Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 2: Level 2

If $z=1+i\sqrt{3}$, then $\left|\arg \left(z\right)\right|+\left|\arg \left(\overline{z}\right)\right|$ is equal to

Let $S=\left\{z\in C : z\left(i{z}_1-1\right)=z_1+1, \left|z_1\right|<1\right\}$. Then, for all $z\in S,$ which one of the following is always true?

If $z_1,z_2,z_3,\dots ,z_n$ lies on a circle $\left|z\right|=5$ then the value of $\left|z_1+z_2+\dots +z_n\right|-25\left|\frac{1}{z_1}+\frac{1}{z_2}+\dots +\frac{1}{z_n}\right|$ is equal to

Let $a$ be a complex number such that $\left|a\right|<1$ and $z_1, z_2, z_3,\dots$ be the vertices of a polygon such that $z_k=1+$$a+a^2+\dots \dots .+a^{k-1}$ for all $k=1, 2, 3,\dots$, then $z_1, z_2,\dots ..$ lie within the circle

Let $\lambda \in R,$ the origin and the non-real roots of $2z^2+2z+\lambda =0$ form the three vertices of an equilateral triangle in the Argand plane, then $\lambda$ is

Let $A=\left\{z\in C : \left|z\right|=25\right\}$ and $B=\left\{z\in C : \left|z+5+12i\right|=4\right\}$, then the minimum value of $|z-\omega |,$ for $z\in A$ and $\omega \in B,$ is

The complex number $z$ satisfying $|z+\overline{z}|+|z-\overline{z}|=2$ and $|iz-1|+|z-i|=2$ is/are

The complex number associated with the vertices $A, B, C$ of $\Delta ABC$ are $e^{i\theta }, \omega , \overline{\omega }$, respectively [where $\omega , \overline{\omega }$ are the complex cube roots of unity and $\cos \theta >Re\left(\omega \right)],$ then the complex number representing the point where angle bisector of $A$ meets the circumcircle of the triangle is