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Question #40477

Difficulty: 8/10 | 🟢 Free

The Question

Let $z$ be a complex number such that $|z| = 1$ . Define the sum of distances from $z$ to the roots of the equation $w^4 - 1 = 0$ as $S(z) = |z - 1| + |z + 1| + |z - i| + |z + i|$ . If $M$ and $m$ are the maximum and minimum values of $S(z)$ respectively, find the value of $M^2 - m^2$ .

Options

A.
2
B.
4
C.
$4\sqrt{2}$
D.
8

Check Your Answer

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

Euler's Form and Rotations|Triangle Inequality Mind Games|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Modulus Function Breakdown' trap within a rotational geometry context.

Students often rely on geometric intuition, incorrectly assuming that the highly symmetric points (e.g., $z = e^{i\pi/4}$ ) minimize the sum of distances to the vertices of a square, analogous to a Fermat-Torricelli point.

However, expressing the distances using half-angle formulas reveals a sum of square roots.

The modulus operator creates non-differentiable 'kinks', and the resulting objective function is concave with respect to the coordinate transformations.

This concavity pushes the minimum to the boundary extremes (the vertices themselves, $t=1$ ) and the maximum to the symmetric points ( $t=\sqrt{2}$ ), completely inverting standard linear geometric intuition and demanding rigorous algebraic substitution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Euler's Form and Rotations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free