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Question #41389

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - 1 = 0$ such that $\text{Im}(z_1) = 0$ and $\text{Im}(z_2) > \text{Im}(z_3)$ . Consider the complex-valued function $S(\theta) = 2z_1 + z_2 e^{i\theta} + z_3 e^{i2\theta}$ Using differentials, the approximate value of $|S(0.02)|$ is:

Options

A.
$1 + 0.01\sqrt{3}$
B.
$1 + 0.02\sqrt{3}$
C.
$1 - 0.01\sqrt{3}$
D.
$1 - 0.02\sqrt{3}$

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

Approximation via Differentials|Roots of Unity Properties|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the differential approximation of the modulus of a complex function, specifically targeting the geometric interpretation of the complex derivative.

A common trap is assuming that the differential of the modulus $d|S|$ is equal to the modulus of the differential $|S'(0)|d\theta$ , which ignores the phase coupling between the terms.

By forcing students to evaluate $\frac{d}{d\theta}|S(\theta)| = \text{Re}(S'(\theta)\overline{S(\theta)}/|S(\theta)|)$ , it exposes the error of treating complex numbers as scalars.

Students who fall for the trap will incorrectly compute $|S'(0)| = \sqrt{3}$ and choose the distractor $1 + 0.02\sqrt{3}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Approximation via Differentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free