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Question #40739

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all complex numbers $z$ that satisfy the system of equations: $(1-i)z + (1+i)\bar{z} = 4$ $z^3 - 6z^2 - 2z\bar{z} + 16z + 6\bar{z} - 16 = 0$ Find the sum of all elements in $S$ .

Options

A.
4
B.
6 + 2i
C.
4 - 2i
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

General Equation Transformations|Vieta’s Formulas in Complex Domain|Algebraic Manipulation of i|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize extraneous solutions when mapping real-plane geometry to the complex domain.

By substituting $\bar{z} = f(z)$ from a line's equation into a curve, students construct a polynomial to find intersection points.

The 'Lethal Collision' occurs because the substitution only guarantees that the complex line $w = f(z)$ intersects the curve, not that $w$ is the actual complex conjugate of $z$ .

Students who blindly apply Vieta's formulas to the resulting cubic will include the extraneous root $2+2i$ and incorrectly answer $6+2i$ .

They must verify each root against the original domain constraint ( $x+y=2$ ).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	General Equation Transformations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free