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Question #40980

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3, z_4$ be the roots of the polynomial equation $z^4 - \alpha z^3 + \beta z^2 - \gamma z + 1 = 0$ , where $\alpha, \beta, \gamma \in \mathbb{C}$ . Given that $|\alpha| = 4$ , the arithmetic mean of the moduli of the roots is $1$ , and $\text{Im}(\alpha) > 0$ . The value of $|\beta + \gamma|^2$ is:

Options

A.
52
B.
100
C.
20
D.
68

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

Properties of Arithmetic Mean|Vieta’s Formulas in Complex Domain|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

The core trap in this problem lies in the Geometrical Interpretation of Modulus, specifically the equality condition of the triangle inequality in the complex plane.

Students often attempt to use Vieta's formulas to find the sum of the roots ( $|\sum z_k| = 4$ ) and then incorrectly treat the complex roots as independent scalars when relating them to the sum of their moduli ( $\sum |z_k| = 4$ ).

The 'Lethal Collision' occurs if they fail to recognize that $|\sum z_k| = \sum |z_k|$ strictly forces all roots to lie on the same ray from the origin ( $z_k = r_k e^{i\theta}$ ).

Bypassing this geometric deduction prevents the subsequent application of the AM-GM inequality ( $\text{AM} = \text{GM} = 1$ ) required to prove that all roots are identical, leaving the polynomial unsolvable.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Properties of Arithmetic Mean
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free