Embibe Experts Solutions for Exercise 8: Assignment

A complex number '$z$' is rotated in anticlockwise direction by an angle '$\alpha$' we get $z_1$ and if same complex number $z$ is rotated by an angle $\alpha$ in clockwise direction we get $z_2$ then

If $1, z_1, z_2,..., z_{n-1}$ be the $n^{\mathrm{th}}$ roots of unity and '$\omega$' be a non-real complex cube root of unity, then sum of all possible values of $\prod _{r=1}^{n-1}\left(\omega -z_r\right)$ will be equal to

If $z_1, z_2, z_3$ are vertices of an equilateral triangle with orthocentre $z_0$ then $z_1^2+z_2^2+z_3^2=k{z}_0^2$, where $k\in N$, then $k$ is equal to

If a complex number $z$ lies on a circle of radius $3$ and centre at $z_0=0$ then the complex number $\left(-3+9z\right)$ lies on a circle of radius

For all the complex numbers $z_1,z_2$ satisfying $\left|z_1-3\right|=3$ and $\left|z_2-3+18i\right|=2$ then the minimum value of $\left|z_1-z_2\right|$ is equal to

The algebraic sum of perpendicular distances from the points $1,{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6,{\alpha }_7$, where ${\alpha }_i$ for $i=1,2,\dots ,7$, are complex $8^{\mathrm{th}}$ roots of unity, from the line $\left(3+2i\right)z+\left(3-2i\right)\overline{z}+5\sqrt{13}=0$, is equal to

If $z_1,z_2,z_3,\dots ,z_n$ lies on a circle $\left|z\right|=5$ then the value of $\left|z_1+z_2+\dots +z_n\right|-25\left|\frac{1}{z_1}+\frac{1}{z_2}+\dots +\frac{1}{z_n}\right|$ is equal to

The eccentricity of conic represented by $\left|z-2\right|+\left|z-2+17 i\right|=19$ is $e$, then ${\left(19e\right)}^2$ is equal to