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Question #40637

Difficulty: 8/10 | 🟢 Free

The Question

Let $n$ be the unique positive integer such that the numerically greatest term in the expansion of $(1 + 5x)^n$ at $x=2$ is the $15^{\text{th}}$ term. Let $X$ be a random variable following a binomial distribution $B(n, p)$ . It is given that the variance of $X$ is $\frac{10}{3}$ and the probability mass function $P(X=k)$ attains its maximum value uniquely at $k=5$ . Find the value of $10 \frac{P(X=5)}{P(X=4)}$ .

Options

A.
11
B.
22
C.
33
D.
44

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

Numerically Greatest Term|Binomial Distribution Tail-Ends|Probability Mass Function (PMF)|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Binomial Theorem's Numerically Greatest Term (NGT) and the properties of the Binomial Distribution.

The first trap requires students to apply the strict inequality condition for a unique NGT to isolate the discrete parameter n=15.

The lethal collision occurs when solving for the probability p using the given variance, which yields a quadratic with two valid roots (p=1/3 and p=2/3).

Students who mechanically solve equations without checking distribution constraints will likely guess or choose the larger probability.

The rigorous path demands mapping the PMF maximum to the mode of the distribution floor((n+1)p) = 5, which uniquely filters out p=2/3.

Failing to verify this mode constraint leads directly to the trap answer 44.

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