Question

Match the functions in List--I with their corresponding properties in List--II:

• A. A--II,\; B--III,\; C--IV,\; D--V
• B. A--V,\; B--I,\; C--II,\; D--III
• C. A--IV,\; B--II,\; C--I,\; D--V
• D. A--IV,\; B--III,\; C--V,\; D--II

Hint:

Equations relating \(\tan^{-1}(x)\) and \(\sin^{-1}(x)\) often convert to classical trigonometric identities once you set \(x=\sin\theta\).
- Carefully handle domain restrictions when inverting trig functions.

Answer 1

The Correct Option is D

(A) \(\sinh x\).
- Domain of \(\sinh x\) is all real numbers, and \(\sinh x\) is an \emph{odd} function.
- Its range is \(\mathbb{R}\). Hence it matches (IV)\;(\text{Range}={R},\;\text{odd function}). (B) \(\sec x\).
- \(\sec x = \frac{1}{\cos x}\). Since \(\cos(-x)=\cos x\), we get \(\sec(-x)=\sec x\), so \(\sec x\) is \emph{even}. Hence it matches (III)\;(\text{Even function}). (C) \(\tanh x\).
- Domain of \(\tanh x\) is all real numbers, and it is an \emph{odd} function.
- Its range is \((-1,1)\). Hence it matches (V)\;(\text{Range}=(-1,1),\;\text{odd function}). (D) \(\mathrm{cosech}^{-1} x\).
- The real inverse hyperbolic cosecant has domain \(\lvert x\rvert \ge 1\), i.e.\([\,1,\infty)\cup(-\infty,-1\,]\) for real values, and it is \emph{neither} even nor odd.
- Focusing on the principal branch often yields domain \([\,1,\infty)\). Hence it matches (II)\;(\text{Domain}=[\,1,\infty),\;\text{neither even nor odd function}).
Therefore the correct mapping is: \[ A \to (IV),\quad B \to (III),\quad C \to (V),\quad D \to (II). \]