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Question #41096

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{z \in \mathbb{C} : z^{12} = 1\}$ . Three distinct complex numbers $z_1, z_2, z_3$ are chosen uniformly at random from $S$ . Let $E$ be the event that $z_1 z_2 z_3 = 1$ . If $\text{Arg}(z) \in (-\pi, \pi]$ denotes the principal argument of $z$ , find the conditional expected value $E[\text{Arg}(z_1) + \text{Arg}(z_2) + \text{Arg}(z_3) \mid E]$ .

Options

A.
$0$
B.
$\frac{4\pi}{19}$
C.
$\frac{2\pi}{19}$
D.
$\frac{6\pi}{19}$

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

Symmetry and Invariance|Roots of Unity Properties|Argument and Principal Value

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between algebraic argument summation and principal value constraints.

Students commonly fall into the trap of assuming that if the product of complex numbers is 1, the sum of their principal arguments must be 0 due to rotational symmetry.

However, the principal argument function $\text{Arg}(z) \in (-\pi, \pi]$ introduces a branch cut discontinuity at the negative real axis.

By forcing students to evaluate the expected value over a constrained subset, the problem demands rigorous tracking of how many roots in each valid triplet cross the branch cut, breaking the naive symmetry and resulting in a non-zero expectation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Symmetry and Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free