Complex Numbers and Quadratic Equations

${\left(\sqrt[]{3}+i\right)}^{100}=2^{100}{\left(\frac{\sqrt[]{3}}{2}+i\frac{1}{2}\right)}^{100}$

$=2^{100}{\left(cos\frac{\pi }{6}+isin\frac{\pi }{6}\right)}^{100}$

$=2^{99}\left(-1+i\sqrt[]{3}\right)=2^{99}\left(p+iq\right)$

$\therefore p=-1\&q=\sqrt[]{3}$              

$\therefore$ Required equation $x^2-\left(\sqrt[]{3}-1\right)x-\sqrt[]{3}=0$

Centre (0, 1)

Radius = $\sqrt[]{2}$

Given equation

$\Rightarrow \frac{cosx}{1+sinx}=\frac{2\left|sinx\right|cosx}{\left|co{s}^{2}x-si{n}^{2}x\right|}$

$\Rightarrow \left|1-2si{n}^{2}x\right|=2\left|sinx\right|\left(1+sinx\right)$

Put sin x = t

then equation,

$\left|1-2t^2\right|=2\left|t\right|\left(1+t\right),t\in \left(-1,1\right)-\left\{\pm \frac{1}{\sqrt[]{2}}\right\}$

Case I : For $\ge \frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

the equation has no solution

Case II : For $0\le t<\frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{\sqrt[]{5}-1}{4}\Rightarrow x=18°$ $\frac{\text{exception 25:}}{}$

Case III : $For\frac{-1}{\sqrt[]{2}}<t<0$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{-1}{2}\Rightarrow x=-30°$ $\frac{}{}$

Case IV : For $T<-\frac{1}{\sqrt[]{2}}$ $\frac{}{\text{exception 25:}}$

Equation t = $\frac{\sqrt[]{5}+1}{4}\Rightarrow$ $\frac{\text{exception 25:}}{}$ x = 54°

Sum of solutions = 18° - 30° - 54° = 66°

$S=21+{\displaystyle \sum _{k=1}^{21}{\left(z^k+\frac{1}{z^k}\right)}^3}=\sum {\left(z^k+{\overline{z}}^k\right)}^3=8\sum {\left(Re\left(z^k\right)\right)}^3$

$\left(As\left|z\right|=1\Rightarrow \frac{1}{z}=\overline{z}\right)$

$S=2{\displaystyle \sum _{k=1}^{21}cosk\pi +6}{\displaystyle \sum _{k=1}^{21}cos\frac{k\pi }{3}+21}$

= $2\left(-1\right)+6Re\left(\alpha +{\alpha }^2+{\alpha }^3\right)+21$

Where $\alpha =e^{i\left(2\pi /6\right)}$

$=2\left(-1\right)+6\left(\frac{1}{2}-\frac{1}{2}-1\right)+21$

$=-8+21=13$

Equation : 2x² – (6 + k)x + 4 + 3k = 0

D < 0 => 6  $-4\sqrt[]{2}<k<6+4\sqrt[]{2}$

Sum of integral values of K is 1+2+3+.+11 = 66

a + b = 1   $\alpha +\gamma =\frac{10}{3}$

$\alpha \beta =2\lambda$ $\alpha \gamma =9\lambda$

$\frac{\beta }{\gamma }=\frac{2}{9},$ $\beta -\gamma =1-\frac{10}{3}=-\frac{7}{3}$

$\Rightarrow \beta =\frac{2}{3}$ $\gamma =3$

$\alpha =\frac{1}{3},\lambda =\frac{1}{9}$

$\therefore \frac{\beta \gamma }{\lambda }=18$                                  

  $\frac{2}{-4}=\frac{-1}{2}$

y – 4 = 2 (x – 3)

y = 2x – 2

x² + (2x – 2)² = 25

  $5x^2-8x-21=0$

$z\left(\frac{-7}{5},\frac{-24}{5}\right)$              

$\left|z-\frac{1}{z}\right|=2$

${\left|z\right|}_{max}=?$

$\left|z-\frac{1}{2}\right|\ge \left|\left|z\right|-\frac{1}{\left|z\right|}\right|$

$2\ge \left|r-\frac{1}{r}\right|$

0 $\le r^2+2r-1\&r^2-2r-1\le 0$

$r=\frac{-2\pm \sqrt[]{8}}{2}$ $r=\frac{2\pm \sqrt[]{8}}{2}$

$=-1\pm \sqrt[]{2}$ $=1\pm \sqrt[]{2}$

r $\ge \sqrt[]{2}-1\&0\le r\le 1+\sqrt[]{2}$

$\sqrt[]{2}-1\le r\le \sqrt[]{2}+1$

${\overline{z}}_1=i{\overline{z}}_2,arg\left(\frac{z_1}{z_2}\right)=\pi$

$\Rightarrow z_1=-z_2,arg\left(\frac{z_1{z}_2}{{\left|z_2\right|}^2}\right)=\pi$

$\Rightarrow arg\left(z_1{z}_2\right)=\pi$

$20-\frac{\pi }{2}=\pi$

$\theta =\frac{3\pi }{4}$

$\theta -\frac{\pi }{2}=\frac{\pi }{4}$

$D=0\Rightarrow b^2=15a,\alpha =\frac{2b\pm 0}{2a}=\frac{b}{a}$  

  $\frac{b}{b^{\frac{2}{15}}}=\frac{15}{b}$

$\alpha +\beta =2b{\alpha }^2+{\beta }^2=4b^2-42$

$\alpha \beta =21$

$a{\left(x-\alpha \right)}^2=a\left(x^2-2\alpha x+{\alpha }^2\right)=a{x}^2-2bx+15$

a = 3                    a = -3

b = 7                    a = -7

a² + b² = 9 + 49 = 58