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Question #40889

Difficulty: 8/10 | 🟢 Free

The Question

Let a point $P_n(x_n, y_n, z_n)$ in a 3D Cartesian coordinate system be defined by: $x_n = \frac{1}{n\sqrt{n}} \sum_{k=1}^n \sqrt{k} \cos\left(\frac{\pi k}{n}\right)$ $y_n = \frac{1}{n\sqrt{n}} \sum_{k=1}^n \sqrt{k} \sin\left(\frac{\pi k}{n}\right)$ $z_n = \frac{1}{n\sqrt{n}} \sum_{k=1}^n \sqrt{n-k}$ As $n \to \infty$ , the point $P_n$ approaches a limiting position $P$ . If $D$ is the distance of $P$ from the origin, which of the following intervals contains $D$ ?

Options

A.
$\left( \frac{2}{3}, \frac{2\sqrt{2}}{3} \right)$
B.
$\left[ \frac{2\sqrt{2}}{3}, 1 \right)$
C.
$\left[ 1, \frac{\sqrt{13}}{3} \right)$
D.
$\left( 0, \frac{2}{3} \right]$

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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

Distance Formula in 3D|Integral as the Limit of a Sum|Inequalities in Definite Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 'Integral as a Limit of a Sum', 3D coordinate geometry, and 'Inequalities in Definite Integrals'.

The primary trap lies in bounding the distance $D$ .

Students who apply the Cauchy-Schwarz inequality to the individual components $x_0$ and $y_0$ will obtain a loose upper bound ( $D^2 \le 17/18$ ), which fails to isolate the correct interval.

Similarly, applying the 3D vector magnitude inequality yields $D \le 1$ .

To succeed, the student must evaluate the $z$ -component exactly and recognize that the $x$ and $y$ components form a 2D vector (or complex number).

By applying the generalized triangle inequality for complex integrals, they obtain the strictly tighter bound ( $D^2 < 8/9$ ), successfully navigating the lethal layer of the problem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Distance Formula in 3D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free