Before You Begin

Question #40482

Difficulty: 8/10 | 🟢 Free

The Question

Let $z = \frac{(1 - i)^8 (\sqrt{3} - i)^6}{(-1 + i\sqrt{3})^3}$ . If $\theta = \text{Arg}(z)$ denotes the principal argument of $z$ , find the value of $2 \sin\left(\frac{\theta}{2}\right) + 4 \sin\left(\frac{\theta}{3}\right)$

Options

A.
$2 + 2\sqrt{3}$
B.
$-2 - 2\sqrt{3}$
C.
$-2 + 2\sqrt{3}$
D.
$2 - 2\sqrt{3}$

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

De Moivre’s Theorem Applications|Argument and Principal Value|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question tests the application of De Moivre's Theorem and the precise definition of the principal argument of a complex number.

Students often compute the raw sum of arguments (here, $-5\pi$ ) and either use it directly or incorrectly reduce it to $-\pi$ by assuming the principal range is $[-\pi, \pi)$ or by simple modulo arithmetic without checking boundary inclusion.

The standard principal argument range is $(-\pi, \pi]$ , meaning negative real numbers have an argument of $\pi$ .

Using $-5\pi$ or $-\pi$ leads to incorrect sign evaluations in the final trigonometric expression, perfectly mapping to the distractor options.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	De Moivre’s Theorem Applications
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free