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Question #73624

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the complex roots of the polynomial equation $z^3 - \alpha z^2 + \beta z - \gamma = 0$ , where $\alpha, \beta, \gamma \in \mathbb{C}$ . Consider the three vectors in $\mathbb{R}^3$ : $\vec{v}_1 = |z_1|\hat{i} + |z_2|\hat{j} + |z_3|\hat{k}$ $\vec{v}_2 = |z_2|\hat{i} + |z_3|\hat{j} + |z_1|\hat{k}$ $\vec{v}_3 = |z_3|\hat{i} + |z_1|\hat{j} + |z_2|\hat{k}$ The volume of the parallelepiped formed by $\vec{v}_1, \vec{v}_2, \vec{v}_3$ is zero. If $|\alpha| = 6$ , $\text{Re}(\alpha) = 3$ , $\text{Im}(\alpha) > 0$ , and the roots satisfy the condition $|\alpha| = |z_1| + |z_2| + |z_3|$ , find the value of $|\beta - \gamma|^2$ .

Options

A.
112
B.
144
C.
96
D.
128

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Scalar Triple Product Volume Logic|Vieta’s Formulas in Complex Domain|Triangle Inequality Mind Games|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of vector algebra and complex numbers.

The primary trap lies in the volume condition: students must recognize that the scalar triple product of the given vectors simplifies to a symmetric polynomial in $|z_i|$ , and setting it to zero forces $|z_1|=|z_2|=|z_3|$ via AM-GM equality.

The secondary trap is the Triangle Inequality equality condition $|\sum z_i| = \sum |z_i|$ , which forces the roots to be collinear and codirectional in the Argand plane.

Failing to combine these geometric constraints leads to an inability to determine that the roots are identical ( $z_1=z_2=z_3$ ), leaving the system seemingly underdetermined.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Triple Product Volume Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free