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Question #40471

Difficulty: 8/10 | 🟢 Free

The Question

Let $x_1 = 1$ , $x_2 = \omega$ , $x_3 = \omega^2$ , and $x_4 = i$ , where $\omega = e^{i2\pi/3}$ . Let $M$ be a $4 \times 4$ matrix whose entries are given by $M_{jk} = (x_j - x_k)^3$ for $1 \le j, k \le 4$ . Find the principal argument of $\det(M)$ .

Options

A.
$\frac{\pi}{2}$
B.
$-\frac{\pi}{2}$
C.
0
D.
$\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Argument and Principal Value|Factor Theorem in Determinants|Skew-Symmetric Determinant

Why This Question Is Tricky

Common Mistakes

This question tests the evaluation of a skew-symmetric determinant using the Pfaffian and the Factor Theorem.

Students often attempt to expand the $4 \times 4$ determinant directly, leading to algebraic exhaustion.

The trap lies in recognizing that the Pfaffian is a cubic polynomial in $x_4$ with roots at the other three variables.

By leveraging the roots of unity, the polynomial simplifies drastically.

Finally, squaring the Pfaffian yields a purely imaginary determinant, forcing a careful check of the principal argument branch.

Quick Facts About This Question

Question Information

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