Question

In the section, each question has some statements $(A,B,C,D,\dots )$ given in Column $-1$ and some statements $(p,q,r,s,t,\dots )$ in column$-2$.

Any given statement is column $-1$ can have correct matching with ONE OR MORE statement(s) in the column $-2$ for example, if for a given questions, statement $\mathrm{B}$ matches with the statements given in $q\&r$, then for that particular question against statement $\mathrm{B}$, darken the bubbles corresponding to $q\&r$ in the ORS. i.e., $r$ answer will be $q\&r$.

In the following $\left[x\right]$ denotes the greatest integer less than or equal to $x$.

Match the functions in Column $-1$ with the properties in the column $-2$.

Column 1	Column 2
$x\left|x\right|$	Continuous in $(-1,1)$
$\sqrt{\left|x\right|}$	Differentiable in $(-1,1)$
$x+\left[x\right]$	Strictly increasing in $(-1,1)$
$[x-1]+[x+1]$	Not differentiable at least one point in $(-1,1)$

Answer 1

Solution text:

Part $\mathrm{A}$:

$f\left(x\right)=x\left|x\right|=\{-x^2,$when $x\le 0$ & $x^2$, when $x\ge 0\}$

$f\left(x\right)$is continuous everywhere

$f\text{'}\left(x\right)=\{-2x,$when $x\le 0$& $2x$when $x\ge 0\}$

$f\left(x\right)$ is differentiable and increasing

Thus, $\left(A\right)-\left(\mathrm{i}\right),\left(\mathrm{ii}\right),\left(\mathrm{iii}\right)$

Part $\mathrm{B}$:

$f\left(x\right)=\sqrt{\left|x\right|}=\{-\sqrt{\left|x\right|},x\le 0\&\sqrt{\left|x\right|},x\ge 0\}$

Clearly, $f\left(x\right)$ is continuous but not differentiable at $x=0$

Thus, $\left(\mathrm{B}\right)-\left(\mathrm{i}\right)\&\left(\mathrm{iii}\right)$

Part $\mathrm{C}$

$f\left(x\right)=x+\left[x\right]=\{x-1$, when $-1<x<0\&x,$when$0\le x\le 1$

$f\left(x\right)$is not continuous at $x=0$ & not differentiable but strictly increasing

Thus, $\left(C\right)-\left(\mathrm{iii}\right)\&\left(\mathrm{iv}\right)$

Part $\mathrm{D}$

$f\left(x\right)=[x-1]+[x+1]$

$\{-2x,$when $x<-1$, $2,$when $-1\le x\le 1,2x$,when $x\ge 1\}$

The function is constant in $(-1,1)$, hence continuous & differentiable.

Thus, $\left(\mathrm{D}\right)-\left(\mathrm{i}\right)\&\left(\mathrm{ii}\right)$