Tag: complex numbers

If $1,{\alpha _1},{\alpha _2}....{\alpha _8}$ are nine, ninth roots of unity (taken in counter-clock wises direction) then $\left| {\left( {2 - {\alpha _1}} \right)\left( {2 - {\alpha _3}} \right)\left( {2 - {\alpha _5}} \right)\left( {2 - {\alpha _7}} \right)} \right|$ is equal to

Number of values of $z$ (real or complex) simultaneously satisfying the system of equations
$1+z+{z}^{2}+{z}^{3}+....+{z}^{17}=0$ and $1+z+{z}^{2}+{z}^{3}+.....+{z}^{13}=0$ is

If $1,{\alpha _1},{\alpha _2},{\alpha _3}$ are the fourth roots of unity, then the value of $\left( {1 + {\alpha _1}} \right)\left( {1 + {\alpha _2}} \right)\left( {1 + {\alpha _3}} \right)$ is equal to

If $n^{th}$ root of unity be $1,a _{1},a _{2},...a _{n-1}$, then $\displaystyle \sum^{n-1} _{r=1}\dfrac {1}{2+a _{r}}$ is equal to

Let $a^{k}$ where $k=0.1.2....2013$ are the $2014^{th}$ roots of unity. If $Z _{1}$ and $Z _{2}$ be any two complex number such that $|Z _{1}|=|Z _{2}|=\dfrac{1}{\sqrt{2014}}$, then the value of $\displaystyle \sum _{ k=0 }^{ 2013 }{ { \left| { Z } _{ 1 }+{ a }^{ k }{ Z } _{ 2 } \right|  }^{ 2 } } $ is equal to

If $x _{1},x _{2},x _{3}$ are three real solutions of the equations $x^{2\ell nx-1}+e^{1/9}=(1+e^{/9})(x^{\ell-0.5})$ none of them being unity. Find $x _{1}x _{2}x _{3}$:

If $1, a _1, a _2,......a _{n-1}$ are the n nth roots of unity, then?

If $z _{1}$ and $z _{2}$ be the $n^{th}$ roots of unity which subtend a right angle at the origin, then $n$  must be of the form

If $A=\begin{bmatrix} a & b\ 0 & a\end{bmatrix}$ is nth root of $I _2$, then choose the correct statements.

The value of $\displaystyle\ \alpha^{4n-1}+\alpha^{4n-3}, n\epsilon\mathbb{N}$ and $\displaystyle\ \alpha$ is a nonreal fourth root of unity is