Cube Root of a Complex Number

Q. If $\omega$ is a cube root of unity, then $(1+\omega {)}^3-(1+{\omega }^2{)}^3=$

1. 0

2. $\omega$

3. ${\omega }^2$
4. None of these

Q. If $\omega$ is a cube root of unity, then $(1+\omega -{\omega }^2)(1-\omega +{\omega }^2)$ =
[MNR 1990; MP PET 1993, 2002]

1. 1

2. 0

3. 2
4. 4

Q. If $\alpha$ and $\beta$ are imaginary cube roots of unity, then ${\alpha }^4+{\beta }^4+\frac{1}{\alpha \beta }$
[IIT 1977]

1. 0

2. 3

3. 1
4. 2

Q.

Sum of the common roots of $z^{2006}+z^{100}+1=0$
and $z^3+2z^2+2z+1=0$ is

1. 0

2. -1

3. 1

4. 2

Q. Let $1,\omega$ and ${\omega }^2$ be the cube roots of unity. The least possible degree of a polynomial with real coefficients, having $2{\omega }^2,3+4\omega ,3+4{\omega }^2$ and $5-\omega -{\omega }^2$ as roots is

Q. The polynomial $x^6+4x^5+3x^4+2x^3+x+1$ is divisible by (where $\omega$ is one of the imaginary cube roots of unity)

1. $x+\omega$
2. $x+{\omega }^2$
3. $(x+\omega )(x+{\omega }^2)$
4. $(x-\omega )(x-{\omega }^2)$

Q. Let $\omega$ is the root of the equation $x^2+x+1=0$ whose imiginary part is postive and $|z-\omega |=|z+\omega |$, then $\arg \left(z\right)$ is

1. $\frac{\pi }{3}$
2. $\frac{5\pi }{6}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{4}$

Q. Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6i{z}_0^{81}-3i{z}_0^{93}$, then $\arg z$ is equal to :

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $0$

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

Q. Let $\alpha$ and $\beta$ be roots of the equation $x^2-x+1=0$. Then the value of $\alpha +\beta +{\alpha }^2+{\beta }^2+\cdots +{\alpha }^{100}+{\beta }^{100}$ is

1. $-3$
2. $-1$
3. $0$
4. $3$