Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #936
KNOWLEDGE BASED
APPLY
2 Marks 2025 VSA
Find the domain of \(f(x)=\sin^{-1}(-x^{2})\).
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Step-by-Step Solution

  1. The domain of \(\sin^{-1}(x)\) is \([-1, 1]\). Therefore, for \(f(x) = \sin^{-1}(-x^2)\) to be defined, we must have \(-1 \leq -x^2 \leq 1\).
  2. Multiplying by -1, we get \(1 \geq x^2 \geq -1\). This can be split into two inequalities: \(x^2 \leq 1\) and \(x^2 \geq -1\).
  3. The inequality \(x^2 \geq -1\) is always true for all real numbers \(x\), since \(x^2\) is always non-negative.
  4. The inequality \(x^2 \leq 1\) can be rewritten as \(-1 \leq x \leq 1\).
  5. Therefore, the domain of \(f(x)\) is \([-1, 1]\).

Correct Answer: [-1, 1]

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply their knowledge of the domain of inverse trigonometric functions and algebraic manipulation to find the domain of the given function.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to execute a procedure, which involves understanding the domain of the arcsin function and then solving the inequality to find the allowed values of x.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the domain of inverse trigonometric functions, a concept covered in the textbook.

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