Properties of Cube Root of a Complex Number

Q. If $C^2+S^2=1,then\frac{1+C+iS}{1+C-iS}$ is equal to

1. C + iS
2. C - iS
3. S + iC
4. S - iC

Q. Square of either of the two imaginary cube roots of unity will be

1. Real root of unity

2. Other imaginary cube root of unity

3. Sum of two imaginary roots of unity

4. None of these

Q.

If $z$ = $(i{)}^{(i{)}^{\left(i\right)}}$, where i = $\sqrt{-1}$, then |z| is equal to:

1. 

2. 

3. 

4. 

Q.

What are the values of the cube roots of unity?

Q. If $z^2+z+1=0$ where z is a complex number, then the value of
${(z+\frac{1}{z})}^2+{(z^2+\frac{1}{z^2})}^2+{(z^3+\frac{1}{z^3})}^2+......+{(z^6+\frac{1}{z^6})}^2$is

1. 18
2. 54
3. 6
4. 12

Q.

If z is any complex number then $\frac{\overline{z}}{|z{|}^2}$ =

1. 

2. 

3. 

4. 

Q.

If $\omega$ = $\frac{-1+i\sqrt{3}}{2}$ , then $Z_1$ = $\frac{-\sqrt{3}-i}{2}$ and $Z_2$ = $\frac{\sqrt{3}-i}{2}$ can be expressed in terms of $\omega$ and ${\omega }^2$ as

1. -i , i

2. i , i

3. i , - i

4. -i , -i

Q.

If $z_{1}and{z}_2$ are two complex number and a, b are two real numbers, then

$|a{z}_1-b{z}_2{|}^2+|b{z}_1+a{z}_2{|}^2$ equals

1. 

2. 2ab

3. 

4. 

Q. If $n$ is an integer and
${(1+i\sqrt{3})}^n+{(1-i\sqrt{3})}^n=2^{n+1}\cos \theta ,$ then $\theta$ is equal to

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

Q. Square of either of the two imaginary cube roots of unity will be

1. Real root of unity

2. Other imaginary cube root of unity

3. Sum of two imaginary roots of unity

4. None of these

Q.

Let f(x) = $(1+x{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$ $x^2$ + ${}^n{C}_2$ $x^3$ ........${}^n{C}_n$ $x^n$.

If f(1) = $S_1$, f(w) = $S_2$ and $f\left(w^2\right)$ = $S_3$, find the value of ${}^n{C}_0$ + ${}^n{C}_3$ + ${}^n{C}_6$ + .......... in terms of $S_1$, $S_2$ and

$S_3$.[W is the complex cube root of unity]

1. 

2. 

3. 

4. 

Q.

$\frac{(cos\theta +isin\theta {)}^4}{(sin\theta +icos\theta {)}^5}$ is equal to

1. 

2. 

3. 

4. 

Q.

Value of the numerically greatest term in the expansion of $\sqrt{3}(1+\frac{1}{\sqrt{3}}{)}^{20}$ is

1. 

2. 

3. 

4. 

Q.

If $\alpha ,\beta and\gamma$ are the roots of the equation $x^3$ + 2$x^2$ + 3x + 1 = 0. Find the constant term of the equation whose roots are $\frac{1}{{\beta }^3}$ + $\frac{1}{{\gamma }^3}$ - $\frac{1}{{\alpha }^3}$, $\frac{1}{{\gamma }^3}$ + $\frac{1}{{\alpha }^3}$ - $\frac{1}{{\beta }^3}$ , $\frac{1}{{\alpha }^3}$ + $\frac{1}{{\beta }^3}$ - $\frac{1}{{\gamma }^3}$.

__

Q.

$\omega$ is a cube root of unity, $\omega \ne 1$.

Match the following.

(1) ${\omega }^{99}$ (p) - 1

(2) 1 + ${\omega }^{33}$ + ${\omega }^{66}$ (q) 0

(3) 1 + ${\omega }^{14}$ + ${\omega }^{28}$ (r) 3

(4) ${\omega }^4$ + ${\omega }^5$ + ${\omega }^6$ (s) 1

1. 1 - p, 2 - q, 3 - R, 4 - s

2. 1 - s, 2 - q, 3 - r, 4 - q

3. 1 - s, 2 - r, 3 - q, 4 - q

4. 1 - p, 2 - q, 3 - q, 4 - q

Q.

