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Question #40618

Difficulty: 8/10 | 🟢 Free

The Question

Let the function $f(x,y)$ be defined by the expansion: $f(x,y) = \sum_{r=0}^{251} \binom{251}{r} x^{251-r} y^r \left( \frac{x^2+y^2}{xy} \right)^r \ln\left( 1 + \frac{2y}{x} \right)$ A curve $C$ in the first quadrant is defined implicitly by the differential equation: $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 251 \cdot 3^{251} \ln(3)$ If $(1, y_0)$ is the unique point on the curve $C$ for $x=1$ , find the value of $\frac{dy}{dx}$ at this point.

Options

A.
$\frac{2 - 251 \ln(3)}{2 + 502 \ln(3)}$
B.
$\frac{1 - 251 \ln(3)}{1 + 251 \ln(3)}$
C.
$\frac{2 + 251 \ln(3)}{2 - 502 \ln(3)}$
D.
$\frac{251 \ln(3)}{1 + 502 \ln(3)}$

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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

General Term Formulation|Homogeneous Function Homogeneity|Implicit Differentiation Implicit Domain

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of binomial expansion, Euler's Theorem for homogeneous functions, and implicit differentiation.

The primary trap is the 'Implicit Differentiation' explosion: students who fail to recognize the homogeneity and attempt to differentiate the summation directly will be overwhelmed algebraically.

Furthermore, the problem enforces an implicit domain check; the logarithmic term ensures the uniqueness of the coordinate y=1 in the first quadrant, preventing ambiguity.

By equating the Euler's expansion to a constant, it forces the student to simplify the curve equation before applying implicit differentiation on the condensed homogeneous form.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	General Term Formulation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free