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