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Question #40467

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, \dots, z_9$ be the distinct roots of the equation $z^9 = 1$ . A $9 \times 9$ matrix $M = [m_{jk}]$ is constructed such that its entries are given by $m_{jk} = |z_j + i z_k|^6 - |z_j - i z_k|^6$ for $1 \le j, k \le 9$ . If $S = \sum_{1 \le j < k \le 9} |z_j - z_k|^2$ , find the value of $\det(M) + S$ .

Options

A.
0
B.
81
C.
162
D.
729

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

Geometrical Interpretation of Modulus|Roots of Unity Properties|Skew-Symmetric Determinant

Why This Question Is Tricky

Common Mistakes

This question exploits a student's tendency to overcomplicate algebraic structures involving complex moduli.

The trap lies in the intimidating 6th-degree modulus polynomial, which acts as a smokescreen.

Students may attempt to use properties of roots of unity to evaluate the determinant directly, leading to an algebraic dead end.

The key insight is recognizing that the modulus difference generates a purely imaginary relation, making the matrix skew-symmetric.

Because the matrix is of odd order (9x9), its determinant is trivially zero, bypassing the matrix entirely and reducing the problem to a standard summation of roots of unity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Geometrical Interpretation of Modulus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free