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Question #40487

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - 5z^2 + 17z - 13 = 0$ . Let $M(z)$ be a $4 \times 4$ matrix whose entries are given by $m_{jk} = |w_j - w_k|^2$ for $1 \le j, k \le 4$ , where $w_1 = z$ , $w_2 = z_1$ , $w_3 = z_2$ , and $w_4 = z_3$ . Let $C$ be the locus of points $z \in \mathbb{C}$ such that $\det(M(z)) = 0$ . If $A$ is the area of the region bounded by $C$ , find the value of $\frac{A}{\pi}$ .

Options

A.
13
B.
16
C.
25
D.
36

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

Locus Problems via Transformation|Determinant Product Rule|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to connect determinant properties with geometric loci.

The primary trap lies in the expansion of the squared distance $|w_j - w_k|^2$ .

A student might incorrectly group the terms into a sum of three products, mistakenly concluding that the $4 \times 4$ matrix $M(z)$ is a product of $4 \times 3$ and $3 \times 4$ matrices.

This would lead to the erroneous conclusion that $\det(M(z)) = 0$ for all $z$ , implying the locus is the entire complex plane.

The correct approach requires recognizing the 4-term expansion, factoring $M(z)$ into two $4 \times 4$ matrices, and identifying that the vanishing determinant yields the standard determinantal equation of a circumcircle.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Locus Problems via Transformation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free