Complex Numbers

For any complex number $w=c+id$, let $arg(w)\in(-\pi,\pi]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying…

Let $\theta_{1},\theta_{2},\ldots,\theta_{10}$ be positive valued angles (in radian) such that $\theta_{1}+\theta_{2}+\cdots+\theta_{10}=2\pi$. Define the complex numbers…

Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i|\geq\sqrt{5}.$ If the complex number $z_{0}$ is such that $\frac{1}{|z_{0}-1|}$ is the maximum of the set ${\frac{1}{|z-1|}:z\in S},$ then the principal…

Let $w⁡=\frac{\sqrt{3}+\text{i}}{2}$ and $\text{P}={w⁡^{\text{n}}:\text{n}=1, 2, 3,\ldots}$.Further $\text{H}_{1}={\text{z}\in\text{C}:\text{Re z}>\frac{1}{2}}$ and $\text{H}_{2}={\text{z}\in\text{C}:\text{Re…

Match the statements of Column I with these in Column II. [Note : Here $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and the real…

A particle $P$ starts from the point $z_0=1+2 i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$ the…

Paragraph: Let $A, B, C$ be three sets of complex numbers as defined below. $A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.Question: The number of elements in…

A man walks a distance of 3 units from the origin towards the North-East $\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)$ direction. From there, he walks a distance of 4 units towards the North-West ($\mathrm{N}…

Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1=1+2 i$ and $z_2=3 i$ be two complex numbers, where $i=\sqrt{-1}$. Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_1\right|=2\left|x+i…

Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

Let $z$ be a complex number satisfying $|z|^{3}+2z^{2}+4z¯-8=0$, where $z¯$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.Match each entry in List-I to the correct entries in List-II.…

Let $z¯$ 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 $(z¯)^{2}+\frac{1}{z^{2}}$ are integers, then which of the following is/are…