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Question #40483

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{z \in \mathbb{C} : z^8 = 1\}$ . A multiset $M$ is formed by applying the transformation $f(z) = z^{\text{sgn}(\text{Im}(z))}$ to each $z \in S$ , where $\text{sgn}(x)$ denotes the signum function. Find the number of distinct linear arrangements of all elements of $M$ such that no two identical elements are adjacent.

Options

A.
864
B.
2220
C.
1656
D.
2520

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

Roots of Unity Properties|String Method (Adjacency)|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of complex number properties and advanced combinatorics (Inclusion-Exclusion Principle).

The primary trap lies in the discontinuity of the signum function at zero.

Students often incorrectly assume $\text{sgn}(\text{Im}(z))$ evaluates to $1$ or $-1$ for all roots, forgetting that the real roots $1$ and $-1$ have an imaginary part of exactly $0$ .

This oversight leads to $f(-1) = -1$ , resulting in a multiset with only 3 pairs instead of 4, yielding the trap answer 2220.

Recognizing that $z^0 = 1$ for both real roots correctly establishes the 4-pair multiset structure required for the String Method.

Quick Facts About This Question

Question Information

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