Complex Numbers

Let $S$ be the set of all complex numbers $z$ satisfying $|z - 2 + i| \ge \sqrt5$. If the complex number $z_0$ is such that $\dfrac{1}{|z_0 - 1|}$ is the maximum of the set $\left\{\dfrac{1}{|z-1|} : z\in S\right\}$, then the principal argument of $\dfrac{4 - z_0 - \bar z_0}{z_0 - \bar z_0 + 2i}$ is

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set

$$\left\{|a + b\omega + c\omega^2|^2 : a, b, c \text{ distinct non-zero integers}\right\}$$

equals ______

For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument with $-\pi < \arg(z) \le \pi$. Then, which of the following statement(s) is (are) FALSE?

Let $s$, $t$, $r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ $(x, y \in \mathbb{R}, i = \sqrt{-1})$ of the equation $sz + t\bar{z} + r = 0$, where $\bar{z} = x - iy$. Then, which of the following statement(s) is (are) TRUE?

Let $a$, $b$, $x$ and $y$ be real numbers such that $a - b = 1$ and $y \ne 0$. If the complex number $z = x + iy$ satisfies $\mathrm{Im}\left(\frac{az + b}{z + 1}\right) = y$, then which of the following is(are) possible value(s) of $x$?

Let ω be a complex number such that $2\omega+1=z$ where $z=\sqrt{-3}$. If
$$\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k,$$
then k is equal to :

Let $z\in\mathbb C$, the set of complex numbers. Then the equation, $2|z+3i|-|z-i|=0$ represents :

The equation $\operatorname{Im}\left(\dfrac{iz-2}{z-i}\right)+1=0$, $z\in\mathbb C$, $z\ne i$ represents a part of a circle having radius equal to :

Let $z=\dfrac{-1+\sqrt3\,i}{2}$, where $i=\sqrt{-1}$, and $r,s\in\{1,2,3\}$. Let $P=\begin{bmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{bmatrix}$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is

Let $a,b\in\mathbb R$ and $a^2+b^2\ne0$. Suppose $S=\left\{z\in\mathbb C:z=\dfrac1{a+ibt},\ t\in\mathbb R,\ t\ne0\right\}$, where $i=\sqrt{-1}$. If $z=x+iy$ and $z\in S$, then $(x,y)$ lies on

A value of θ for which $\dfrac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary, is :

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $2\sqrt2$ units in the south-westwards direction. Then its new position in the Argand plane is at the point represented by :

Let z = 1 + ai be a complex number, a > 0, such that $z^3$ is a real number. Then the sum $1+z+z^2+\dots+z^{11}$ is equal to :

For any integer $k$, let $\alpha_k=\cos\left(\dfrac{k\pi}7\right)+i\sin\left(\dfrac{k\pi}7\right)$, where $i=\sqrt{-1}$. The value of the expression $\dfrac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}$ is

A complex number $z$ is said to be unimodular if $|z|=1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\dfrac{z_1-2z_2}{2-z_1\bar z_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a :

The largest value of r for which the region represented by the set $\{\omega\in\mathbb C/|\omega-4-i|\le r\}$ is contained in the region represented by the set $\{z\in\mathbb C/|z-1|\le|z+i|\}$, is equal to :

If $2+3i$ is one of the roots of the equation $2x^3-9x^2+kx-13=0$, $k\in\mathbb R$, then the real root of this equation :

If $z$ is a non-real complex number, then the minimum value of $\dfrac{\text{Im }z^5}{(\text{Im }z)^5}$ is :

The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x))=0$$
has

Let $z_k=\cos\left(\frac{2k\pi}{10}\right)+i\sin\left(\frac{2k\pi}{10}\right)$; $k=1,2,\ldots,9$.