G Tewani Solutions for Chapter: Complex Numbers, Exercise 2: Archives

If $\left|z-\frac{4}{z}\right|=2,$ then the maximum value of $\left|z\right|$ is equal to

Let $\alpha , \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct roots on the line $Rez=1,$ then it is necessary that

If $\omega \left(\ne 1\right)$ is a cube root of unity and $\left(1+\omega \right){}^7=A+B\omega .$ Then, $\left(A, B\right)$ equals

A complex number $z$ is said to be unimodular, if $\left|z\right|=1$ Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-z_1{\overline{z}}_2}$ is unimodular and $z_2$ is not unimodular. Then, the point $z_1$ lies on a

Let complex numbers $\mathrm{\alpha }$ and $\frac{1}{\overline{\mathrm{\alpha }}}$ lies on circles ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2=r^2$ and ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2=4r^2$, respectively. If $z_0=x_0+i{y}_0$ satisfies the equation $2{\left|z_0\right|}^2=r^2+2$, then $\left|\alpha \right|=$

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$. If the complex number $z=x+iy$ satisfies $\mathrm{Im}\left(\frac{az+b}{z+1}\right)=y$, then which of the following is (are) possible value(s) of $x$?

If $z$ is a complex number of unit modulus and argument $\theta ,$ then $arg\left(\frac{1+z}{1+\overline{z}}\right)$ equals

If $z$ a complex number such that $\left|z\right|\ge 2$, then the minimum value of $\left|z+\frac{1}{2}\right|$