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Let \( \mathrm{f}_{1}: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{f}_{2}:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathrm{R}, \mathrm{f}_{3}:\left(-1, \mathrm{e}^{\frac{\pi}{2}}-2\right) \rightarrow \mathrm{R} \)
|JEE Advanced-2018| and \( f_{4}: R \rightarrow R \) be functions defined by
(P) The function \( f_{1} \) is List-II
(Q) The function \( \mathrm{f}_{2} \) is
(1) NOT continuous at \( x=0 \)
\( P \)
(i) \( \mathrm{f}_{1}(\mathrm{x})=\sin \left(\sqrt{1-\mathrm{e}^{-\mathrm{x}^{2}}}\right) \)
(R) The function \( \mathrm{f}_{3} \) is (3) differentiable at \( \mathrm{x}=0 \) and
W
2) continuous at \( x-0 \) and its derivative is \( \quad \) NOT continuous at \( x=0 \)
(S) The function \( f_{4} \) is
(4) differentiable at \( x=0 \) and
(ii) \( f_{2}(x)=\left\{\begin{array}{ll}\frac{|\sin x|}{\tan ^{-1} x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array}\right. \), where the inverse its derivative is continuous at \( x=0 \)
The correct option is :
trigonometric function \( \tan _{\mathrm{x}}^{-1} \) assumes values in
(A) \( \mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 3 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 4 \)
(B) \( \mathrm{P} \rightarrow 4 ; \mathrm{Q} \rightarrow 1 ; \mathrm{R} \rightarrow 2 ; \mathrm{S} \rightarrow 3 \)
\( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
(C) \( \mathrm{P} \rightarrow 4 ; \mathrm{Q} \rightarrow 2 ; \mathrm{R} \rightarrow 1 ; \mathrm{S} \rightarrow 3 \)
(D) \( \mathrm{P} \rightarrow 2 ; \mathrm{Q} \rightarrow 1 ; \mathrm{R} \rightarrow 4 ; \mathrm{S} \rightarrow 3 \)
(iii) \( f_{3}(x)=\left[\sin \left(\log _{c}(x+2)\right]\right. \), where for \( t \in \mathbb{R},[t] d e \) notes the greatest integer less than or equal to \( t \),
iv) \( f_{4}(x)=\left\{\begin{array}{cc}x^{2} \sin \left(\frac{1}{x}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right. \)
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