Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #588
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
If \(y=\sin^{-1}x\), \(-1 \le x \le 0\), then the range of y is
(A) \((\frac{-\pi}{2}, 0)\)
(B) \([\frac{-\pi}{2}, 0]\)
(C) \([\frac{-\pi}{2}, 0)\)
(D) \((\frac{-\pi}{2}, 0]\)
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Correct Answer: B
Explanation
Given \(y=\sin^{-1}x\) with \(-1 \le x \le 0\).

For the principal branch of the inverse sine function,
\[
-\frac{\pi}{2} \le \sin^{-1}x \le \frac{\pi}{2}.
\]

At the endpoints:
\[
x=-1 \;\Rightarrow\; y=\sin^{-1}(-1)=-\frac{\pi}{2},
\]
\[
x=0 \;\Rightarrow\; y=\sin^{-1}(0)=0.
\]

Since \(\sin^{-1}x\) is increasing on \([-1,1]\), the range is
\[
\boxed{-\frac{\pi}{2} \le y \le 0}.
\]

AI Tutor Explanation

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Detailed Solution

Step 1: Understanding the domain of \( \sin^{-1}x \)

The domain of the inverse sine function, \( \sin^{-1}x \), is \( -1 \le x \le 1 \). In this problem, we are given that \( -1 \le x \le 0 \).

Step 2: Evaluating \( \sin^{-1}x \) at the endpoints of the given domain

We need to find the values of \( \sin^{-1}x \) at \( x = -1 \) and \( x = 0 \). $$ \sin^{-1}(-1) = -\frac{\pi}{2} $$ $$ \sin^{-1}(0) = 0 $$

Step 3: Determining the range of \( y \)

Since \( \sin^{-1}x \) is a continuous and increasing function on the interval \( [-1, 0] \), the range of \( y = \sin^{-1}x \) for \( -1 \le x \le 0 \) is \( [-\frac{\pi}{2}, 0] \).

Final Answer: \([-\frac{\pi}{2}, 0]\)<\/span>

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires the student to comprehend the concept of the inverse sine function and its range over a specific domain.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of the inverse sine function and its range, rather than just recalling a fact or performing a procedure.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the syllabus content on inverse trigonometric functions and their properties.

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