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Question #40446

Difficulty: 8/10 | 🟢 Free

The Question

Let $[ \cdot ]$ denote the greatest integer function. Find the solution set of the equation $\left[ \sin^{-1} x \right]^2 + \left[ \cos^{-1} x \right]^2 + \left[ \sin^{-1} x \right] \left[ \cos^{-1} x \right] = 3$

Options

A.
$[-\sin 1, \cos 2]$
B.
$(\cos 3, 0)$
C.
$[-\sin 1, \cos 2] \cup [\sin 1, 1]$
D.
$[-1, \cos 2]$

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

Extraneous Solution Rejection|Domain Restrictions|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reject extraneous solutions arising from algebraic manipulation of the Greatest Integer Function (GIF) applied to inverse trigonometric functions.

The trap lies in solving the quadratic equation for the GIF values to find multiple integer pairs (such as $(1,1)$ and $(-2,1)$ ) and blindly assuming they all yield valid $x$ -values.

Students must independently intersect the domain restrictions imposed by the principal branches of $\sin^{-1} x$ and $\cos^{-1} x$ for each pair.

Failing to recognize that the intervals for the extraneous pairs are disjoint leads to incorrect unions of intervals or boundary errors.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Extraneous Solution Rejection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free