Modulus and argument

Let $z_1=1+2i$, $z_2=2+i$, $\dfrac1{z_1}+\dfrac1{z_2}=\dfrac6w$, and $z=\dfrac{iw}{2-\bar w}$.

Consider the statements

(S$_1$) $|z|=\dfrac5{\sqrt{17}}$

(S$_2$) $\arg(z)+\arg(w)=\tan^{-1}\left(\dfrac53\right)$.

Then

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2 + z + 1| = 1$. Then which of the following statements is/are TRUE?

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

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?

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 :

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 :

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 $\omega=e^{i\pi/3}$, 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

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 $z$ be any point in $A\cap B\cap C$ and let $w$ be any point satisfying $|w-2-i|<3$. Then, $|z|-|w|+3$ lies between

If $|z|=1$ and $z\ne\pm1$, then all the values of $\dfrac{z}{1-z^2}$ lie on