Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $\mathrm{g}: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by
$$
f(\mathrm{x})=\left\{\begin{array}{ll}
\mathrm{x}|\mathrm{x}| \sin \left(\frac{1}{\mathrm{x}}\right), & \mathrm{x} \neq 0, \
0, & \mathrm{x}=0,
\end{array} \text { and } g(\mathrm{x})= \begin{cases}1-2 \mathrm{x}, & 0 \leq \mathrm{x} \leq \frac{1}{2}, \
0, & \text { otherwise } .\end{cases}\right.
$$
Let $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in \mathbb{R}$. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ by
$$
h(\mathrm{x})=\mathrm{a} f(\mathrm{x})+b\left(\mathrm{~g}(\mathrm{x})+\mathrm{g}\left(\frac{1}{2}-\mathrm{x}\right)\right)+\mathrm{c}(\mathrm{x}-\mathrm{g}(\mathrm{x}))+\mathrm{dg}(\mathrm{x}), \mathrm{x} \in \mathbb{R}
$$
Match each entry in List-I to the correct entry in List-II.

| List-I | | List-II | |
| --- | --- | --- | --- |
| (P) | If $\mathrm{a}=0, \mathrm{~b}=1, \mathrm{c}=0$ and $\mathrm{d}=0$, then | (1) | $h$ is one-one. |
| (Q) | If $\mathrm{a}=1, \mathrm{~b}=0, \mathrm{c}=0$ and $\mathrm{d}=0$, then | (2) | $h$ is onto. |
| (R) | If $\mathrm{a}=0, \mathrm{~b}=0, \mathrm{c}=1$ and $\mathrm{d}=0$, then | (3) | $h$ is differentiable on $\mathbb{R}$. |
| (S) | If $\mathrm{a}=0, \mathrm{~b}=0, \mathrm{c}=0$ and $\mathrm{d}=1$, then | (4) | the range of $h$ is [0,1]. |
| | | (5) | the range of $h$ is \{0,1\}. |

The correct option is :

Accepted Answer

(C) $(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(3) \quad(\mathrm{R}) \rightarrow(2) \quad(\mathrm{S}) \rightarrow(4)$

Answer

(A) $(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(3) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(4)$

Answer

(B) $(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(4) \quad(\mathrm{S}) \rightarrow(3)$

Answer

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

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Continuity and Differentiability

Solution

More JEE Advanced questions on Continuity and Differentiability

List-I		List-II
(P)	If $a = 0, b = 1, c = 0$ and $d = 0$ , then	(1)	$h$ is one-one.
(Q)	If $a = 1, b = 0, c = 0$ and $d = 0$ , then	(2)	$h$ is onto.
(R)	If $a = 0, b = 0, c = 1$ and $d = 0$ , then	(3)	$h$ is differentiable on $R$ .
(S)	If $a = 0, b = 0, c = 0$ and $d = 1$ , then	(4)	the range of $h$ is [0,1].
		(5)	the range of $h$ is {0,1}.