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Question #41299

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_k = x_k + i y_k$ ( $k=1,2,3$ ) be three distinct complex numbers lying on the curve $\text{Im}(z) = (\text{Re}(z))^3$ . Let $\vec{v}_k = \hat{i} + x_k \hat{j} + y_k \hat{k}$ for $k=1,2,3$ . It is given that the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ are coplanar. Let $S$ be the set of all real numbers $r > 0$ such that there exist such points $z_1, z_2, z_3$ satisfying $|z_k - (4+3i)| \le r$ for all $k \in \{1,2,3\}$ . Which of the following represents the set $S$ ?

Options

A.
$(5, \infty)$
B.
$[5, \infty)$
C.
$(0, 5)$
D.
$[0, 5]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Factor Theorem in Determinants|Geometric Interpretation of Cross Product|Geometry of Complex Inequalities

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of vector algebra, determinant factorization, and complex geometry.

The primary trap lies in the boundary condition of the complex inequality.

Students must first translate the coplanarity of the given vectors into a determinant and use the Factor Theorem to derive the algebraic constraint $x_1+x_2+x_3=0$ .

The lethal trap occurs when minimizing the radius $r$ : students might incorrectly assume the minimum radius allows the points to collapse to the origin, yielding $r \ge 5$ .

However, because the points must remain distinct, at least one point must have a strictly negative real part, pushing its distance from $4+3i$ strictly greater than 5.

Thus, the boundary is open, making the correct set $(5, \infty)$ rather than $[5, \infty)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Determinants - Class 12
Concept	Factor Theorem in Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free