Complex numbers practice · Mixed difficulty

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$, then value of $\left|z_2-z_3\right|^2+$ $\left|z_3-z_1\right|^2+\left|z_1-z_2\right|^2$ cannot exceed

Let $\left(-2-\frac{1}{3} \mathrm{i}\right)^3=\frac{\mathrm{x}+\mathrm{iy}}{27}(\mathrm{i}=\sqrt{-1})$, where $\mathrm{x}$ and $y$ are real numbers, then $y-x$ equals :

If the cube roots of unity are $1,\;\omega ,\;{\omega ^2}$, then the roots of the equation ${\left( {x - 2} \right)^3} + 27 = 0$ are

If ${z_1},{z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right| = 1$ and $i\;{z_1} = k\;{z_2}$ where $k \in R,$ then the angle between ${z_1} - {z_2}$ and ${z_1} + {z_2}$ is

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

The values of $(16)^{1 / 4}$ are

If $x$ is an integer satisfying ${x^2} - 6x + 5 \le 0$ and ${x^2} - 2x > 0,$ then the number of positive values of $x$, is

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is :

Let $z=x+i y$ be a complex number, where $x$ and $y$ are integers and $i=\sqrt{-1}$. Then, the area of the rectangle whose vertices are the roots of the equation $z \bar{z}^3+\bar{z} z^3=350$, is

Let $z = x + iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $z{z^{ - 3}} + \bar z{z^3} = 350$ is

Let $w_1$ be the point obtained by the rotation of $z_1=5+4 i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5 i$ about the origin through a right angle in the clockwise direction. Then the principal argument $w_1-w_2$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x+1=0$, $\alpha^{2009}+\beta^{2009}$ is equal to

In which quadrant of the complex plane, the point $\frac{{1 + 2i}}{{1 - i}}$ lies?

The real part of ${\left( {1 - \cos \mathrm{\theta } + 2i\sin \mathrm{\theta }} \right)^{ - 1}}$ is

For two complex numbers ${z_1},\;{z_2}$ the relation $\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$ holds, if

The solution set of $\left| {{x^2} - 10} \right| \le 6,$ is

If $x, y, a, b \in \mathbf{R}, a \neq 0$ and $(a+i b)(x+i y)=\left(a^2+b^2\right) i$, then $(x, y)$ equals

If $z^{2}+z+1=0$, where $\mathrm{z}$ is complex number, then the value of $\left(z+\frac{1}{z}\right)^{2}+\left(z^{2}+\frac{1}{z^{2}}\right)^{2}+\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\ldots \ldots .+\left(z^{6}+\frac{1}{z^{6}}\right)^{2}$ is

Let $z,w$ be complex numbers such that $\bar z + \overline {iw} = 0$ and $\mathrm{\;arg}\left( {zw} \right) = \pi$. Then $\arg \left( z \right)$ equals