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Question #40642

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha = \lim_{n \to \infty} \sum_{k=1}^n \frac{12n}{(n+k)^2}$ , $\beta = \lim_{n \to \infty} \sum_{k=1}^n \frac{3\pi}{10n} \sin\left(\frac{k\pi}{2n}\right)$ , and $\gamma = \lim_{n \to \infty} \sum_{k=1}^n \frac{k^2}{9n^3}$ . If $C$ is the coefficient of $x^3$ in the expansion of $(1 + \beta x + \gamma x^2)^\alpha$ , find the value of $[C]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
3
B.
4
C.
5
D.
6

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

Multinomial Theorem|Integral as the Limit of a Sum|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

The problem tests the integration of Riemann sum limits with the Multinomial Theorem, culminating in a precision trap using the greatest integer function.

Students must evaluate three separate limits to find the parameters $\alpha, \beta, \gamma$ of the expansion.

The lethal trap lies in the final calculation of the coefficient $C = 374/75 \approx 4.986$ .

Because this value is extremely close to 5, students who prematurely round intermediate fractions (e.g., $2/3 \approx 0.7$ ) or assume the value is exactly 5 due to calculation fatigue will incorrectly evaluate $[C] = 5$ .

Rigorous exact rational arithmetic is required before applying the floor function.

Quick Facts About This Question

Question Information

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