Complex numbers practice · Medium

If $z$ lies on the circle $|z|=1$, then $2 / z$ lies on a

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

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

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

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

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 $f\left( x \right)$ is a polynomial of degree $n$ with rational coefficients and $1 + 2i,\;2 - \sqrt 3$ and 5 are three roots of $f\left( x \right) = 0$, then the least value of $n$ is

Let $S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$. Then $\sum\limits_{{z \in S}}|z|^{2}$ is equal to

The greatest and the least value of $\left| {{z_1} + {z_2}} \right|,$ if ${z_1} = 24 + 7i$ and $\left| {{z_2}} \right| = 6,$ are respectively

The modulus of $\frac{(3+2 i)^2}{(4-3 i)}$ is

If $|z|=1, z \neq 1$, then value of $\arg \left(\frac{1}{1-z}\right)$ cannot exceed

If $\alpha$ and $\beta$ are the roots of $x^2-2 x+2=0$, then $\alpha^n+\beta^n=$

If $x=\omega-\omega^2-2$, then the value of $x^4+3 x^3+2 x^2-11 x-6$ is

If one vertex of a square whose diagonals intersect at the origin is $3\left( {\cos \mathrm{\theta } + i\sin \mathrm{\theta }} \right),$ then the two adjacent vertices are

If $\arg \left(\frac{z-2}{z+2}\right)=\frac{\pi}{4}$ for any complex number $z$, then $|z-2 i|=$

If $\left(\frac{1+i}{1-i}\right)^3-\left(\frac{1-i}{1+i}\right)^3=x+i y$, then $(x, y)$ is equal to

If $z = x + iy$ and $\left| {\frac{{1 - iz}}{{z - i}}} \right| = 1$, the locus of $z$ is

If $z_{1}$ and $z_{2}$ are two non- zero complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$, then $\arg z_{1}-\arg z_{2}$ is equal to

If $\arg \left( {{z_1}{z_2}} \right) = 0$ and $\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1,$ then

If $\mathrm{Re\;}\left( {\frac{{z + 4}}{{2z - i}}} \right) = \frac{1}{2},$ then $z$ is represented by a point lying on

If $a,b,c$ and $u,\upsilon ,\omega$ are complex numbers representing the vertices of two triangles such that $c = \left( {1 - r} \right)\;a + rb$ and $\omega = \left( {1 - r} \right)u + r\upsilon ,$ where $r$ is a complex number, then the two triangles

If the points $z,-i z$ and 1 are collinear then $z$ lies on

If $i=\sqrt{-1}$, then $\frac{e^{x i}+e^{-x}}{2}=$

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