Complex Numbers

Let $\bar z$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$(\bar z)^2 + \frac{1}{z^2}$$
are integers, then which of the following is/are possible value(s) of $|z|$?

Let $A=\{z\in\mathbb C:1\le|z-(1+i)|\le2\}$ and $B=\{z\in A:|z-(1-i)|=1\}$. Then, $B$ :

Let $S=\{z\in\mathbb C:|z-3|\le1$ and $z(4+3i)+\bar z(4-3i)\le24\}$. If $\alpha+i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha+\beta)$ is equal to ______.

Let a circle C in complex plane pass through the points $z_1=3+4i$, $z_2=4+3i$ and $z_3=5i$. If $z\,(\ne z_1)$ is a point on C such that the line through $z$ and $z_1$ is perpendicular to the line through $z_2$ and $z_3$, then $\arg(z)$ is equal to :

Let $z_1$ and $z_2$ be two complex numbers such that $\bar z_1=i\bar z_2$ and $\arg\left(\dfrac{z_1}{\bar z_2}\right)=\pi$. Then

Let $A=\left\{z\in\mathbb C:\left|\dfrac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in\mathbb C:\arg\left(\dfrac{z-1}{z+1}\right)=\dfrac{2\pi}3\right\}$.

Then $A\cap B$ is :

If $z^2+z+1=0$, $z\in\mathbb C$, then $\left|\displaystyle\sum_{n=1}^{15}\left(z^n+(-1)^n\frac1{z^n}\right)^2\right|$ is equal to ______.

The area of the polygon, whose vertices are the non-real roots of the equation $\bar z=iz^2$ is :

The value of $\cos\left(\dfrac{2\pi}7\right)+\cos\left(\dfrac{4\pi}7\right)+\cos\left(\dfrac{6\pi}7\right)$ is equal to :

The number of points of intersection of $|z-(4+3i)|=2$ and $|z|+|z-4|=6$, $z\in\mathbb C$, is

Let for some real numbers $\alpha$ and $\beta$, $a=\alpha-i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3\right)x+\bar ay=0$ has more than one solution, then $\dfrac\alpha\beta$ is equal to

The number of elements in the set $\left\{z=a+ib\in\mathbb C:a,b\in\mathbb Z\text{ and }1<|z-3+2i|<4\right\}$ is ______.

Let $A=\begin{pmatrix}1+i & 1\\ -i & 0\end{pmatrix}$ where $i=\sqrt{-1}$. Then, the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n=A\right\}$ is ______.

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar z=iz^2+z^2-z$ is equal to ______.

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2i-1)=0$. Then, the value of $\left|\alpha^8+\beta^8\right|$ is equal to :

Let $S=\left\{z\in\mathbf C:|z-2|\le1,\ z(1+i)+\bar z(1-i)\le2\right\}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at $z_1\in S$ and $z_2\in S$. If $5\left(|z_1|^2+|z_2|^2\right)=\alpha+\beta\sqrt5$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha+\beta$ is equal to ______.

Let $\alpha$ be a root of the equation $1+x^2+x^4=0$. Then the value of $\alpha^{1011}+\alpha^{2022}-\alpha^{3033}$ is equal to :

Let $\arg(z)$ represent the principal argument of the complex number $z$. Then, $|z|=3$ and $\arg(z-1)-\arg(z+1)=\dfrac\pi4$ intersect

The real part of the complex number $\dfrac{(1+2i)^8\cdot(1-2i)^2}{(3+2i)\cdot\overline{(4-6i)}}$ is equal to :

If $\alpha,\beta,\gamma,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :