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Question #40561

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be a multiset containing 3 identical red balls, 2 identical blue balls, and 1 green ball. Let $a_r$ denote the number of ways to select exactly $r$ balls from $S$ . Consider the differential equation: $x \left(\frac{d^m y}{dx^m}\right)^n + (m+1) \frac{d^{m-1} y}{dx^{m-1}} = a_5 x^{a_1 - 1}$ where $m = a_5 - 1$ and $n = a_6$ . If $y(x)$ is a solution to this differential equation satisfying $y(1) = a_2$ and $y'(1) = a_3$ , then the value of $40 y(2)$ is:

Options

A.
337
B.
327
C.
347
D.
317

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

Fundamental Principle of Counting (FPC)|Selection from Identical Items|Order and Degree Definition|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Combinatorics (FPC with identical items) and Differential Equations.

The primary trap lies in the 'Order and Degree' definition acting as a choke point: if a student incorrectly uses combinations for distinct items (e.g., $\binom{6}{5}$ instead of 3 for $a_5$ ), they generate a 5th-order DE, which is unsolvable with the given two boundary conditions, forcing them to re-evaluate their combinatorics.

The 'Integration Constant Trap' is deployed during the DE solution; the equation $x y'' + 3y' = 3x^2$ requires an integrating factor or exact differential recognition $(x^3 y')' = 3x^4$ .

Students often forget the first constant $C_1$ entirely, which breaks the dependency between the discrete selection counts ( $a_3=6$ ) and the continuous function, leading to an incorrect final scalar.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Fundamental Principle of Counting (FPC)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free