let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if & x List – I List – II (A) f _ 4 is (P) Onto but not one – one (B) f _ 3 is (Q) neither continuous

Correct answer: B. a-p;b-r;c-q;d-s; f _ 2 ( f _ 1 ) = \ cases x ^ 2 & , & x f _ 4 : R → [ 0, ∞ ) f _ 4 (x) = \ cases f _ 2 ( f _ 1 (x)) & , & x = \ cases x ^ 2 & , & x f _ 2 (x) is continuous and one-one. f _ 3 (x) is differentiable but not one-one. f _ 2 ( f _ 1 (x)) is neither one one nor continuous. f _ 4 (x) is onto but not one-one.

JEE Advanced 2014Paper 2

Exam: JEE Advanced
Year: 2014
Paper: Paper 2
Subject: Mathematics
Chapter: Functions

JEE Advanced 2014 · Paper 2 · Mathematics · Functions

let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R →...

let

{f}_{1}: R \to R, {f}_{2}: \left[0, \infty\right) \to R, {f}_{3}: R \to Rand{f}_{4}: R \to [ 0, \infty )

be defined by

{f}_{1}\left(x\right) = \left\{

\begin{cases}\left|x\right| & if & x < 0 \\ {e}^{x} & if & x \ge 0\end{cases}

\right; {f}_{2}\left(x\right) = {x}^{2}; {f}_{3}\left(x\right) = \left\{

\begin{cases}sin x & if & x < 0 \\ x & if & x \ge 0\end{cases}

\rightand{f}_{4}\left(x\right) = \left\{

\begin{cases}{f}_{2}\left({f}_{1}\left(x\right)\right) & if & x < 0 \\ {f}_{2}\left({f}_{1}\left(x\right)\right) - 1 & if & x \ge 0\end{cases}

\right)<br /> <table border="1"> <tbody> <tr> <td> </td> <td>List – I</td> <td> </td> <td>List – II</td> </tr> <tr> <td>(A)</td> <td>

{f}_{4}

is</td> <td>(P)</td> <td>Onto but not one – one</td> </tr> <tr> <td>(B)</td> <td>

{f}_{3}

is</td> <td>(Q)</td> <td>neither continuous nor one-one</td> </tr> <tr> <td>(C)</td> <td>

{f}_{2}

of

{f}_{1}

is</td> <td>(R)</td> <td>differentiable but not one-one</td> </tr> <tr> <td>(D)</td> <td>

{f}_{2}$ is (S) continuous and one-one $let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$ $let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$ $let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$ $let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$ $let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$

Options

Aa-q;b-r;c-s;d-p;
Ba-p;b-r;c-q;d-s;
Ca-q;b-s;c-r;d-p;
Da-q;b-p;c-r;d-s;

Correct answer

B. a-p;b-r;c-q;d-s;

Step-by-step solution

${f}_{2}\left({f}_{1}\right) = \left\{$ \begin{cases}{x}^{2} & , & x < 0 \\ {e}^{2x} & , & x \ge 0\end{cases} $\right) $ {f}_{4}: R \to [ 0, \infty ) $ $ {f}_{4}\left(x\right) = \left\{ $\begin{cases}{f}_{2}\left({f}_{1}\left(x\right)\right) & , & x < 0 \\ {f}_{2}\left({f}_{1}\left(x\right)\right) - 1 & , & x \ge 0\end{cases}$ \right)
$= \left\{$ \begin{cases}{x}^{2} & , & x < 0 \\ {e}^{2x} - 1 & , & x \ge 0\end{cases} $\right) <img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/image2-b02e0f52-9809-46c1-bd93-0d6a920101fb.jpg"/> <img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/image3-a37b2d3e-7ac2-4f93-a117-7d3a8b7b04fc.jpg"/>$ {f}_{2}\left(x\right) $is continuous and one-one. <img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/image4-66b7280a-9b26-4633-ab53-ea0fe347f513.jpg"/>$ {f}_{3}\left(x\right) $is differentiable but not one-one. <img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/image5-411eee2e-6be6-4460-8885-46e17ada6323.jpg"/>$ {f}_{2}\left({f}_{1} $\left(x\right)\right)$ is neither one one nor continuous.
$let f _ 1 : R → R, f _ 2 : [0, ∞) → R, f _ 3 : R → Rand f _ 4 : R → [ 0, ∞ ) be defined by f _ 1 (x) = \ cases |x| & if$

${f}_{4}\left(x\right)$ is onto but not one-one.

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