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Question #40301

Difficulty: 8/10 | 🟢 Free

The Question

Let $D = [a, b)$ be the domain of the function $f(x) = \sqrt{ \log_{1/2} \left( \frac{ [\sin^{-1} x] + 2 }{ [\cos^{-1} x] } \right) }$ where $[ \cdot ]$ denotes the greatest integer function. Then the value of the integral $\int_{a}^{b} \frac{x}{\sqrt{1 - x^2}} \, dx$ is:

Options

A.
$\cos 1 - 1$
B.
$1 - \cos 1$
C.
$-1$
D.
$0$

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

Universal Set Relativity|Domain Constraint Intersections|Greatest Integer Function Traps|Compound Function Boundaries

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of domain constraints with the restricted ranges of inverse trigonometric functions (Universal Set Relativity).

Students often solve the algebraic inequality involving the Greatest Integer Function (GIF) but fail to intersect it with the inherent bounds of $\sin^{-1}x$ and $\cos^{-1}x$ .

A lethal trap is neglecting the strict positivity required for the argument of the logarithm, which filters out $[ \sin^{-1} x ] = -2$ .

If missed, students incorrectly extend the domain to $[-1, 0)$ , leading to the trap answer $-1$ .

The disjointed nature of GIF outputs forces a precise case-by-case analysis of the complementary angles $u+v=\pi/2$ .

Quick Facts About This Question

Question Information

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