 - \frac{k}{3} x^2}$ . Let $L$ be the ray in the first quadrant consisting of the points where $H(x, y)$ is non-differentiable. A curve $y = f(x)$ (for $x \ge 1$ ) satisfies the differential equation $x \frac{dy}{dx} = y + x H\left(1, \frac{y}{x}\right)$ and intersects the ray $L$ at $x = 1$ . Find the area of the region bounded by the curve $y = f(x)$ , the ray $L$ , and the vertical line $x = 3$ .

Options

A.
$\frac{8}{3}$
B.
$\frac{14}{3}$
C.
$\frac{10}{3}$
D.
$\frac{4}{3}$

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

Circular Permutations Constraint|Homogeneous Function Homogeneity|Critical Point vs Stationary Point

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of discrete combinatorics and continuous analysis by embedding a circular permutation constraint as a parameter in a homogeneous differential equation.

The primary trap lies in the 'Critical Point' analysis of the homogeneous function $H(x,y)$ .

Students often incorrectly assume that critical points must be isolated stationary points where the gradient is zero.

Here, the gradient is never zero; instead, the critical points form a line of non-differentiability ( $y=2x$ ) at the boundary of the function's domain.

Failing to recognize this line of discontinuity prevents students from identifying the boundary condition $y(1)=2$ required to solve the differential equation, effectively trapping them between the discrete parameter calculation and the continuous integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Circular Permutations Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free