Tag: complex numbers

If $1,{ \alpha  } _{ 1 },{ \alpha  } _{ 2 },{ \alpha  } _{ 3 }$ and $\alpha _4$ be the roots of $x^5-1=0$, then $\displaystyle \frac { \omega -{ \alpha  } _{ 1 } }{ { \omega  }^{ 2 }-{ \alpha  } _{ 1 } } .\frac { \omega -{ \alpha  } _{ 2 } }{ { \omega  }^{ 2 }-{ \alpha  } _{ 2 } } .\frac { \omega -{ \alpha  } _{ 3 } }{ { \omega  }^{ 2 }-{ \alpha  } _{ 3 } } .\frac { \omega -{ \alpha  } _{ 4 } }{ { \omega  }^{ 2 }-{ \alpha  } _{ 4 } } =$ 

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 from 

If 1, $a _{1},a _{2},.....a _{n-1} $ are  $n^{th} $ roots of unity then $\frac{1}{1-a _{1}} +\frac{1}{1-a _{2}}+...+\frac{1}{1-a _{n-1}}$ equals

If $\omega$ is a complex cube root of unity, then the equation $\left|z-\omega\right|^{2}+\right|z-\omega^{2}\right|^{2}=\lambda$ will represent a circle if

Let $\displaystyle z _{1}$ and $\displaystyle z _{2}$ be the $n^{th}$ roots of unity, which are ends of a line segment that subtends a right angle at the origin. Then, $n$ must be of the form

Which one is not a root of the fourth root of unity

If $z _{1},z _{2}$be two $nth$ roots of unity such that they represent two point $A,B$ in the Argand plane where $\angle AOB=60^{\circ}$ and $O$ is the orgin then the positive integer $n$ is of the form 

If $\left| { a } _{ 1 } \right| <1,\lambda _{ 1 }\ge 0$ for $i=1,2,3....n$, and ${ \lambda  } _{ 1 }+{ \lambda  } _{ 2 }+{ \lambda  } _{ 3 }+...+\lambda _{ n }=1$ then the value ....$+\left| \lambda _{ n }{ a } _{ n } \right| $ is

If ${ z } _{ 1 },{ z } _{ 2 }$ are two complex numbers and ${ \omega  }^{ k },k=0,1,...,n-1$ are the nth roots of unity, then $\displaystyle \sum _{ k=0 }^{ n-1 }{ { \left| { z } _{ 1 }+{ z } _{ 2 }{ \omega  }^{ k } \right|  }^{ 2 } } $

If $\displaystyle \alpha$ is a non-real root of $\displaystyle x^{5}+1=0$ then $\displaystyle \alpha ^{10n+2}+\alpha ^{5n+2}+\alpha ^{5n}$, where n is an odd positive integer,has the value