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Question #40376

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the domain of the real-valued function $f(x) = \sin^{-1}\left( \frac{x^3 - 1}{2(x - 1)} - \frac{x^2}{2} \right) + \cos^{-1}\left( \frac{x^2 - x}{x} \right)$ Find the number of integers in the set $S$ .

Options

A.
0
B.
1
C.
2
D.
3

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

Real-Valued Function Domain|Indeterminate Forms|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' in the context of finding the domain of real-valued functions.

Students frequently simplify rational expressions like $(x^3-1)/(x-1)$ and $(x^2-x)/x$ by cancelling common factors before applying the domain constraints of the outer functions (like inverse sine and cosine).

By doing so, they incorrectly assume the simplified function is identical to the original, failing to exclude the removable discontinuities at $x=1$ and $x=0$ .

This leads them to an erroneous domain of $[0, 1]$ , which contains two integers, rather than the correct domain of $(0, 1)$ , which contains none.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Real-Valued Function Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free