Complex numbers practice · Mixed difficulty

The argument of $\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}$ is

Two conjugate complex numbers $z_1$ and $\overline{z_1}$ represent two adjacent vertices of a polygon of $\mathrm{n}$ sides, whose centre is origin. If $\frac{\operatorname{Im}\left(z_1\right)}{\operatorname{Re}\left(\bar{z}_1\right)}=\sqrt{2}-1$ then ratio of the areas of the circles inscribed and circumscribed the polygon is

If $z$ is a complex number in the Argand plane, then the equation ${\left| {z - 2} \right|^2} + {\left| {z + 2} \right|^2} = 8$ represents

If the number $\frac{(1-i)^n}{(1+i)^{n-2}}$ is real and positive, then $n$ is

If $z \in \mathbf{C}$ and $2 z=|z|+i$, then $z$ equals

If $(1-i)^n=2^n$, then the value of $n$ is

If $\omega$ is a complex cube root of unity, then the value of ${\omega ^{99}} + {\omega ^{100}} + {\omega ^{101}}$ is

If $n$ is a positive integer, then ${\left( {1 + i\sqrt 3 } \right)^n} + {\left( {1 - i\sqrt 3 } \right)^n}$ is equal to

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$, then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to:

If $z = r(\cos \theta + i\sin \theta )$, then the value of $\frac{z}{{\bar z}} + \frac{{\bar z}}{z}$ is

If $|z+4|=3$, then greatest possible value of $|z-7| / 7$ is

If $\sqrt{3}+i=(a+i b)(c+i d)$, then $\tan ^{-1} \frac{b}{a}+\tan ^{-1} \frac{d}{c}$ is equal to

A particle $P$ starts from the point $z_0=1+2 i, i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$, the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and then it moves through an angle $\frac{\pi}{2}$ in anti-clockwise direction on a circle with centre at origin, to reach a point $z_2$, then the point $z_2$ is given by

If $m$ and $x$ are two real numbers, then $e^{2 m i \cot ^{-1} x}\left(\frac{x i+1}{x i-1}\right)^m$ (where $i=\sqrt{-1}$ ) is equal to

If $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then one of the values of $\arg \left(z_2 / z_1\right)$ is

If $z$ is a non-real ${7^{th}}$ root of $- 1$, then ${z^{86}} + {z^{175}} + {z^{289}}$ is equal to

If $\left| {\frac{{z + i}}{{z - i}}} \right| = \sqrt 3$, then radius of the circle is

For complex numbers $z_1=x_1+i y$, and $z_2$ $=x_2+i y_2$, we write $z_1 \cap z_2$ if $x_1 \leq x_2$ and $y_1 \leq y_2$. Let $z$ be a complex number such that $1 \cap z$, then

If the $5^{\text {th }}$ roots of unity are $1, z_1, z_2, z_3, z_4$ then $z_1{ }^2+z_2{ }^2+z_3{ }^2+z_4{ }^2$ is equal to

If $\alpha, \beta$ are distinct complex numbers with $|\beta|=1$, then value of $\left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right|$ equals

Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

For any two complex numbers ${z_1}$ and ${z_2}$ and any real numbers $a$ and $b$, ${\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2}$ is equal to

Let ${\omega _n} = \cos \left( {\frac{{2\pi }}{n}} \right) + i\sin \left( {\frac{{2\pi }}{n}} \right),\;{i^2} = - 1,$ then $\left( {x + y{\omega _3} + z\omega _3^2} \right)\left( {x + y\omega _3^2 + z{\omega _3}} \right)$ is equal to

If $\omega(\neq 1)$ is a cube root of unity and $(1+\omega)^7=A+B \omega$, then $(A, B)$ equals to

If the cube roots of unity are $1, \omega, \omega^2$, then the roots of the equation $x^3-3 x^2+3 x+63=0$ are

Locus of point $z$ so that $z, i$, and $i z$ are collinear, is

Let $\alpha ,\;\beta$ be the roots of ${x^2} - 2x\cos \phi + 1 = 0$, then the equation whose roots are ${\alpha ^n},\;{\beta ^n},\;$ is

If $\frac{{3\left( {x - 2} \right)}}{5} \ge \frac{{5\left( {2 - x} \right)}}{3},$ then $x$ belongs to the interval

If $i = \sqrt { - 1}$, then $4 + 5{\left( { - \frac{1}{2} + \frac{{i\sqrt 3 }}{2}} \right)^{334}} + 3{\left( { - \frac{1}{2} + \frac{{i\sqrt 3 }}{2}} \right)^{365}}$ is equal to

The area of the triangle whose vertices are $i,\omega$ and ${\omega ^2},$, where $i = \sqrt { - 1}$ and $\omega ,{\omega ^2}$ are complex cube roots of unity, is