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Question #40333

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers in the domain of the real-valued function $f(x) = \sin^{-1}\left(\frac{|x^2 - 9x + 20| + |x^2 - 9x + 18|}{2}\right) + \ln(x-3) + \ln(6-x)$ The total number of relations that can be defined on the set $S$ is:

Options

A.
$2^4$
B.
$2^9$
C.
$2^{12}$
D.
$2^{16}$

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

Total Number of Relations|Real-Valued Function Domain|Modulus Function Breakdown|Domain Constraint Intersections

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to carefully intersect multiple domain constraints, specifically targeting the common trap of ignoring strict inequalities at boundary points.

Students often solve the modulus inequality $|A| + |B| \le |A-B|$ to find the closed intervals $[3, 4] \cup [5, 6]$ , yielding four integer points $\{3, 4, 5, 6\}$ .

The 'Lethal Collision' occurs when they fail to apply the strict domain constraints of the logarithmic functions ( $x > 3$ and $x < 6$ ), which exclude the boundary points 3 and 6.

Missing this reduces the set size from $n=4$ to $n=2$ , drastically changing the power set calculation from $2^{16}$ to $2^4$ .

This punishes rote calculation of relations without rigorous domain intersection.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Total Number of Relations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free