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Question #40490

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 - 2 e^{i\pi/4} z + 2i| = 1$ . What is the maximum value of $|z|$ for $z \in S$ ?

Options

A.
$\sqrt{2+\sqrt{2}}$
B.
$\sqrt{2} + 1$
C.
$\sqrt{3}$
D.
$\sqrt{2-\sqrt{2}}$

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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

Quadratic Equations with Complex Coefficients|Euler's Form and Rotations|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of complex quadratic loci and optimization.

The primary trap is algebraic: students reflexively apply the triangle inequality $|z| \le |z - e^{i\pi/4}| + |e^{i\pi/4}| \le \sqrt{2} + 1$ , assuming the maximum modulus of the shifted variable aligns with the shift vector.

However, the locus is a Lemniscate of Bernoulli, and the point furthest from the origin does not lie on the axis of the lemniscate.

By forcing a polar coordinate transformation and calculus-based optimization, the problem bypasses standard algebraic bounds, rewarding students who decompose the complex coefficients into geometric rotations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Quadratic Equations with Complex Coefficients
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free