Complex Number

Let $\alpha , \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Rez=1,$ then it is necessary that

Given $z$ is a complex number with modulus $1.$ Then the equation ${\left[\frac{\left(1+ia\right)}{\left(1-ia\right)}\right]}^4=z$ in $\text{'}a\text{'}$ has

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t) z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If $\arg \left(w\right)$ denotes the principal argument of a non-zero complex number $w$, then

A complex number $z$ is rotated in anticlockwise direction by an angle $\alpha$ and we get $z^{\text{'}}$ and if the same complex number $z$ is rotated by an angle $\alpha$ in clockwise direction and we get $z^{"}$, then 

If $n$ is a natural number $\ge 2$, such that $z^n={\left(z+1\right)}^n$, then

$z_1$ and $z_2$ lie on a circle with center at the origin. The point of intersection $z_3$ of the tangents at $z_1$ and $z_2$ is given by

Given that the two curves $\arg \left(z\right)=\frac{\pi }{6}$ and $\left|z-2\sqrt{3} i\right|=r$ intersect in two distinct points, then 

Let $z_1, z_2, z_3$ be the three nonzero complex numbers such that $z_2\ne 1, a=\left|z_1\right|, b=\left|z_2\right|$ and $c=\left|z_3\right|$. Let $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$, then

If $z_1, z_2 \text{and} z_3$ are the vertices of an equilateral triangle $ABC$ such that $\left|z_1-i\right|=\left|z_2-i\right|=\left|z_3-i\right|,$ then $\left|z_1+z_2+z_3\right|$ equal to

If $z_1, z_2$ and $z_3$ are three points lying on the circle $\left|z\right|=2$, then minimum of ${\left|z_1+z_2\right|}^2+{\left|z_2+z_3\right|}^2+{\left|z_3+z_1\right|}^2$ is

The roots of the cubic equation ${\left(z+ab\right)}^3=a^3$, such that $a\ne 0,$ represent the vertices of a triangle of sides of length

If $\left|z_1\right|=15$ and $\left|z_2-3-4i\right|=5$, then

If $z_1$ is a root of the equation $a_0{z}^n+a_1{z}^{n-1}+...+a_{n-1}z+a_n=3,$ where $\left|a_i\right|<2$ for $i=0,1,...,n,$ then

The complex number $z$ satisfies $z+\left|z\right|=2+8i$. The value of $\left(\left|z\right|\right)$ is _________

If complex number $z\left(z\ne 2\right)$ satisfies the equation $z^2=4z+{\left|z\right|}^2+\frac{16}{{\left|z\right|}^3},$ then the value of ${\left|z\right|}^4$ is _____.

If the expression ${\left(1+ir\right)}^3$ is of the form of $s\left(1+i\right)$ for some real number $s$, where $r$ is also a real number, then sum of all possible values of $r$ is ______.

Modulus of non-zero complex number $z$ satisfying $\overline{z}+z=0$ and ${\left|z\right|}^2-4zi=z^2$ is _______.

Modulus of non-zero complex number $z$ satisfying $\overline{z}+z=0$ and ${\left|z\right|}^2-4zi=z^2$ is _______.

Let $z=9+bi,$where $b$ is a non-zero real number and $i^2=-1.$ If the imaginary part of $z^2 \& z^3$ are equal, then $b$ is _________.

If $x=\omega -{\omega }^2-2,$ then the value of $x^4+3x^3+2x^2-11x-6$ is (where $\omega$ is cube root of unity)__________.