Exemplar Solution

$\begin{array}{l}WecanwrittenthegivensetsisRoasterform\\ R=\left\{\dots ,-8,-6,-4,-2,0,2,4,6,8,\dots \right\}\\ andT=\left\{\dots ,-18,-12,-6,0,6,12,18\dots \right\}\\ SinceeveryelementofTispresentinR.So,T\subset R.\\ Hence,thestatementis\text{'}True\text{'}.\end{array}$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat:sin\theta +cos\theta =1\\ Dividingbothsidesby\sqrt[]{{\left(1\right)}^2+{\left(1\right)}^2}=\sqrt[]{2}weget\\ \frac{1}{\sqrt[]{2}}sin\theta +\frac{1}{\sqrt[]{2}}cos\theta =\frac{1}{\sqrt[]{2}}\dots \left(i\right)\\ \Rightarrow sin\frac{\pi }{4}sin\theta +cos\frac{\pi }{4}cos\theta =\frac{1}{\sqrt[]{2}}\\ \Rightarrow cos\left(\theta -\frac{\pi }{4}\right)=cos\frac{\pi }{4}\\ \Rightarrow \theta -\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4},n\in Z\left[\begin{array}{l}Ifcos\theta =cos\alpha \\ \theta =2n\pi \pm \alpha \end{array}\right]\\ \Rightarrow \theta =2n\pi +\frac{\pi }{4}+\frac{\pi }{4}or\theta =2n\pi -\frac{\pi }{4}+\frac{\pi }{4}\\ \therefore \theta =2n\pi +\frac{\pi }{2}or\theta =2n\pi ,n\in Z\\ Hence,thegeneralvaluesof\theta are2n\pi +\frac{\pi }{2}and2n\pi .\end{array}$

$\begin{array}{l}True,Sinceeveryintegerisarationalnumber\\ \therefore Z\subset Q,SoQ\cup Z=Q.\end{array}$

$False,Sincethetwosetsdonotcontainthesameelements.$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat:tan\theta =\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha }\\ \Rightarrow tan\theta =\frac{tan\alpha -1}{tan\alpha +1}=\frac{tan\alpha -tan\frac{\pi }{4}}{1+tan\frac{\pi }{4}tan\alpha }\\ \Rightarrow tan\theta =tan\left(\alpha -\frac{\pi }{4}\right)\\ \therefore \theta =\alpha -\frac{\pi }{4}\Rightarrow cos\theta =cos\left(\alpha -\frac{\pi }{4}\right)\\ \Rightarrow cos\theta =cos\alpha cos\frac{\pi }{4}+sin\alpha sin\frac{\pi }{4}\\ \Rightarrow cos\theta =cos\alpha .\frac{1}{\sqrt[]{2}}+sin\alpha .\frac{1}{\sqrt[]{2}}\\ \Rightarrow \sqrt[]{2}cos\theta =cos\alpha +sin\alpha \\ \Rightarrow cos\alpha +sin\alpha =\sqrt[]{2}cos\theta .Henceproved.\end{array}$

$\begin{array}{l}M=\left\{1,2,3,4,5,6,7,8,9\right\}\\ andB=\left\{1,2,3,4,5,6,7,8,9\right\}\\ SinceMandBhassameelements\\ \therefore M=B,SoB\not\subset M\text{?}\text{?}is\text{'}False\text{'}\end{array}$

$Sinceeverysetisasubsetofitself.So,itis\text{'}True\text{'}.$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat:\frac{sin\left(x+y\right)}{sin\left(x-y\right)}=\frac{a+b}{a-b}\\ \Rightarrow \frac{sin\left(x+y\right)+sin\left(x-y\right)}{sin\left(x+y\right)-sin\left(x-y\right)}=\frac{a+b+a-b}{a+b-a+b}\left[Usingcomponendoanddividendotheorem\right]\\ \Rightarrow \frac{2sin\left(\frac{x+y+x-y}{2}\right)cos\left(\frac{x+y-x+y}{2}\right)}{2cos\left(\frac{x+y+x-y}{2}\right)sin\left(\frac{x+y-x+y}{2}\right)}=\frac{2a}{2b}\\ \Rightarrow \frac{sinx.cosy}{cosx.siny}=\frac{a}{b}\Rightarrow tanx.coty=\frac{a}{b}\\ \Rightarrow \frac{tanx}{tany}=\frac{a}{b}.Henceproved.\end{array}$

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l}Giventhat:f\left(x\right)=3x^2-1andg\left(x\right)=3+x\\ Sincef\left(x\right)=g\left(x\right)\left(given\right)\\ \Rightarrow 3x^2-1=3+x\Rightarrow 3x^2-x-4=0\\ \Rightarrow 3x^2-4x+3x-4=0\Rightarrow x\left(3x-4\right)+1\left(3x-4\right)=0\\ \Rightarrow \left(3x-4\right)\left(x+1\right)=0\Rightarrow 3x-4=0orx+1=0\\ \Rightarrow 3x=4orx=-1\\ \therefore x=\frac{4}{3}\\ Hence,thevalueofxare-1and\frac{4}{3}.\end{array}$