Before You Begin

Question #40638

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be the integer $k \in [1, 270]$ that maximizes the sequence $a_k = k^2 (271-k)^3$ . If the $M^{\text{th}}$ term is the unique numerically greatest term in the binomial expansion of $(1+x)^{271}$ , then the range of positive values of $x$ is:

Options

A.
$\left( \frac{107}{165}, \frac{27}{41} \right)$
B.
$\left( \frac{27}{41}, \frac{109}{163} \right)$
C.
$\left( \frac{106}{166}, \frac{107}{165} \right)$
D.
$\left( \frac{108}{165}, \frac{28}{41} \right)$

Check Your Answer

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

Numerically Greatest Term|Arithmetic Mean Inequality|Non-Differentiable Extrema|Approximation via Differentials

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of discrete optimization and binomial expansion properties.

The primary trap lies in the non-differentiable nature of the discrete sequence index.

Students often assume the maximum occurs at the nearest integer to the continuous critical point ( $108.4 \to 108$ ).

While true here, rigorous verification requires second-order binomial approximation (differentials), as the first-order margin is extremely tight ( $\sim 10^{-4}$ ) and the cross-terms are negative.

A secondary trap is the NGT index shift: the $M^{\text{th}}$ term corresponds to $r = M-1$ .

Forgetting this shifts the entire inequality, leading directly to Option B.

Quick Facts About This Question

Question Information

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