Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #1672
KNOWLEDGE BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined?
(A) $\tan^{-1}:R\rightarrow(-\frac{\pi}{2},\frac{\pi}{2})$
(B) $\sec^{-1}:R-(-1,1)\rightarrow[0,\pi]-\{\frac{\pi}{2}\}$
(C) $\cot^{-1}:R\rightarrow(0,\pi)$
(D) $\text{cosec}^{-1}:R-(-1,1)\rightarrow[-\frac{\pi}{2},\frac{\pi}{2}]$
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Detailed Solution

Step 1: Analyze the Principal Value Branch of $\tan^{-1}$

The function $\tan^{-1} x$ has a domain of $R$ and a range of $(-\frac{\pi}{2}, \frac{\pi}{2})$. Option (A) is correctly defined.

Step 2: Analyze the Principal Value Branch of $\sec^{-1}$

The function $\sec^{-1} x$ has a domain of $R - (-1, 1)$ and a range of $[0, \pi] - \{\frac{\pi}{2}\}$. Option (B) is correctly defined.

Step 3: Analyze the Principal Value Branch of $\cot^{-1}$

The function $\cot^{-1} x$ has a domain of $R$ and a range of $(0, \pi)$. Option (C) is correctly defined.

Step 4: Analyze the Principal Value Branch of $\text{cosec}^{-1}$

The function $\text{cosec}^{-1} x$ has a domain of $R - (-1, 1)$ and a range of $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$. Option (D) states the range as $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which is incorrect because $\text{cosec} \theta$ is undefined at $\theta = 0$. Therefore, $0$ must be excluded from the range.

Final Answer: D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must recall the specific definitions of principal value branches and apply that knowledge to identify the incorrect mathematical statement among the given options.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the definitions and constraints of inverse trigonometric functions, which are fundamental concepts in the CBSE Class 12 curriculum.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's grasp of the NCERT textbook definitions regarding the domain and range of inverse trigonometric functions.

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