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Question #59649

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in the Argand plane defined by $|z - c| + |z + c| \le D$ , where $c$ and $D$ are positive real constants with $D > 2c$ . Let $d$ be the minimum distance between any two parallel lines that completely enclose $S$ . Let $R = d/2$ . Consider the set $X$ consisting of the real parts of the roots of the equation $z^8 - R^8 = 0$ . What is the mean deviation of the elements of $X$ about their median?

Options

A.
$\frac{\sqrt{2}+1}{8} \sqrt{D^2 - 4c^2}$
B.
$\frac{\sqrt{2}+1}{4} \sqrt{D^2 - 4c^2}$
C.
$\frac{\sqrt{2}+1}{8} D$
D.
$0$

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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

Mean Deviation about Mean vs. Median|Distance Between Parallel Lines|Geometry of Complex Inequalities

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 intersection of complex geometry and statistics, specifically targeting the conceptual understanding of symmetric distributions and bounding regions.

The primary trap lies in the statistical calculation: students often misread 'real parts' and calculate the mean deviation of the complex moduli, which trivially yields 0 (Option D), failing to realize the distribution of real parts is symmetric about a median of 0.

Additionally, a geometric trap is embedded in finding the 'minimum distance between parallel lines enclosing S'.

Students may incorrectly use the major axis D (leading to Option C) instead of identifying that the minimum bounding distance corresponds to the minor axis of the ellipse defined by the complex inequality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Mean Deviation about Mean vs. Median
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free