Complex Numbers

If $z$ is a complex number such that $|z|\ge2$, then the minimum value of $\left|z+\dfrac12\right|$ :

Let $w$ ($\operatorname{Im}w\ne0$) be a complex number. Then the set of all complex numbers $z$ satisfying the equation $w-\bar wz=k(1-z)$, for some real number $k$, is :

If $z_1$, $z_2$ and $z_3$, $z_4$ are 2 pairs of complex conjugate numbers, then $\arg\left(\dfrac{z_1}{z_4}\right)+\arg\left(\dfrac{z_2}{z_3}\right)$ equals :

Let $z\ne-i$ be any complex number such that $\dfrac{z-i}{z+i}$ is a purely imaginary number. Then $z+\dfrac1z$ is :

For all complex numbers $z$ of the form $1+i\alpha$, $\alpha\in\mathbb R$, if $z^2=x+iy$, then :

Let complex numbers $\alpha$ and $\dfrac1{\bar\alpha}$ lie on circles $(x-x_0)^2+(y-y_0)^2=r^2$ and $(x-x_0)^2+(y-y_0)^2=4r^2$, respectively. If $z_0=x_0+iy_0$ satisfies the equation $2|z_0|^2=r^2+2$, then $|\alpha|=$

Let $w=\dfrac{\sqrt3+i}2$ and $P=\{w^n:n=1,2,3,\ldots\}$. Further $H_1=\left\{z\in\mathbb C:\operatorname{Re}z>\frac12\right\}$ and $H_2=\left\{z\in\mathbb C:\operatorname{Re}z<\frac{-1}2\right\}$, where $\mathbb C$ is the set of all complex numbers. If $z_1\in P\cap H_1$, $z_2\in P\cap H_2$ and $O$ represents the origin, then $\angle z_1Oz_2=$

Let $\omega$ be a complex cube root of unity with $\omega\ne1$ and $P=[p_{ij}]$ be a $n\times n$ matrix with $p_{ij}=\omega^{i+j}$. Then $P^2\ne0$, when $n=$

Area of $S=$

$\displaystyle\min_{z\in S}|1-3i-z|=$

Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying (E), then the value of
$$\frac3{\omega^a}+\frac1{\omega^b}+\frac3{\omega^c}$$
is equal to

If $z$ is any complex number satisfying $|z-3-2i|\le2$, then the minimum value of $|2z-6+5i|$ is

Let $\omega\ne1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form
$$\begin{bmatrix}1&a&b\\\omega&1&c\\\omega^2&\omega&1\end{bmatrix},$$
where each of $a$, $b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Let $\omega=e^{i\pi/3}$, and $a$, $b$, $c$, $x$, $y$, $z$ be non-zero complex numbers such that
$$a+b+c=x,\qquad a+b\omega+c\omega^2=y,\qquad a+b\omega^2+c\omega=z.$$
Then the value of $\dfrac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is

Match the statements given in Column I with the values given in Column II

Match the statements given in Column I with the intervals/union of intervals given in Column II

Let $\omega$ be a complex cube root of unity with $\omega\ne1$. A fair die is thrown three times. If $r_1$, $r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t)z_1+tz_2$ for some real number t with $0<t<1$. If Arg(w) denotes the principal argument of a nonzero complex number w, then

Let $\omega$ be the complex number $\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3$. Then the number of distinct complex numbers z satisfying $\begin{vmatrix}z+1&\omega&\omega^2\\\omega&z+\omega^2&1\\\omega^2&1&z+\omega\end{vmatrix}=0$ is equal to