Polar Form of Complex Number

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 $\lambda \in R,$ 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$\text{'}z\text{'}$ 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 $\left|z-\frac{4}{z}\right|=2,$ then the maximum value of $\left|z\right|$ is equal to

The greatest positive argument of complex number satisfying $\left|z-4\right|=Re\left(z\right)$ is

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 $x$?

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 $\left|\alpha \right|=$

Match the statements given in Column I with the values given in Column II.

Number of ordered pairs(s) $\left(\mathrm{a},\mathrm{b}\right)$ of real numbers such that ${\left(\mathrm{a}+\mathrm{ib}\right)}^{2008} = \mathrm{a}-\mathrm{ib}$ holds good is

Let $z=x+iy$ be a complex number where, $x$ and $y$ are integers. Then, the area ( in square units ) of the rectangle whose vertices are the root of the equation $z{\overline{z}}^3+\overline{z}{z}^3=350,$ is

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

For a non-zero complex number $z$, let $arg \left(z\right)$ denote the principal argument with $-\pi <\arg \left(z\right)\le \pi$.Then, which of the following statement(s) is (are) FALSE? 

If all the three roots of $a{z}^3+b{z}^2+cz+d=0$ have negative real parts $\left(a,b,c\in R\right)$, then

Consider three distinct complex numbers $a,b$ and $c$ such that $\left|a\right|=\left|b\right|=\left|c\right|=1$. Also, $z_1$ and $z_2$ are the roots of the equation $a{z}^2+bz+c=0$ with $\left|z_1\right|=1$. If  $\mathrm{P}$ and $Q$ represent the complex numbers $z_1$ and $z_2$ in the Argand plane with $\angle POQ=\theta , 0<\theta <180°$ (where $O$ being the origin), then

Which of the following is true?

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

Let $z$ be a complex number satisfying equation $z^p={\overline{z}}^q$, where $p,q\in N$, then

Which of the following represents a point in an Argand plane, equidistant from the roots of the equation ${\left(z+1\right)}^4=16z^4?$

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