22. To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

  $x^2+y^2-2\sqrt[]{2}x-6\sqrt[]{2}y+14=0$

 centre $\left(\sqrt[]{2},3\sqrt[]{2}\right)$

radius  ${\left({\left(\sqrt[]{2}\right)}^2+{\left(3\sqrt[]{2}\right)}^2-14\right)}^{1/2}$

$={\left(2+18-14\right)}^{1/2}=\left(\sqrt[]{6}\right)$

${\left(x-2\sqrt[]{2}\right)}^2+{\left(y-2\sqrt[]{2}\right)}^2=r^2$

centre  $\left(2\sqrt[]{2},2\sqrt[]{2}\right)$

OA = $\sqrt[]{{\left(2\sqrt[]{2}-\sqrt[]{2}\right)}^2+{\left(2\sqrt[]{2}-3\sqrt[]{2}\right)}^2}=\sqrt[]{2+2}=2$                  

r² = ${\left(\sqrt[]{6}\right)}^2+{\left(2\right)}^2=6+4=10$  

Let y = mx + c is the common tangent

$soc=\frac{1}{m}=\pm \frac{3}{2}\sqrt[]{1+m^2}\Rightarrow m^2=\frac{1}{3}$

so equation of common tangents will be

$y=\pm \frac{1}{\sqrt[]{3}}x\pm \sqrt[]{3}$

which intersects at Q (3, 0)

Major axis and minor axis of ellipse are 12 and 6. So eccentricity

$e^2=1-\frac{1}{4}=\frac{3}{4}$

and length of latus rectum

$=\frac{2b^2}{a}=3$

Hence

$\frac{\mathcal{l}}{e^2}=\frac{3}{3/4}=4$

$z^2+z+1=0\Rightarrow z=w,w^2$

$\left|{\displaystyle \sum _{n=1}^{15}{\left(z^n+{\left(-1\right)}^n\frac{1}{z^n}\right)}^2}\right|=\left|{\displaystyle \sum _{n=1}^{15}\left(z^{2n}+\frac{1}{z^{2n}}+2{\left(-1\right)}^n\right)}\right|=\left|{\displaystyle \sum _{n=1}^{15}{w}^{2n}+\frac{1}{w^{2n}}+2{\left(-1\right)}^n}\right|$

$=\left|\frac{w^2\left(1-1\right)}{1-w^2}+\frac{\frac{1}{w^2}\left(1-1\right)}{1-\frac{1}{w^2}}-2\right|=2$

This is a Fill in the blanks Type Questions as classified in NCERT Exemplar

$\begin{array}{l}\left(i\right)-4x\ge 12\Rightarrow x\le -3\\ Hence,thefilleris\left(\le \right)\\ \left(ii\right)If-\frac{3}{4}x\le -3\Rightarrow x\ge 3x\times \frac{4}{3}\Rightarrow x\ge 4\\ Hence,thefilleris\left(\ge \right)\\ \left(iii\right)If\frac{2}{x+2}>0\Rightarrow x>-2\\ Hence,thefilleris\left(>\right)\\ \left(iv\right)Ifx>-5\Rightarrow 4x>-20\\ Hence,thefilleris\left(>\right)\\ \left(v\right)Ifx>yandz<0,then\\ xz<yz\Rightarrow -xz>-yz\\ Hence,thefilleris\left(>\right)\\ \left(vi\right)Ifp>0andq<0,then\\ p-q>p\\ Hence,thefilleris\left(>\right)\\ \left(vii\right)If\left|x+2\right|>5then\\ x+2<-5orx+2>5\\ \Rightarrow x<-5-2orx>5-2\\ \Rightarrow x<-7orx>3\\ So,x\in \left(-\infty ,-7\right)\cup \left(3,\infty \right)\\ Hence,thefilleris\left(<\right)or\left(>\right)\\ \left(viii\right)If-2x+1\ge 9then\\ \Rightarrow -2x\ge 9-1\Rightarrow -2x\ge 8\Rightarrow 2x\le -8\Rightarrow x\le -4\\ Hence,thefilleris\left(\le \right)\end{array}$

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

$\begin{array}{l}\left(i\right)Ifx<yandb<0\\ \Rightarrow \frac{x}{b}>\frac{y}{b}\\ Hence,statement\left(i\right)isFalse.\\ \left(ii\right)Ifx,y>0thenx>0,y>0orx<0,y<0\\ Hence,statement\left(ii\right)isFalse.\\ \left(iii\right)Ifxy>0thenx<0,andy<0\\ Hence,statement\left(iii\right)isTrue.\\ \left(iv\right)Ifxy<0thenx<0,y>0orx>0,y<0\\ Hence,statement\left(iv\right)isFalse.\\ \left(v\right)Ifx<-5andx<-2\Rightarrow x\in \left(-\infty ,-5\right)\\ Hence,statement\left(v\right)isTrue.\\ \left(vi\right)Ifx<-5andx>2thenxhasnovalue.\\ Hence,statement\left(vi\right)isFalse.\\ \left(vii\right)Ifx>-2andx<9\Rightarrow x\in \left(-2,9\right)\\ Hence,statement\left(vii\right)isTrue.\\ \left(viii\right)If\left|x\right|>5thenx<-5orx>5\\ \Rightarrow x\in \left(-\infty ,-5\right)\cup \left(5,\infty \right)\\ Hence,statement\left(viii\right)isFalse.\\ \left(ix\right)If\left|x\right|\le 4then-4\le x\le 4\\ \Rightarrow x\in \left[-4,4\right]\\ Hence,statement\left(ix\right)isTrue.\\ \left(x\right)Thegivengraphrepresentsx\le 3\\ Hence,statement\left(x\right)isFalse.\\ \left(xi\right)Thegivengraphrepresentsx\ge 0\\ Hence,statement\left(xi\right)isTrue.\\ \left(xii\right)Thegivengraphrepresentsy\ge 0\\ Hence,statement\left(xii\right)is\text{\&thins}\end{array}$