Embibe Experts Solutions for Exercise 9: Assignment

If a complex number $z$ satisfies ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{{\left|z\right|}^2+2\left|z\right|+6}{2{\left|z\right|}^2-2\left|z\right|+1}\right)<0$, then locus/region of the point represented by $z$ is

$S=1+\frac{{\omega }^2{x}^2}{2!}+\frac{{\omega }^4{x}^4}{4!}+...$ and 

$S^{\text{'}}=-x-\frac{{\omega }^2{x}^3}{3!}-\frac{{\omega }^4{x}^5}{5!}+.......$, where $\omega$ is non-real cube root of unity, then

The possible values of the real parameter $m$ for which $\left|z-\left(m^2-7m+11+i\right)\right|=1$ and $\arg \left(z\right)\ge \frac{\pi }{2}$ is satisfied by at least one $z$ is/are

If $\beta =e^{i\frac{2\pi }{7}}$, and $\left|\sum _{r=0}^{3n-1}{\beta }^{2^r}\right|=4\sqrt{2}$ then $n$ is equal to

_{If area of the region on complex plane by complex number $z$, such that $\frac{\pi }{4}\le \arg \left(\frac{z-2}{z+2}\right)\le \frac{3\pi }{4}$ is $a\pi +b$ then $\left(b-a\right)$ equals}

Let $a,b\in \mathrm{ℝ}$ and $a^2+b^2\ne 0$. Suppose $S=\left\{z\in \mathrm{ℂ}:Z=\frac{1}{a+ibt}, t\in \mathrm{ℝ},t\ne 0\right\}$, where $i=\sqrt{-1}$. If $z=x+iy$ and $z\in S$, then $\left(x,y\right)$ lies on

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

Let $\omega =e^{i\left(\frac{\mathrm{\pi }}{3}\right)}$ and $a,b,c,x,y,z$ be non-zero complex number such that

$a+b+c=x$

$a+b\omega +c{\omega }^2=y$

$a+b{\omega }^2+c\omega =z$

Then the value of $\frac{{\left|x\right|}^2+{\left|y\right|}^2+{\left|z\right|}^2}{{\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2}$ is