Appeared in: 2024 (5 sets)
Assertion (A) : Domain of $y = \cos^{-1}(x)$ is $[-1, 1]$. Reason (R) : The range of the principal value branch of $y = \cos^{-1}(x)$ is $[0, \pi] - \{\frac{\pi}{2}\}$.
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Official Evaluation Steps
Step 1 (Formula / Governing Law):
According to the NCERT Mathematics Textbook for Class 12 (Chapter 2: Inverse Trigonometric Functions), the function $y = \cos^{-1}(x)$ is defined such that:
- The domain is the set of all possible input values for $x$, which is $[-1, 1]$.
- The range of the principal value branch is the set of all possible output values for $y$, which is $[0, \pi]$.
Mark Allocation: 0.5 Mark for Assertion analysis, 0.5 Mark for Reason analysis.
Step 2 (Step-by-Step Working):
- Analysis of Assertion (A): The domain of $y = \cos^{-1}(x)$ is indeed $[-1, 1]$. Therefore, Assertion (A) is True.
- Analysis of Reason (R): The range of the principal value branch of $y = \cos^{-1}(x)$ is defined as $[0, \pi]$. The set $[0, \pi] - \{\frac{\pi}{2}\}$ is the range of the principal value branch of $y = \sec^{-1}(x)$, not $\cos^{-1}(x)$. Therefore, Reason (R) is False.
Step 3 (Final Answer):
Since Assertion (A) is True but Reason (R) is False, the correct option is:
Assertion (A) is true but Reason (R) is false.