Properties of nth Root of a Complex Number

Q.

The numerically greatest term in the expansion of $(2x-3y{)}^{12}$) when $x=1,y=\frac{5}{3}$, is

1. $9^{\text{th}}$ term

2. ${10}^{\text{th}}$ term

3. ${11}^{\text{th}}$ term

4. ${12}^{\text{th}}$ term

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 $\alpha =\cos \left(\frac{8\pi }{11}\right)+i\sin \left(\frac{8\pi }{11}\right),$ then $Re(\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5)$ is equal to

1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $0$
4. $1$

Q.

If $z_1$ and $z_2$ be the $n$th roots of unity which subtend right angle at the origin. Then $n$ must be of the form

1. $4k+1$

2. $4k+2$

3. $4k+3$

4. $4k$

Q. In a $G.P.$ first term and common ratio both are equal to $\frac{1}{2}(\sqrt{3}+i).$ Then the modulus value of $n^{th}$ term of the $G.P.$ is

1. $2^n$
2. $4^n$
3. $2^{n-1}$
4. $1$

Q. Let $p$ be an integer and let $1,\alpha ,\dots \dots {\alpha }^{n-1}$ be the $n^{\text{th}}$ roots of unity. Then $1+{\alpha }^p+({\alpha }^2{)}^p+\dots \dots +({\alpha }^{n-1}{)}^p=$

1. $n$ if $p$ is a multiple of $n$
2. $n$ if $p$ is not a multiple of $n$
3. $0$ if $p$ is a multiple of $n$
4. $0$ if $p$ is not a multiple of $n$

Q.

If the fourth roots of unity are$z_1,z_2,z_3$ and $z_4$ , then ${z_1}^2+{z_2}^2+{z_3}^2+{z_4}^2$ is equal to

1. $0$

2. $2$

3. $3$

4. $4$

5. None of these

Q. If $z_1$ and $\overline{z_2}$ represent adjacent vertices of a regular polygon of n sides and if $\frac{Im\left(z_1\right)}{Re\left(z_1\right)}=\sqrt{2}-1$, Then n is equal to

1. $8$
2. $16$
3. $24$
4. $18$

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

Q. The value of $\sum _{k=1}^{10}(\sin \frac{2k\pi }{11}-i\cos \frac{2k\pi }{11})$ is

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

Q. If $1,\omega ,{\omega }^2,\cdots \cdots {\omega }^{n-1}$ are the $n^{\text{th}}$ roots of unity and $z_1$ and $z_2$ are any two complex numbers, then $\sum _{k=0}^{n-1}|z_1+{\omega }^k{z}_2{|}^2$ is equal to

1. $n[|z_1{|}^2+|z_2{|}^2]$
2. $(n+1)[|z_1{|}^2+|z_2{|}^2]$
3. $(n+1)[|z_1{|}^2-|z_2{|}^2]$
4. $(n-1)[|z_1{|}^2+|z_2{|}^2]$

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. If $2\cos \alpha =x+\frac{1}{x},2\cos \beta =y+\frac{1}{y}$ then $\frac{x^{10}}{y^{12}}-\frac{y^{12}}{x^{10}}=$

1. $\pm 2i\sin (10\alpha +12\beta )$
2. $\pm i\sin (10\alpha +12\beta )$
3. $\pm 2i\sin (10\alpha -12\beta )$
4. $\pm i\sin (10\alpha -12\beta )$

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. The value of integral $\underset{0}{\overset{10\pi }{\int }}\frac{\cos 6x\cos 7x\cos 8x\cos 9x}{1+e^{{\sin }^{3}4x}}dx$ is equal to

1. $\frac{\pi }{8}$
2. $\frac{3\pi }{8}$
3. $\frac{5\pi }{8}$
4. $\frac{7\pi }{8}$

Q. The centre of regular polygon of $n$ sides is located at $z=0$ and one of its vertices is $z_1$. If $z_2$ is vertex adjacent to $z_1$, then $z_2=$

1. $z_1(\cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n})$
2. $z_1(\cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n})$
3. $z_1(\cos \frac{\pi }{2n}\pm i\sin \frac{\pi }{2n})$
4. $z_1(\cos \frac{\pi }{3n}\pm i\sin \frac{\pi }{3n})$

Q. Sum of roots of the equation $(z-1{)}^4=16$ is

Q.

Let $S=\{a\in N,a\le 100\}$. If the equation $\left[Ta{n}^{2}x\right]-Tanx-a=0$ has real roots then number of elements in S is (where [] is step functin).

1. 10

2. 8

3. 9

4. 0

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. $1$

Q. Let $z_1,z_2,z_3,z_4,z_5$ and $z_6$ be complex numbers lying on a unit circle with centre $(0,0).$ If $\omega =\left(\sum _{k=1}^6{z}_k\right)\left(\sum _{k=1}^6\frac{1}{z_k}\right),$ then

1. $\omega$ is purely real.
2. $\omega$ is purely imaginary.
3. Maximum possible value of $\left|\omega \right|$ is $6.$
4. Maximum possible value of $\left|\omega \right|$ is $36.$

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 coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3).....(1+x^{100})$ is

Q. If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to:

1. $11$
2. $5$
3. $0$
4. $4$

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. Suppose $z$ is any root of $11z^8+20i{z}^7+10iz–22=0$, where $i=\sqrt{-1}$ . Then $S=|z{|}^2+|z|+1$ satisfies

1. $S\le 3$
2. $3<S<7$
3. $S\ge 13$
4. $7\le S<13$

Q. If the fourth roots of unity are $z_1,z_2,z_3,z_4$ then $z_1^2+z_2^2+z_3^2+z_4^2$ is equal to

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 $\alpha$ 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. ${\alpha }^4$