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Question #40665

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2 \in \mathbb{C}$ be non-zero complex numbers such that $|z_1|^2 + |z_2|^2 = 32$ and $\arg(z_1) = \arg(z_2)$ . Let $w_1 = z_1 e^{i\alpha}$ and $w_2 = z_2 e^{i\beta}$ , where $\alpha \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ and $\beta \in \left[\frac{5\pi}{6}, \frac{7\pi}{6}\right]$ . What is the maximum possible value of $|w_1 + w_2|$ ?

Options

A.
$4\sqrt{3}$
B.
$8$
C.
$4\sqrt{5}$
D.
$6\sqrt{2}$

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

Arithmetic Mean Inequality|Euler's Form and Rotations|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of algebraic inequalities (AM-GM) and geometric properties of complex numbers (Euler's form and rotations).

The primary trap is the 'Modulus Trap,' where students blindly apply the triangle inequality $|w_1 + w_2| \le |w_1| + |w_2|$ to conclude the maximum is $|z_1| + |z_2| \le 8$ .

This assumes the equality condition $\arg(w_1) = \arg(w_2)$ is achievable.

However, the domain constraints on the rotation angles $\alpha$ and $\beta$ strictly prevent collinearity, forcing the student to evaluate the exact phase difference and maximize the cosine term alongside the AM-GM bound.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Arithmetic Mean Inequality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free