Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\ array ll x|x| ((1)/(x)), & x ≠ 0, 0, & x=0, array . and g(x)= cases 1-2 x, & 0 ≤ x ≤ (1)/(2) 0, & otherwise cases Let a, b, c, d ∈ ℝ . Define the function

Correct answer: C. (P) arrow(5) (Q) arrow(3) (R) arrow(2) (S) arrow(4) f(x)=\ array ccc x|x| (1)/(x) & ; & x ≠ 0 0 & ; & x=0 array g(x)=\ array ccc 1-2 x & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array .. g((1)/(2)-x)=\ array ccc 2 x & ; & 0 ≤ (1)/(2)-x ≤ (1)/(2) 0 & ; & otherwise array =\ array ccc 2 x & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array \ . g(x)+g((1)/(2)-x)=\ array lll 1 & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array \ (P) Now a=0, ~b=1, c=0, ~d=0 h(x)=g(x)+g((1)/(2)-x)=\ array lll 1 & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array . Hence Range of h(x) is \ 0,1\ (Q) aligned & a=1, ~b=0, c=0, ~d=0 & ~h(x)=f(x)=\ array cll x|x| (1)/(x) & ; & x ≠ 0 0 & ; & x=0 array . & RHD = _ x arrow 0 (x^2 (1)/(x)-0)/(x)=0 & LHD = _ x arrow 0 (-x^2 (1)/(x)-0)/(x)=0 aligned Hence h(x) is differentiable on R (R) aligned & a=0, ~b=0, c=1, ~d=0 & h(x)=x-g(x)=\ array ccc 3 x-1 & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array . aligned h(x) is ONTO (S) aligned & a=0, ~b=0, c=0, ~d=1 & ~h(x)=g(x)=\ array ccl 1-2 x & ; & 0 ≤ x ≤ (1)/(2) 0 & ; & otherwise array . aligned Range of h(x) is [0,1]

JEE Advanced 2024Paper 1

Exam: JEE Advanced
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Functions

JEE Advanced 2024 · Paper 1 · Mathematics · Functions

Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\...

Let

f: \mathbb{R} \rightarrow \mathbb{R}

and

g: \mathbb{R} \rightarrow \mathbb{R}

be functions defined by

f(x)=\left\{

\begin{array}{ll}x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0,\end{array}

\quad\right.

and

g(x)=

\begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2} \\ 0, & \text{ otherwise }\end{cases}$

Let

a, b, c, d \in \mathbb{R}

. Define the function

h: \mathbb{R} \rightarrow \mathbb{R}

by

h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R}$ Match each entry in List-I to the correct entry in List-II. $Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\ array ll x|x| ((1)/(x)), & x ≠ 0, 0, & x=0, array .$ The correct option is :

Options

A $(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(3) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(2)$
B $(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(4) \quad(\mathrm{S}) \rightarrow(3)$
C $(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(3) \quad(\mathrm{R}) \rightarrow(2) \quad(\mathrm{S}) \rightarrow(4)$
D $(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(3)$

Correct answer

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

Step-by-step solution

f(x)=\left\{

\begin{array}{ccc}x|x| \sin \frac{1}{x} & ; & x \neq 0 \\ 0 & ; & x=0\end{array}

\quad g(x)=\left\{

\begin{array}{ccc}1-2 x & ; & 0 \leq x \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}

\right.\right.

g\left(\frac{1}{2}-x\right)=\left\{

\begin{array}{ccc}2 x & ; & 0 \leq \frac{1}{2}-x \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}

=\left\{

\begin{array}{ccc}2 x & ; & 0 \leq x \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}

\right\}\right.

g(x)+g\left(\frac{1}{2}-x\right)=\left\{

\begin{array}{lll}1 & ; & 0 \leq x \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}

\right\}

(P) Now

\mathrm{a}=0, \mathrm{~b}=1, \mathrm{c}=0, \mathrm{~d}=0

\because \mathrm{h}(\mathrm{x})=\mathrm{g}(\mathrm{x})+\mathrm{g}\left(\frac{1}{2}-\mathrm{x}\right)=\left\{

\begin{array}{lll}1 & ; & 0 \leq \mathrm{x} \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}

\right.

$Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\ array ll x|x| ((1)/(x)), & x ≠ 0, 0, & x=0, array .$ Hence Range of

\mathrm{h}(\mathrm{x})

\{0,1\}

(Q)

\begin{aligned} & \mathrm{a}=1, \mathrm{~b}=0, \mathrm{c}=0, \mathrm{~d}=0 \\ & \mathrm{~h}(\mathrm{x})=\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cll}\mathrm{x}|\mathrm{x}| \sin \frac{1}{x} & ; & x \neq 0 \\ 0 & ; & x=0\end{array}\right. \\ & \text{ RHD }=\lim _{x \rightarrow 0} \frac{x^2 \sin \frac{1}{x}-0}{x}=0 \\ & \text{ LHD }=\lim _{x \rightarrow 0} \frac{-x^2 \sin \frac{1}{x}-0}{x}=0\end{aligned}

Hence

\mathrm{h}(\mathrm{x})

is differentiable on

\mathrm{R}

(R)

\begin{aligned} & \mathrm{a}=0, \mathrm{~b}=0, \mathrm{c}=1, \mathrm{~d}=0 \\ & h(x)=x-g(x)=\left\{\begin{array}{ccc}3 \mathrm{x}-1 & ; & 0 \leq x \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}\right.\end{aligned}

$Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\ array ll x|x| ((1)/(x)), & x ≠ 0, 0, & x=0, array .$

\therefore \mathrm{h}(\mathrm{x})

is ONTO (S)

\begin{aligned} & \mathrm{a}=0, \mathrm{~b}=0, \mathrm{c}=0, \mathrm{~d}=1 \\ & \mathrm{~h}(\mathrm{x})=\mathrm{g}(\mathrm{x})=\left\{\begin{array}{ccl}1-2 \mathrm{x} & ; & 0 \leq \mathrm{x} \leq \frac{1}{2} \\ 0 & ; & \text{ otherwise }\end{array}\right.\end{aligned}

$Let f: ℝ arrow ℝ and g: ℝ arrow ℝ be functions defined by f(x)=\ array ll x|x| ((1)/(x)), & x ≠ 0, 0, & x=0, array .$ Range of

h(x)

[0,1]

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