Exercises

If$\text{'}z\text{'}$ is complex number then the locus of $\text{'}z\text{'}$ satisfying the condition $|2z-1|=|z-1|$ is

If $|z-1|\le 2$ and $\left|\omega z-1-{\omega }^2\right|=a$ (where $\omega$ is a cube root of unity), then complete set of values of $a$ is

If $|2z-1|=|z-2|$ and $z_1, z_2$ are complex numbers such that $\left|z_1-\alpha \right|<\alpha , \left|z_2-\beta \right|<\beta ,$ then $\left|\frac{z_1+z_2}{\alpha +\beta }\right|$

Let $\left|z_r-r\right|\le r,\forall r=1, 2, 3,$....$n.$ Then, $\left|\sum _{r=1}^n{z}_r\right|$ is less than

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

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$ and $z_1+z_2+z_3=0,$ then the area of the triangle whose vertices are $z_1, z_2, z_3$ is (in sq. units)

If $\left|z_2+i{z}_1\right|=\left|z_1\right|+\left|z_2\right|$ and $\left|z_1\right|=3, \left|z_2\right|=4,$ then area (in sq. units) of $\Delta ABC,$ if affixes of $A, B$ and $C$ are $z_1, z_2$ and $\left[z_2-i{z}_1\right]/\left[\right(1-i\left)\right]$, respectively, is

Let $z$ be a non-real complex number which satisfies the equation $z^{23}=1$. Then the value of $\sum _{k=1}^{22}\frac{1}{1+z^{8k}+z^{16k}}$