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Question #40985

Difficulty: 8/10 | 🟢 Free

The Question

A dataset consists of an infinite number of subgroups indexed by $n \in \mathbb{N}$ . The weight of the $n$ -th subgroup is given by $N_n = \left| \alpha^n \right|$ , where $\alpha = \int_{0}^{1} x \ln x \, dx$ . The mean of the $n$ -th subgroup is $\bar{x}_n = \beta^n$ , where $\beta = \int_{0}^{\infty} x e^{-2x} \, dx$ . What is the combined mean of the entire dataset?

Options

A.
$\frac{1}{5}$
B.
$\frac{5}{17}$
C.
$\frac{1}{4}$
D.
$\frac{3}{17}$

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

Combined Mean of Subgroups|Infinite GP Convergence|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the concept of combined mean for an infinite number of subgroups, integrating calculus to find the parameters.

The core trap lies in the modulus sign dependency of the subgroup weights $N_n = |\alpha^n|$ .

Since $\alpha = -1/4$ , students who mechanically apply the infinite GP sum formula without resolving the absolute value will incorrectly compute $\sum N_n = \sum (-1/4)^n = -1/5$ and $\sum N_n \bar{x}_n = \sum (-1/16)^n = -1/17$ , leading to the trap answer $5/17$ .

This punishes blind formula application and enforces strict adherence to the definition of absolute values in sequences.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Combined Mean of Subgroups
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free