Question

Let $f_{1}: \mathbb{R} \rightarrow \mathbb{R}, f_{2}:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R}, f_{3}:\left(-1, e^{\frac{\pi}{2}}-2\right) \rightarrow \mathbb{R}$ and $f_{4}: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by
(i) $f_{1}(x)=\sin \left(\sqrt{1-e^{-x^{2}}}\right)$,
(ii) $f_{2}(x)=\left\{\begin{array}{cl}\frac{|\sin x|}{\tan ^{-1} x} & \text { if } x \neq 0 \ 1 & \text { if } x=0\end{array}\right.$, where the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,
(iii) $f_{3}(x)=\left[\sin \left(\log _{e}(x+2)\right)\right]$, where, for $t \in \mathbb{R},[t]$ denotes the greatest integer less than or equal to $t$,
(iv) $f_{4}(x)=\left\{\begin{array}{cl}x^{2} \sin \left(\frac{1}{x}\right) & \text { if } x \neq 0 \ 0 & \text { if } x=0\end{array}\right.$.

| 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 $x=0$ and its derivative is continuous at $x=0$ |

The correct option is:

Accepted Answer

(D) $\mathbf{P} \rightarrow \mathbf{2} ; \mathbf{Q} \rightarrow \mathbf{1} ; \mathbf{R} \rightarrow \mathbf{4} ; \mathbf{S} \rightarrow \mathbf{3}$

Answer

(A) $\mathbf{P} \rightarrow \mathbf{2} ; \mathbf{Q} \rightarrow \mathbf{3} ; \mathbf{R} \rightarrow \mathbf{1} ; \mathbf{S} \rightarrow \mathbf{4}$

Answer

(B) $\mathbf{P} \rightarrow \mathbf{4 ;} \mathbf{Q} \rightarrow \mathbf{1} ; \mathbf{R} \rightarrow \mathbf{2 ; ~ S} \rightarrow \mathbf{3}$

Answer

(C) $\mathbf{P} \rightarrow \mathbf{4} ; \mathbf{Q} \rightarrow \mathbf{2} ; \mathbf{R} \rightarrow \mathbf{1} ; \mathbf{S} \rightarrow \mathbf{3}$

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Continuity and Differentiability

Solution

More JEE Advanced questions on Continuity and Differentiability

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 $x = 0$ and its derivative is continuous at $x = 0$