nth Root of a Complex Number

Q.

Prove ${10}^{2n-1}+1\text{is divisible by 11.}$

Q.

Fill all the roots of the equation.

$x^6$ - $x^5$ + $x^4$ - $x^3$ + $x^2$ - x + 1 = 0 1 + x $\ne 0$

1. + ; n = 0, 1, 2, 3, 4, 5, 6

2. + ; n = 0, 1, 2, 3, 4, 5, 6

3. + ; n = 0, 1, 2, 3, 4, 5, 6

4. + ; n = 0, 1, 2, 3, 4, 5, 6

Q. The root(s) of the equation $x^4=16$ is/are :

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

Q. The sum of roots of the equation $x^8=1$ whose real part is positive is

1. $1+\sqrt{2}$
2. $1-\sqrt{2}$
3. $0$
4. $1$

Q. If $\omega$ is an n^th root of unity, other than unity, then the value of $1+\omega +{\omega }^2+....+{\omega }^{n-1}$ is
[Karnataka CET 1999]

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

Q. The equation $z^{10}+(13z-1{)}^{10}=0$ has $5$ pairs of complex roots $a_1,b_1,a_2,b_2,a_3,b_3,a_4,b_4,a_5,b_5$. If each pair $a_i,b_i$ are complex conjugates, then

1. $\sum _{i=1}^5(\frac{1}{a_i}+\frac{1}{b_i})=130$
2. $\sum _{i=1}^5(\frac{1}{a_i}+\frac{1}{b_i})=260$
3. $\sum _{i=1}^5\left(\frac{1}{a_i{b}_i}\right)=1700$
4. $\sum _{i=1}^5\left(\frac{1}{a_i{b}_i}\right)=850$

Q.

If $\alpha$ is a root of $4x^2+2x-1$ = 0. then root is:

1. 

2. 

3. 

4. 

Q.

Find the set of values of $\alpha$ for which point the $P(\alpha ,-\alpha )$ is inside
$\frac{x^2}{16}+\frac{y^2}{9}=1$

1. $(\frac{-12}{5},\frac{12}{5})$
2. $[\frac{-12}{5},\frac{12}{5}]$
3. $(\frac{-13}{5},\frac{13}{5})$
4. $(\frac{-13}{5},\frac{13}{5}]$

Q. If $1,\omega ,{\omega }^2,{\omega }^3.....,{\omega }^{n-1}$ are the n, $n^{th}$ roots of unity, then $(1-\omega )(1-{\omega }^2).....(1-{\omega }^{n-1})$ equals

1. $n^2$
2. 0
3. 1
4. n

Q. If $1,\omega ,{\omega }^2,{\omega }^3.....,{\omega }^{n-1}$ are the n, $n^{th}$ roots of unity, then $(1-\omega )(1-{\omega }^2).....(1-{\omega }^{n-1})$ equals

1. \N
2. 1
3. n
4. $n^2$

Q. Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin, then $n$ must be of the form

1. $4p,p\in Z^{+},p\le n-1$
2. $4p+3,p\in Z^{+},p\le n-1$
3. $4p+2,p\in Z^{+},p\le n-1$
4. $4p+1,p\in Z^{+},p\le n-1$

Q. The sum of roots of the equation $x^8=1$ whose real part is positive is

1. $1+\sqrt{2}$
2. $1-\sqrt{2}$
3. $0$
4. $1$

Q. Let $z_1$ and $z_2$ be n^th roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form
[IIT Screening 2001; Karnataka 2002]

1. 4k + 1

2. 4k + 2

3. 4k
4. 4k + 3

Q. Let $z_1$ and $z_2$ be n^th roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form
[IIT Screening 2001; Karnataka 2002]

1. 4k + 1

2. 4k + 2

3. 4k + 3
4. 4k

Q.

If ${\alpha }_0,{\alpha }_1,{\alpha }_2\dots ,{\alpha }_{n-1}$ be the n, nth roots of the unity, then the value of ${\sum }_{i=0}^{n-1}\frac{{\alpha }_i}{(3-{\alpha }_i)}$ is equal to

1. $\frac{n-1}{3^n-1}$

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

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

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

Q. The complex number $\omega$ which satisfies the equation $z^3=8i$ and lying in the second quadrant on the complex plane.

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