If a function is of form f(x)=$\frac{g\left(Z\right)}{h\left(z\right)}$, then f'(x) - $\frac{\frac{\left(h\right(z\left)\right)d\left(g\right(z\left)\right)}{dz}+g\left(z\right)\frac{d\left(h\right(z\left)\right)}{dz}}{\left(h\right(z){)}^2}$

1. True

2. False

Q.

If $\frac{(1-3x{)}^{\frac{1}{2}}+(1-x{)}^{\frac{5}{3}}}{\sqrt{4-x}}$ is approximately equal to

a + bx for small values of x, then (a, b) =

1. 

2. 

3. 

4. 

Q. If $1,\omega ,{\omega }^2$ are the cube roots of unity, then their product is
[Karnataka CET 1999, 2001]

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

Q.

For any two complex number $z_1$ and $z_2$ and any

real numbers a and b; $|(a{z}_1-b{z}_2){|}^2$ + $|(b{z}_1-a{z}_2){|}^2$ =

1. 

2. 

3. 

4. None of these

Q. If $i=\sqrt{-1}$, then $4+5{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{334}+3{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{365}$ is equal to

1. $-i\sqrt{3}$
2. $1-i\sqrt{3}$
3. $-1+i\sqrt{3}$
4. $i\sqrt{3}$

Q. If $\omega$ is a complex cube root of unity, then (1 + $\omega$)(1 + ${\omega }^2$) $(1+{\omega }^4$) $(1+{\omega }^8$)... to 2n factors =
[AMU 2000]

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

Q. If $z^2+z+1=0$ where z is a complex number, then the value of
${(z+\frac{1}{z})}^2+{(z^2+\frac{1}{z^2})}^2+{(z^3+\frac{1}{z^3})}^2+......+{(z^6+\frac{1}{z^6})}^2$is

1. 18
2. 54
3. 6
4. 12

Q. If $\alpha ,\beta$ be the roots of the equation $u^2-2u+2=0$ and if $\cot \theta =x+1$, then $\frac{(x+\alpha {)}^n-(x+\beta {)}^n}{(\alpha -\beta )}$ is equal to

1. $\frac{\sin n\theta }{{\sin }^n\theta }$
2. $\frac{\cos n\theta }{{\cos }^n\theta }$
3. $\frac{\sin n\theta }{{\cos }^n\theta }$
4. $\frac{\cos n\theta }{{\sin }^n\theta }$

Q. FB cube root of unity yaar FB cube root of unity are 1 Omega omega square then the root of the equation x minus 1 whole cube + 8 equal to zero Hour

Q. If $z^2+z+1=0$ where z is a complex number, then the value of
${(z+\frac{1}{z})}^2+{(z^2+\frac{1}{z^2})}^2+{(z^3+\frac{1}{z^3})}^2+......+{(z^6+\frac{1}{z^6})}^2$is

1. 18
2. 54
3. 6
4. 12

Q. The sum
$1+\left(1+x\right)+\left(1+x+x^2\right)+\left(1+x+x^2+x^3\right)+\dots n$ terms equals

1. $\frac{1-x^n}{1-x}$
2. $\frac{x\left(1-x^n\right)}{1-x}$
3. $\frac{n\left(1-x\right)-x\left(1-x^n\right)}{{\left(1-x\right)}^2}$
4. None of these

Q. Let $z$ be a complex number satisfying $z+z^{-1}=1$. A possible value of $n$ when $z^n+z^{-n}$ is minimum, is

1. $12$
2. $15$
3. $10$
4. $11$

Q. The function $f:R\to R$ defined by $f\left(x\right)=[x{]}^2+[x+1]-3,$ (where [.] represents the greatest integer function). Then which of the following is/are true?$

1. $f\left(x\right)$ many-one
2. $f\left(0.5\right)=-2$
3. $f\left(0.7\right)=0$
4. $f\left(x\right)$ one-one

Q. If the four roots of the equation $z^4+z^3+2z^2+z+1=0$ form a quadrilateral on the Argand plane, then the area of the quadrilateral is

1. $\frac{\sqrt{3}+2}{8}$
2. $\frac{\sqrt{3}+2}{4}$
3. $\frac{\sqrt{3}+2}{2}$
4. $\frac{\sqrt{3}+2}{3}$

Q.

what is the cube root of 57