Properties of Cube Root of a Complex Number

Q. Let $Z$ be the set of integers. If
$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

1. $2^{10}$
2. $2^{12}$
3. $2^{18}$
4. $2^{15}$

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

Q. The value of $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$ is

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

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

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

Q. 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 $k$ is equal to:

1. $-z$
2. $z$
3. $-1$
4. $1$

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

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

Q. The value of the expression $1\cdot (2-\omega )(2-{\omega }^2)+2\cdot (3-\omega )(3-{\omega }^2)+....+(n-1)\cdot (n-\omega )(n-{\omega }^2)$, 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. None of the above

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. \N
2. 1
3. -1
4. None of these

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