Relations and Functions

This is a True or False Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Here,m=kn(wherekisaninteger)\\ Ifk=1m=n,sozisreflexive.\\ Clearlyzisnotsymmetricbutzistransitive.\\ Hence,thestatementisfalse.\end{array}$

This is a True or False Type Question as classified in NCERT Exemplar

Sol:

This is a True or False Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat,f(x)=sin(3x+2),\forall x\in R\\ f(x)isnotone-onefunction.\\ Hence,thestatementisfalse.\end{array}$

This is a True or False Type Question as classified in NCERT Exemplar

Sol:

This is a Fill in the Blanks Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}f(x)=\left[4-{(x-7)}^3\right]\\ Lety=\left[4-{(x-7)}^3\right]\\ \Rightarrow {(x-7)}^3=4-y\\ \Rightarrow x-7={(4-y)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\Rightarrow x={(4-y)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+7\\ Hence,f^{-1}(x)={(4-y)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+7\end{array}$

This is a Fill in the Blanks Type Question as classified in NCERT Exemplar

This is a Fill in the Blanks Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{}\end{array}Giventhat, f=\{(1,2),(3,5),(4,1)\}andg=\{(2,3),(5,1),(1,3)\}$

$gof\left(1\right)=g\left\{f\left(1\right)\right\}=g\left(2\right)=3$

$gof\left(3\right)=g\left\{f\left(3\right)\right\}=g\left(5\right)=1$

$gof\left(4\right)=g\left\{f\left(4\right)\right\}=g\left(1\right)=3$

$\begin{array}{l}\therefore gof=\{(1,3),(3,1),(4,3)\}\end{array}$

$\begin{array}{l}\end{array}fog\left(2\right)=f\left\{g\left(2\right)\right\}=f\left(3\right)=5$

$fog\left(5\right)=f\left\{g\left(5\right)\right\}=f\left(1\right)=2$

$fog\left(1\right)=f\left\{g\left(1\right)\right\}=f\left(3\right)=5$

$\therefore fog=\{(2,5),(5,2),\left(1,5\right)\}$

This is a Fill in the Blanks Type Question as classified in NCERT Exemplar

Sol:

This is a Fill in the Blanks Type Question as classified in NCERT Exemplar

Sol:

This is a Objective Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhatf(x)=tanx\\ Letf(x)=y=tanx\Rightarrow x=ta{n}^{-1}y\\ \Rightarrow f^{-1}(x)=ta{n}^{-1}(x)\\ \Rightarrow f^{-1}(1)=ta{n}^{-1}(1)\\ \Rightarrow f^{-1}(1)=ta{n}^{-1}\left[tan\left(\frac{\pi }{4}\right)\right]=\frac{\pi }{4}\\ Hence,thecorrectansweris(a).\end{array}$