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Question #60467

Difficulty: 8/10 | 🟢 Free

The Question

Let $T_r(n)$ denote the $r$ -th term ( $0 \le r \le n$ ) in the binomial expansion of $(u+v)^n$ , where $u, v > 0$ and $p = \frac{v}{u+v}$ is an irrational number. Let $k_n$ be the index of the numerically greatest term in this expansion. Define $A = \limsup_{n \to \infty} n \left( 1 - \frac{T_{k_n+1}(n)}{T_{k_n}(n)} \right)$ . Find the value of $A$ .

Options

A.
$\frac{(u+v)^2}{uv}$
B.
$\frac{u^2+v^2}{uv}$
C.
$\frac{u+v}{uv}$
D.
$\frac{uv}{(u+v)^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Infinite Limit Behavior|Numerically Greatest Term|Denominator Restriction

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the 'Denominator Restriction' and 'Infinite Limit Behavior' traps by forcing students to evaluate an indeterminate form $n(1-R_n)$ where the ratio $R_n$ approaches 1.

A common mistake is to use the continuous approximation $k_n \approx np$ prematurely, which incorrectly simplifies $1 - R_n$ to 0, leading students to falsely conclude the limit is 0.

By demanding the exact algebraic expansion, the hidden $O(1/n)$ terms in the numerator reveal a dependency on the fractional part $\epsilon_n$ .

Because the index fluctuates, the sequence does not converge, requiring the rigorous application of the limit superior and the density of fractional parts of irrational multiples to find the maximum accumulation point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Infinite Limit Behavior
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free