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Question #40385

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_1 = \\{ x \\in \\mathbb{R} : \\sec^{-1}(\\text{sgn}(x^2 - 24x + 144)) = \\cos^{-1}(\\text{sgn}(x^2 - 24x + 144)) \\}$ and $S_2 = \\{ x \\in \\mathbb{R} : \\csc^{-1}(\\text{sgn}(x^2 - 24x + 205)) = \\sin^{-1}(\\text{sgn}(x^2 - 24x + 205)) \\}$ . Define the sets $A = (\\mathbb{Z} \\setminus S_1) \\cup (\\mathbb{Z} \\setminus S_2)$ , $B = \\{ y \\in \\mathbb{Z} : \\sin^{-1}(\\sin y) = y \\}$ , and $C = \\{ z \\in \\mathbb{Z} : \\cos^{-1}(\\cos z) = z \\}$ . Find the cardinality of the Cartesian product $A \\times B \\times C$ .

Options

A.
12
B.
0
C.
36
D.
\text{Infinite}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Cartesian Product Cardinality|Bijectivity and Inverse|Composition Inverse Identity Traps|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This problem disguises an infinite set logic puzzle as a finite Cartesian product.

The core trap lies in the domain restriction of $\\sec^{-1}(x)$ and $\\csc^{-1}(x)$ , which are undefined at $0$ .

Students often overlook that $\\text{sgn}(0)=0$ makes the inverse trigonometric function undefined, incorrectly assuming $S_1 = \\mathbb{R}$ and concluding $A = \\emptyset$ (leading to an answer of 0).

Furthermore, the bijectivity constraints for $\\sin^{-1}(\\sin x) = x$ and $\\cos^{-1}(\\cos x) = x$ restrict the integers to their principal domains, trapping students who assume these identities hold for all $\\mathbb{Z}$ , which would incorrectly yield an 'Infinite' 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