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Question #40486

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all complex numbers $z$ satisfying $|z - \sqrt{3} - i| = 2$ . Let $A$ be defined as $A = \left\{ z \in S : \left( \frac{z}{|z|} \right)^{12} = 1 \right\}$ Find the value of $\sum_{z \in A} |z - \sqrt{3} - i|^2$

Options

A.
16
B.
20
C.
24
D.
28

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

Locus Problems via Transformation|Position Vector Origin Dependence|Unit Vector Normalization

Why This Question Is Tricky

Common Mistakes

This question tests the 'Origin Dependence' trap in complex locus transformations.

Students frequently simplify the condition $\left( \frac{z}{|z|} \right)^{12} = 1$ algebraically to $z^{12} = |z|^{12}$ , which introduces an extraneous solution at $z=0$ .

Because the circle $S$ passes through the origin, $z=0$ satisfies the algebraic simplification but is a singularity (division by zero) in the original vector normalization $\frac{z}{|z|}$ .

Failing to exclude the origin leads to counting 6 points instead of 5, resulting in the trap answer of 24 instead of the correct 20.

Quick Facts About This Question

Question Information

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