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Question #40316

Difficulty: 8/10 | 🟢 Free

The Question

Let $U = \{x \in \mathbb{Z} : -5 \leq x \leq 5\}$ be the universal set. Let $f: U \to U$ be a function defined by $f(x) = -|x|$ , and let $R_f$ denote the range of $f$ . Let $X = \{x \in U : x \text{ is an even integer}\}$ . Define set $B = (X \cup R_f) \cap (X \cup R_f^c) \cap R_f$ , where $R_f^c$ is the complement of $R_f$ in $U$ . Let $A = \{ x \in \mathbb{Z} : (x^2 - 9)(x^2 - 4x + 3) = (x^2 - 9)(x - 1) \}$ . Find the cardinality of the Cartesian product $A \times B$ .

Options

A.
6
B.
10
C.
12
D.
20

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

Set Identity Laws|Cartesian Product Cardinality|Codomain vs Range Distinction|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question tests two critical JEE Advanced traps: the 'Zero-Factor Cancellation' trap and the 'Codomain vs Range' distinction.

In determining set A, students often algebraically cancel the common factors $(x^2-9)$ and $(x-1)$ from both sides of the equation, which leads to the 'lost root' phenomenon and an incorrect cardinality of $n(A)=2$ or $n(A)=1$ .

In determining set B, students must apply distributive set identity laws to simplify the expression to $X \cap R_f$ .

A common mistake here is assuming the function is surjective (Range = Codomain), leading them to use $U$ instead of $R_f$ , which incorrectly yields $n(B)=5$ .

Falling for either or both traps leads directly to the incorrect options provided.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Identity Laws
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free