Properties of nth Root of a Complex Number

Q. The $n^{th}$ roots of unity are in
[Orissa JEE 2004]

1. None of these

2. G.P.

3. H.P.
4. A.P.

Q. If $z_1,z_2,z_3.....n_n$ are n^th, roots of unity, then for k = 1, 2, ....., n

1. $|z_{k+1}|=k|z_k|$
2. $|z_{k+1}|=|z_k|+|z_{k+1}|$
3. $|z_k|=|z_{k+1}|$
4. $|z_k|=k|z_{k+1}|$

Q. ${\alpha }_1,{\alpha }_2,{\alpha }_3,........{\alpha }_{100}$ are all the 100th roots of unity. The numerical value of is
$\sum \sum _{1\le i<<j\le 100}({\alpha }_i{\alpha }_j{)}^5$

1. 20
2. 0
3. none of these
4. 

Q. If $1,z_1,z_2,...,z_{n-1}$ are the roots of $z^n-1=0,$ then $\frac{1}{3-z_1}+\frac{1}{3-z_2}+\frac{1}{3-z_3}+...+\frac{1}{3-z_{n-1}}$ is equal to

1. $\frac{\sum _{r=1}^{n-1}r\cdot 3^{r-1}}{\sum _{r=1}^n{3}^{r-1}}$
2. $\frac{\sum _{r=1}^{n}r\cdot 3^{r-1}}{\sum _{r=1}^n{3}^{r-1}}$
3. $\frac{n\cdot 3^{n-1}}{3^n-1}-\frac{1}{2}$
4. $\frac{n\cdot 3^n}{3^n-1}-\frac{1}{2}$

Q. If $\alpha$ is a non real root of $z=(1{)}^{1/5},$ then the value of $(1+\alpha +{\alpha }^2+{\alpha }^{-2}-{\alpha }^{-1})$ is

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

Q. Let $A_1,A_2,\cdots ,A_n$ be the vertices of a regular polygon of $n$ sides in a circle of radius unity and
$a=|A_1{A}_2{|}^2+|A_1{A}_3{|}^2+\cdots |A_1{A}_n{|}^2,$
$b=|A_1{A}_2||A_1{A}_3|\cdots |A_1{A}_n|,$ then $\frac{a}{b}=$

Q. If $1,z_1,z_2,z_3,\cdots z_{n-1}$ are $n$ roots of unity then the value of $\frac{1}{3-z_1}+\frac{1}{3-z_2}+\cdots +\frac{1}{3-z_{n-1}}$ is equal to :

1. $\frac{n\cdot 3^{n-1}}{3^n-1}-\frac{1}{2}$
2. $\frac{n\cdot 3^{n-1}}{3^n-1}+\frac{1}{2}$
3. $\frac{n\cdot 3^{n-1}}{3^n-1}-1$
4. $\frac{n\cdot 3^{n-1}}{3^n-1}+1$

Q. The sum of the roots of equation $z^6+64=0$ whose real part is positive is

1. $2\sqrt{3}$
2. $2$
3. $4$
4. $\sqrt{3}$

Q. If $1,d_1,d_2,d_3\cdots ,d_{n-1}$ be $n^{th}$ roots of unity then which among the following are true

1. $\frac{1}{1-d_1}+\frac{1}{1-d_2}+\cdots \frac{1}{1-d_{n-1}}=\frac{n-1}{2}$
2. $(2+d_1)(2+d_2)\cdots (2+d_{n-1})=\frac{2^n+1}{3},$if $n$ is odd
3. $(2-d_1)(2-d_2)\cdots (2-d_{n-1})=2^n-1$
4. $(2+d_1)(2+d_2)\cdots (2+d_{n-1})=\frac{2^n+1}{3},$if $n$ is even

Q. The value of $\sum _{k=1}^6(\sin \frac{2k\pi }{7}-\cos \frac{2k\pi }{7})$ is

1. $i$
2. $-1$
3. $-i$
4. $1$