Cube Root of a Complex Number

Q. If $\alpha ,\beta$ are complex cube roots of unity, then the value of $\frac{a+b\alpha +c\beta }{a\alpha +b\beta +c}$ can be

1. $\alpha$
2. $\beta$
3. ${\alpha }^2$
4. $\frac{1}{\beta }$

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

1. 3

2. \N
3. 1
4. 2

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. If the polynomial $5x^3+Mx+N$ is divisible by $x^2+x+1$, then $|M+N|=$

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 $z=-1$, then principal value of the $\arg \left(z^{2/3}\right)$ is/are

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

Q. If $\omega$ is a complex cube root of unity, then the value of $(a+b{)}^2+{(a\omega +b{\omega }^2)}^2+{(a{\omega }^2+b\omega )}^2$ is

1. $6ab$
2. $3ab$
3. $12ab$
4. $ab$

Q. If $1,\omega ,{\omega }^2$ be the three cube roots of unity, then
$(1+\omega )\prod _{n=1}^{2n-1}(1+{\omega }^{2^n})=$

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.

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. If $1,\omega ,{\omega }^2$ be the three cube roots of unity, then
$(1+\omega )\prod _{n=1}^{2n-1}(1+{\omega }^{2^n})=$

Q. If $(x+1{)}^n-x^n-1$ is divisible by $x^3+x^2+x,$ then which of the following is true :

1. $n$ is not a multiple of $3$ and odd number
2. $n$ is a multiple of $3$ and odd number
3. $n$ is a multiple of $3$ and even number
4. $n$ is not a multiple of $3$ and even number

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 $\omega$ is a complex cube root of unity, then the equation whose roots are $2\omega$ and $2{\omega }^2$ is

1. $x^2+2x+4=0$
2. $x^2+2\omega x+4=0$
3. $x^2+2{\omega }^{2}x+4=0$
4. $x^2-2\omega x+4=0$

Q. If $z=-2+2\sqrt{3}i,$ then $z^{2n}+2^{2n}{z}^n+2^{4n}$ may be equal to

1. $3\cdot 4^{2n}$
2. $2^{2n}$
3. $2\cdot 4^{2n}$
4. $0$

Q. If $\omega$ is a non-real cube root of unity, then the value of $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+\cdots$ sum upto $19$ terms is

1. $56680$
2. $44080$
3. $84390$
4. $64580$

Q. If $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }+\frac{1}{d+\omega }=\frac{2}{\omega },$ where $a,b,c$ are real and $\omega$ is non real cube root of unity, then:

1. $\frac{1}{a+{\omega }^2}+\frac{1}{b+{\omega }^2}+\frac{1}{c+{\omega }^2}+\frac{1}{d+{\omega }^2}=\frac{2}{{\omega }^2}$
2. $abc+bcd+abd+acd=2$
3. $a+b+c+d=2abcd$
4. $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}=2$

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. The value of the expression $1(2-\omega )(2-{\omega }^2)+2(3-\omega )(3-{\omega }^2)+\cdots \cdots +(n-1)(n-\omega )(n-{\omega }^2)$ is where $\omega$ is an imaginary cube root of unity is

1. ${\left(\frac{n(n+1)}{2}\right)}^2$
2. ${\left(\frac{n(n+1)}{2}\right)}^2-n$
3. ${\left(\frac{n(n+1)}{2}\right)}^2+n$
4. ${\left(\frac{n(n+1)}{2}\right)}^2+2n$