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Question #40612

Difficulty: 8/10 | 🟢 Free

The Question

Let $R$ be the rank of the word 'SCORE' in the dictionary. A curve $y=f(x)$ in the first quadrant satisfies the differential equation $x \frac{dy}{dx} = y(\ln y - \ln x + R)$ and passes through the point $(1, R)$ . Find the value of $f(2)$ .

Options

A.
$20000 e^{99}$
B.
$20000 e^{100}$
C.
$2 e^{101}$
D.
$2 e^{-97}$

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

Rank of a Word|Substitution Logic in Homogeneous DE|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

The problem tests the integration of combinatorics (rank of a word) with solving a homogeneous differential equation.

The specific trap lies in the substitution $y=vx$ , where students frequently forget to subtract $v$ from the RHS, leading to $x \frac{dv}{dx} = v(\ln v + R)$ instead of $v(\ln v + R - 1)$ .

Furthermore, students often fail to properly evaluate the arbitrary constant $C$ using the boundary condition $(1, R)$ , mistakenly treating $R$ as a variable or assuming $C=0$ , which results in incorrect exponential scaling in the final answer.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Rank of a Word
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free