Before You Begin

Question #40327

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{ z \in \mathbb{Z}[i] \mid |z|^2 \le 5 \}$ and $B = \{ w \in \mathbb{Z}[i] \mid |w|^2 \le 5 \}$ , where $\mathbb{Z}[i]$ denotes the set of Gaussian integers. Let $S$ be the subset of the Cartesian product $A \times B$ defined by: $S = \left\{ (z, w) \in A \times B \mid \text{Im}(z) = \text{Re}(w) \text{ and } d(P, \Pi) \le \frac{1}{\sqrt{3}} \right\}$ where $d(P, \Pi)$ is the perpendicular distance from the point $P(\text{Re}(z), \text{Im}(z), \text{Im}(w))$ to the plane $\Pi: x+y+z=1$ . Find the cardinality of $S$ .

Options

A.
14
B.
41
C.
43
D.
93

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

Cartesian Product Cardinality|Locus Problems via Transformation|Distance from point to plane|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map complex number loci into a 3D Cartesian space (a Steinmetz solid intersection) and apply spatial constraints.

The primary trap lies in the distance formula: students frequently forget the denominator ( $\sqrt{3}$ ) when calculating the distance from a point to a plane, leading them to solve $|x+y+z-1| \le 1/\sqrt{3}$ .

Since $x,y,z$ are integers, this erroneously forces the strict equality $x+y+z=1$ , yielding exactly 14 solutions.

Furthermore, failing to link the sets via the shared imaginary/real component results in calculating the unconstrained Cartesian product cardinality.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Cartesian Product Cardinality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free