Algebraic Equations in Complex Numbers

Let $S=\left\{z\in \mathrm{ℂ}:\overline{z}=i\left(z^2+Re\left(\overline{z}\right)\right)\right\}$. Then $\sum _{z\in S}|z{|}^2$ is equal to

Let $Z_1, {\overline{Z}}_1, Z_2, {\overline{Z}}_2, Z_3, {\overline{Z}}_3, Z_4, {\overline{Z}}_4, Z_5, {\overline{Z}}_5$ be the roots of the equation $x^{10}+{\left(13x-1\right)}^{10}=0$. If $M=\sum _{i=1}^5\frac{1}{{\left|Z_i\right|}^2}$, then the digit in the $10\text{'}s$ place of $M$ is

All the solutions of $z^{2018}+\frac{1}{z^{2018}}+z^{2016}+\frac{1}{z^{2016}}=0$ lie on the curve

Let ${\mathrm{z}}_1,{\mathrm{z}}_2,{\mathrm{z}}_3,{\mathrm{z}}_4\in C$ (none of ${\mathrm{z}}_1$ are zero and no two of ${\mathrm{z}}_3$ are equal for $i=1,2,3,4$) be such that

$z_1\left(z_1-z_2\right)\left(z_1-z_3\right)\left(z_1-z_4\right)=4$

$z_2\left(z_2-z_1\right)\left(z_2-z_3\right)\left(z_2-z_4\right)=4$

$z_3\left(z_3-z_1\right)\left(z_3-z_2\right)\left(z_3-z_4\right)=4$

$z_4\left(z_4-z_1\right)\left(z_4-z_2\right)\left(z_4-z_3\right)=4$

Then :

The area of polygon whose vertices are non-real roots of the equation $\overline{z}=i{z}^2$ is

If $S=\left\{z\in C:\overline{z}=i{z}^2\right\},$ then the maximum value of $|z-\sqrt{3}-i{|}^2$  on $S$ is

For all complex numbers $z$ of the form $1+i\alpha , \alpha \in R$, if $z^2=x+iy$ then which of the following is correct

${\left(\frac{1+\cos \frac{\pi }{8}-i\sin \frac{\pi }{8}}{1+\cos \frac{\pi }{8}+i\sin \frac{\pi }{8}}\right)}^{12}=$

$$\begin{gathered} \mathrm{Let} {\mathrm{x}}_1, {\mathrm{x}}_2, {\mathrm{x}}_3, {\mathrm{x}}_4 \mathrm{be} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{equation} 4{\mathrm{x}}^4 + 8{\mathrm{x}}^3 – 17{\mathrm{x}}^2 – 12\mathrm{x} + 9 = 0 \mathrm{and} \\ \left(4+{\mathrm{x}}_1^2\right) \left(4+{\mathrm{x}}_2^2\right) \left(4+{\mathrm{x}}_3^2\right) \left(4+{\mathrm{x}}_4^2\right) =\frac{125}{16}\mathrm{m}. \\ \mathrm{Then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{m} \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_}. \end{gathered}$$

Consider the following two statements :
Statement I : For any two non-zero complex numbers z₁, z₂

_{$\left(\left|{\mathrm{z}}_1\right|+\left|{\mathrm{z}}_2\right|\right)\left|\frac{{\mathrm{Z}}_1}{\left|{\mathrm{Z}}_1\right|}+\frac{{\displaystyle {\mathrm{Z}}_2}}{{\displaystyle \left|{\mathrm{Z}}_2\right|}}\right|\le 2\left(\left|{\mathrm{Z}}_1\right|+\left|{\mathrm{Z}}_2\right|\right)$and}

_{Statement II : If x, y, z are three distinct complex
numbers and a, b, c are three positive real numbers
such that $\frac{\mathrm{a}}{\left|\mathrm{y}-\mathrm{z}\right|}=\frac{{\displaystyle \mathrm{b}}}{{\displaystyle \left|\mathrm{y}-\mathrm{x}\right|}}=\frac{{\displaystyle \mathrm{c}}}{{\displaystyle \left|\mathrm{x}-\mathrm{y}\right|}},$ then}

_{$\frac{{\mathrm{a}}^2}{\mathrm{y}- \mathrm{z}}+\frac{{{\displaystyle \mathrm{b}}}^2}{\mathrm{z}- \mathrm{x}}+\frac{{{\displaystyle \mathrm{c}}}^2}{\mathrm{x}- \mathrm{y}}=1.$}

_{Between the above two statements,}

If ${\mathrm{\alpha }}_1$ , ${\mathrm{\alpha }}_2$ , ${\mathrm{\alpha }}_3$ , ${\mathrm{\alpha }}_4$ , ${\mathrm{\alpha }}_5$ , ${\mathrm{\alpha }}_6$ are the roots of the equation 7z⁶ + 6z⁵ + 5z⁴ + 4z³ + 3z² + 2z + 1 = 0, where z is a complex number then the value of $\sum _{i=1}^6\frac{1}{1-{\alpha }_i}$ is equal to

यदि ${\mathrm{\alpha }}_1$ , ${\mathrm{\alpha }}_2$ , ${\mathrm{\alpha }}_3$ , ${\mathrm{\alpha }}_4$ , ${\mathrm{\alpha }}_5$ , ${\mathrm{\alpha }}_6$ समीकरण 7z⁶ + 6z⁵ + 5z⁴ + 4z³ + 3z² + 2z + 1 = 0 के मूल हैं, जहाँ z एक सम्मिश्र संख्या है, तब $\sum _{i=1}^6\frac{1}{1-{\alpha }_i}$ का मान है

If $x=2+5i,$ then the value of $x^3-5x^2+33x-19$ is

If $\mathrm{z}=\mathrm{x}+\mathrm{i}\mathrm{y},\mathrm{ }{\mathrm{z}}^{1/3}=\mathrm{a}\mathrm{ }-\mathrm{ }\mathrm{i}\mathrm{b},\mathrm{ }\mathrm{a}\ne \mathrm{ }\pm \mathrm{b}\mathrm{a},\mathrm{ }\mathrm{b}\ne \mathrm{ }0$ then $\frac{\mathrm{x}}{\mathrm{a}}-\frac{\mathrm{y}}{\mathrm{b}}=\mathrm{k}\left({\mathrm{a}}^2-\mathrm{ }{\mathrm{b}}^2\right)$ , where $\mathrm{k}=\_\_\_\_\_\_\_$

If the complex number $z$ satisfy the equation $\left(i-z\right)\left(1+2i\right)+\left(1-iz\right)\left(3-4i\right)=1+7i,$ then $z+\overline{z}+z\overline{z}$ is equal to -

If $z=4+i\sqrt{7},$ then value of $z^3- 4z^2- 9z+91$ equals

The solutions of the equation ${\mathrm{z}}^4+4{\mathrm{z}}^3\mathrm{i}-6{\mathrm{z}}^2-\mathrm{ }4\mathrm{z}\mathrm{i}\mathrm{ }-\mathrm{ }\mathrm{i}=0$ are the vertices of a convex polygon in the complex plane. The area of the polygon is :

If complex number $z$ and its conjugate $\overline{z}$ satisfy $z\overline{z}+2i \left(z+\overline{z}\right)=12+8i$ then value of $z$ is

If $x=\left(\frac{1+i}{2}\right), \left(\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i=\sqrt{-1}\right)$, then the expression $2x^4-2x^2+x+3$ equals

If $i{z}^4+1=0$, then $z$ can take the value

The number of the solutions of the equation $z^2+\overline{z}=0$ is