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Question #40493

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the roots of the cubic equation $z^3 - p z^2 + q z - 26 = 0$ , where $p, q \in \mathbb{R}$ . For each root $z_k$ , let $P_k$ be a point in 3D space with position vector $\vec{v}_k = \text{Re}(z_k) \hat{i} + \text{Im}(z_k) \hat{j} + \hat{k}$ . It is given that the volume of the parallelepiped formed by $\vec{v}_1, \vec{v}_2, \vec{v}_3$ is zero. If the cubic equation has non-real roots and the distance between the points corresponding to these non-real roots is $6$ , find the value of $p + q$ .

Options

A.
21
B.
27
C.
33
D.
15

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

Conjugate Root Theorem|Scalar Triple Product Volume Logic|Distance Formula in 3D

Why This Question Is Tricky

Common Mistakes

This question tests the isomorphism between 3D vector constraints and complex plane geometry, specifically targeting the Conjugate Root Theorem.

The primary trap lies in the interpretation of the zero-volume condition (Scalar Triple Product = 0) for vectors with a constant $\hat{k}$ component.

Students often attempt to solve this using purely 3D algebraic distance formulas without realizing that $[\vec{v}_1, \vec{v}_2, \vec{v}_3] = 0$ geometrically forces the roots to be collinear in the 2D Argand plane.

Combined with the Conjugate Root Theorem (which dictates that non-real roots of a real-coefficient polynomial must be conjugates), this collinearity strictly requires the real root to share the same real part as the complex roots ( $\alpha = \beta$ ).

Failing to synthesize these domains leaves students lost in underdetermined 3D distance equations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Conjugate Root Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free