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Question #41711

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of distinct complex roots of the polynomial $P(z) = z^6 - z^5 + z^4 - z^3 + z^2 - z + 1$ . For any $z \in S$ , let $\arg z \in (-\pi, \pi]$ denote its principal argument. Define the set $T \subset S$ as: $T = \left\{ z \in S \ \Big| \ \sum_{n=1}^{\infty} \left( \frac{\arcsin(\sin(\arg z))}{\arg z} \right)^n \text{ diverges} \right\}$ A pair of distinct complex numbers $\{z, w\}$ is chosen uniformly at random from $T$ . Let $p$ be the probability that $\text{Im}(z + w) > 0$ . Find the value of: $p \sum_{z \in S} |z^2 - z + 1|^2$

Options

A.
2
B.
4
C.
6
D.
8

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

Conditional Probability Constraints|Sigma Notation Properties|Conjugate Root Theorem|Domain Constraint Intersections

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of complex roots, inverse trigonometric domain constraints, and conditional probability.

The primary trap lies in the divergence condition of the geometric series: students often incorrectly assume $\arcsin(\sin x) = x$ for all $x$ , which would lead them to include all roots in $T$ and obtain an incorrect probability of $2/5$ .

Furthermore, calculating the algebraic sum requires recognizing the roots as 7th roots of unity (excluding -1) and correctly applying Newton's sums or symmetric polynomial properties, where forgetting to exclude the extraneous root $z=-1$ yields a sum of 21 instead of 12.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Conditional Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free