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Question #40680

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the roots of the cubic equation $z^3 - 3(1+i)z^2 + pz + q = 0$ , where $p, q \in \mathbb{C}$ . If $z_1, z_2, z_3$ form an arithmetic progression and satisfy $|z_1| + |z_3| = 2\sqrt{2}$ , then the difference between the maximum and minimum possible values of $|p|$ is:

Options

A.
2
B.
4
C.
6
D.
8

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

Selection of AP Terms|Vieta’s Formulas in Complex Domain|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of algebraic sequences and complex geometry.

The primary trap lies in the assumption that complex coefficients in an AP will yield discrete algebraic solutions.

By setting the sum of the moduli of the roots equal to the modulus of their sum ( $|z_1| + |z_3| = |z_1 + z_3|$ ), the problem forces the use of the Triangle Inequality's equality condition.

Students who fail to recognize this geometric constraint will be unable to bound the common difference $d$ in the Argand plane, missing that $d$ must lie on a specific line segment, which is essential to finding the range of $|p|$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Selection of AP Terms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free