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Question #40509

Difficulty: 8/10 | 🟢 Free

The Question

Let $f_n(x) = x^2 - 2nx + n^2 - 1$ for $n \in \mathbb{N}$ , and let $\alpha_n, \beta_n$ ( $\alpha_n < \beta_n$ ) be the roots of $f_n(x) = 0$ . For $x \in (\alpha_2, \beta_2)$ , a differentiable function $y = y(x)$ satisfies the differential equation $|f_2(x)| \frac{dy}{dx} + f'_2(x) y = f_2(x) \sqrt{f_2(x) + 1}$ If $y(2) = 0$ , then the value of $y\left(\frac{5}{2}\right)$ is

Options

A.
$\frac{3}{8} \ln\left(\frac{3}{4}\right)$
B.
$\frac{7}{48}$
C.
$\frac{3}{8} \ln\left(\frac{4}{3}\right)$
D.
$\frac{3}{4} \ln\left(\frac{3}{4}\right)$

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

Maximum/Minimum of Quadratic Expressions|Quadratic Roots in Sequences|Integrating Factor (IF) Sign Sensitivity

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding the sign of quadratic expressions within differential equations, specifically when determining the Integrating Factor (IF).

The trap lies in the absolute value $|f_2(x)|$ .

Since the interval of definition is strictly between the roots of the quadratic, $f_2(x)$ is negative.

Students who mechanically drop the absolute value or ignore the negative sign will compute an incorrect IF of $x^2-4x+3$ instead of $-(x^2-4x+3)^{-1}$ .

This 'Lethal Collision' fundamentally alters the RHS integral from a logarithmic function to a polynomial one, leading directly to the trap answer 7/48.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Maximum/Minimum of Quadratic Expressions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free