Cube Root of a Complex Number

Q.

If $1,$ $\omega and{\omega }^2$ are the cube roots of unity, then $\omega \left({\left(1+\omega \right)}^3-{\left(1+\omega \right)}^2\right)$is equal to

1. $1$

2. $-1$

3. $i$

4. $0$

Q.

If $\omega \ne 1$ is a cube root of unity and $(1+\omega {)}^7=A+B\omega$ then $A\text{and}B$ are respectively

1. $0,1$

2. $1,1$

3. $1,0$

4. $-1,1$

Q.

If $\omega$ is an imaginary cube root of unity and $x=a+b,y=a\omega +b{\omega }^2,z=a{\omega }^2+b\omega$, then $x^2+y^2+z^2$ is equal to

1. $6ab$

2. $3ab$

3. $6a^{2b^2}$

4. $3a^{2b^2}$

Q.

If $\alpha$ is a complex number satisfying the equation ${\alpha }^2+\alpha +1=0$, then ${\alpha }^{31}$ is equal to

1. $\alpha$

2. ${\alpha }^2$

3. $1$

4. $i$

Q.

Then eccentricity of the Ellipse $4x^2+y^2-8x-2y+1$ = 0

1. 

2. 

3. 

4. 

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 $(3+5\omega +3{\omega }^2{)}^2+(3+3\omega +5{\omega }^2{)}^2$ is equal to

1. 4
2. 0
3. -4
4. None of these

Q. By using binomial theorem, expand the following:
${\left(101\right)}^4$

Q.

1, $\omega$, ${\omega }^2$ are the cube roots of unity then the roots of $(x-1{)}^3+8=0$

1. 

2. 

3. 

4. 

Q.

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

1. 1

2. 2

3. -1

4. 0

Q. Let $R$ be the set of real numbers. If $f:R\to R$ is a function defined by $f\left(x\right)=x^4$, then $f$ is

1. a surjective but not an injective function
2. a bijective function
3. an injective function but not a surjective function
4. Neither an injective nor a surjective function

Q.

The value of
$(1-\omega +{\omega }^2)(1-{\omega }^2+{\omega }^4)(1-{\omega }^4+{\omega }^8)(1-{\omega }^8+{\omega }^{16})$ is, where $\omega$ is the cube root of unity

Q.

If $\alpha$ and $\beta$ are imaginary cube roots of unity and x = a+b, y = a$\alpha$ + b$\beta$, z= a$\beta$ + b$\alpha$, then xyz =

1. 

2. 

3. 

4. 

Q. Let $z=\frac{\sqrt{3}}{2}-\frac{i}{2}.$ Then the smallest positive integer $n$ such that $(z^{95}+i^{67}{)}^{94}=z^n$ is

Q. If $|z-\frac{4}{z}|$ = 2, then the maximum value of $\left|z\right|$ is equal to

1. -1
   

2. 5

3. 2

4. +1
   

Q.

If $\omega$ is imaginary cube root of unity, then arg(i $\omega$) + arg(i ${\omega }^2$) =

1. π/2

2. π

3. -π

4. 0

Q.

If |$\frac{a+ib}{c+id}$| = |4+3i| , then the value of |$\frac{a^2+b^2}{c^2+d^2}$| is _________

Q.

If $A(a{t}^2,2at),B(\frac{a}{t^2},\frac{-2a}{t})$ and C(a, 0), then 2a is equal to

1. A.M. of CA and CB

2. H.M. of CA and CB

3. G.M. of CA and CB

4. None of these

Q. Let $f:N\to N$ be a function such that $f(m+n)=f\left(m\right)+f\left(n\right)$ for every $m,n\in N.$ If $f\left(6\right)=18,$ then $f\left(2\right)\cdot f\left(3\right)$ is equal to

1. $18$
2. $6$
3. $54$
4. $36$

Q.

Write the factors of the following terms

$-13p^3{q}^2{r}^2$

Q. If $z^2-z+1=0,$ then possible value(s) of $z^n-z^{-n},$ where $n$ is even number

1. $\sqrt{3}i$
2. $0$
3. $-\sqrt{3}i$
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. If $f\left(x\right)=2x+1$ and $g\left(x\right)=x^2+1$, then $go\left(fof\right)\left(2\right)=$

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. 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 $\omega (\ne 1)$ is a cube root of unity, and $(1+\omega {)}^7=A+B\omega$. Then (A, B) equals?

1. $(1,1)$
2. $(-1,1)$
3. $(1,0)$
4. $(0,1)$

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

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

1. $\omega$

2. None of these

3. 0

4. ${\omega }^2$

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. $3$

Q. If $x=\omega -{\omega }^2-2,$ then the value of $x^4+3x^3+2x^2-11x-6$ is