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Question #41823

Difficulty: 8/10 | 🟢 Free

The Question

A subset of two distinct complex numbers $\{z_1, z_2\}$ is chosen uniformly at random from the set $S = \{ z \in \mathbb{C} : z^{15} = 1 \}$ . Let $w = (z_1 + 1)(z_2 + 1)$ . What is the probability that the principal argument of $w$ lies in the interval $\left(0, \frac{\pi}{2}\right)$ ? \ (Note: The principal argument $\text{arg}(z)$ takes values in $(-\pi, \pi]$ ).

Options

A.
$\frac{16}{105}$
B.
$\frac{37}{105}$
C.
$\frac{74}{225}$
D.
$\frac{1}{3}$

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

Sampling With/Without Replacement|Locus Problems via Transformation|Euler's Form and Rotations|The Quadrant Adjustment Logic

Why This Question Is Tricky

Common Mistakes

This problem tests the 'Quadrant Adjustment Logic' in the Argand plane during transformations.

Students often use the standard polar form $z = e^{i\phi}$ with $\phi \in [0, 2\pi)$ and incorrectly assume $\text{arg}(z+1) = \phi/2$ .

However, for $\phi \in (\pi, 2\pi)$ , $\cos(\phi/2) < 0$ , which shifts the principal argument by $\pi$ .

By defining the sample space using $\theta \in (-\pi, \pi)$ , the modulus $2\cos(\theta/2)$ remains strictly positive, preserving the argument as exactly $\theta/2$ .

Failing to account for this branch cut behavior leads to a severe undercounting of the valid configurations (yielding 16/105 instead of 37/105).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Sampling With/Without Replacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free