Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #1490
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2026 AISSCE(Board Exam)65/1/1 MCQ SINGLE
If $2 \cos^{-1} x = y$, then
(A) $0 \leq y \leq \pi$
(B) $-\pi \leq y \leq \pi$
(C) $0 \leq y \leq 2\pi$
(D) $-\pi \leq y \leq 0$
Prev Finish
Correct Answer: C

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Detailed Solution

Step 1: Determine the range of $\cos^{-1} x$

The range of the inverse cosine function, $\cos^{-1} x$, is $[0, \pi]$. This means that $0 \leq \cos^{-1} x \leq \pi$.

Step 2: Determine the range of $2 \cos^{-1} x$

Given that $2 \cos^{-1} x = y$, we need to find the range of $y$. Since $0 \leq \cos^{-1} x \leq \pi$, we can multiply the inequality by 2: $$2(0) \leq 2 \cos^{-1} x \leq 2(\pi)$$ $$0 \leq y \leq 2\pi$$

Final Answer: $0 \leq y \leq 2\pi$

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires the student to understand the range of the inverse cosine function and apply a simple transformation to find the range of the given expression.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of the range of inverse trigonometric functions and how it transforms under scalar multiplication.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the properties of inverse trigonometric functions, a standard topic in the syllabus.
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