The $n^{th}$ roots of unity are in

Important Questions on Complex Numbers and Quadratic Equations

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\omega =\frac{-1+\sqrt{3}i}{2}$, then ${\left(3+\omega +3{\omega }^2\right)}^4$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\omega$ is a cube root of unity, then ${\left(3+5\omega +3{\omega }^2\right)}^2+{\left(3+3\omega +5{\omega }^2\right)}^2$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set $\{{\left|a+b\omega +c{\omega }^2\right|}^2:a,b,c$ are distinct non-zero integers $\}$ equals ________

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Imaginary part of $(\sqrt{3}-\mathrm{i}{)}^{2016}+(-\sqrt{3}-\mathrm{i}{)}^{2019}$is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

The imaginary part of ${\left(i-\sqrt{3}\right)}^{13}$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\omega$ is a complex cube root of unity, then ${\left(1-\omega +{\omega }^2\right)}^6+{\left(1-{\omega }^2+\omega \right)}^6=$

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

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

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\alpha$ and $\beta$ be the roots of the equation $x^2+x+1=0$. Then, the equation whose roots are ${\alpha }^{19}$ and ${\beta }^7$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $z=e^{\frac{i4\pi }{3}}$$,$ then ${\left(z^{192}+z^{194}\right)}^3$ is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let $\omega$ be a cube root of unity not equal to $1$. Then the maximum possible value of $\left|a+bw+c{w}^2\right|$ where $a,b,c\in \left\{1,-1\right\}$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

On any given arc of positive length on the unit circle $\left|z\right|=1$ in the complex plane.

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$ . If

$\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\omega }^2-1& {\omega }^2\\ 1& {\omega }^2& {\omega }^7\end{array}\right|=3k,$

Then $k$ can be equal to:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\omega$ is a complex cube root of unity, then the value of $\frac{p+q\omega +r{\omega }^2}{r+p\omega +q{\omega }^2}+\frac{p+q\omega +r{\omega }^2}{q+r\omega +p{\omega }^2}, \left(p,q,r\in R\right)$  is equal to

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

The $n^{\mathrm{th}}$ roots of unity are in

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\mathrm{\omega }$ is an imaginary cube root of unity, then ${\left(1+\mathrm{\omega }-{\mathrm{\omega }}^2\right)}^7$ equals

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\omega$ is an imaginary cube root of unit, then ${\left(1+\omega -{\omega }^2\right)}^7$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

The sum of ${162}^{th}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is ______.

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

If $\alpha , \beta \in C$ are the distinct roots of the equation $x^2-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

The least positive integer $n$ for which ${\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n=1$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>De Moivre's Theorem

Let $z_0$ be a root of quadratic equation, $x^2+x+1=0.$ If $z=3+6i{z}_0^{81}-3i{z}_0^{93}$ , then $\mathrm{a}\mathrm{r}\mathrm{g}$ $\left(z\right)$ is equal to: