G Tewani Solutions for Chapter: Complex Numbers, Exercise 6: DPP

If $\left|z-1-i\right|=1,$then the locus of a point represented by the complex number $5\left(z-i\right)-6$ is a

Let  $z\in C$ and if $A= \left\{z, arg\left(z\right)=\frac{\pi }{4}\right\} B= \left\{z, arg(z-3-3i)=\frac{2\pi }{3}\right\}$. Then $n\left(A\cap B\right)$ is equal to

$\theta \in \left[0,2\pi \right]$ and ${\mathrm{z}}_1, {\mathrm{z}}_2, {\mathrm{z}}_3$ are three complex numbers such that they are collinear and $\left(1+\left|\mathrm{sin\theta }\right|\right){\mathrm{z}}_1+\left(\left|\mathrm{cos\theta }\right|-1\right){\mathrm{z}}_2-\sqrt{2}{\mathrm{z}}_3=0.$ If at least one of the complex numbers ${\mathrm{z}}_1, {\mathrm{z}}_2, {\mathrm{z}}_3$ is non-zero, then number of possible values of $\theta$ is

Let ${}^{\text{'}}z^{\text{'}}$ be a complex number and ${}^{\text{'}}a^{\text{'}}$ be a real parameter such that $z^2+az+a^2=0$, then which is of the following is not true?

Let $z=x+iy$. Then locus of moving point $P\left(z\right)$ such that $\frac{1+\overline{z}}{z}\in R$ is (where $i^2=-1$) 

Let $A\left(z_1\right)$ and $B\left(z_2\right)$ are two distinct non-real complex numbers in the argand plane such that $\frac{z_1}{z_2}+\frac{{\displaystyle \overline{{\mathit{z}}_{\mathit{1}}}}}{{\displaystyle \overline{z_2}}}=2$. Then the value of $|\angle ABO|$ is

Complex numbers $z_1$ and $z_2$ satisfy $\left|z_1\right|=2$ and $\left|z_2\right|=3.$ If the included angle of their corresponding vectors is $60°$ then the value of $19{\left|\frac{z_1-z_2}{z_1+z_2}\right|}^2$ is