Properties of nth Root of a Complex Number

Q. Given that $\alpha ,\beta ,a,b$ are in A.P., $\alpha ,\beta ,c,d$ are in G.P. and $\alpha ,\beta ,e,f$ are in H.P. If $b,d,f$ are in G.P., then the value of $\frac{2({\alpha }^6-{\beta }^6)}{\alpha \beta ({\alpha }^4-{\beta }^4)},0<\alpha <\beta$ is

Q. Let $f\left(x\right)$ and $g\left(x\right)$ are polynomial of degree 4 such that
$g\left(\alpha \right)=g^{\prime }\left(\alpha \right)=g^{\prime \prime }\left(\alpha \right)=0$.
If $\underset{x\to \alpha }{lim}\frac{f\left(x\right)}{g\left(x\right)}=0,$ then number of different real solutions of equation $f\left(x\right)g^{\prime }\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)=0$ is equal to

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

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

Q. In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$. If $x^2-c^2=y$, where $c$ is the length of the third side of the triangle, then the circumradius of the triangle is:

1. $\frac{c}{3}$
2. $\frac{c}{\sqrt{3}}$
3. $\frac{y}{\sqrt{3}}$
4. $\frac{3}{2}y$

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

1. A.P.

2. G.P.

3. H.P.
4. None of these

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

Q. If $C_0,C_1,\cdots C_n$ are the coefficient of $x$ in expansion of $(1+x{)}^n$, then $C_0-C_2+C_4-C_6+\cdots +(-1{)}^n{C}_n=$

1. $(2{)}^{n/2}\cos \frac{n\pi }{4}$
2. $(2{)}^{n/2}\sin \frac{n\pi }{4}$
3. $(2{)}^n\sin \frac{n\pi }{4}$
4. $(2{)}^n\cos \frac{n\pi }{4}$

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. 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 $C_0,C_1,\cdots C_n$ are the coefficient of $x$ in expansion of $(1+x{)}^n$, then $C_0-C_2+C_4-C_6+\cdots +(-1{)}^n{C}_n=$

1. $(2{)}^{n/2}\cos \frac{n\pi }{4}$
2. $(2{)}^{n/2}\sin \frac{n\pi }{4}$
3. $(2{)}^n\sin \frac{n\pi }{4}$
4. $(2{)}^n\cos \frac{n\pi }{4}$

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. 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. 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. ${\alpha }_1,{\alpha }_2,{\alpha }_3,........{\alpha }_{100}$ 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. $\left(20\right)\frac{1}{20}$
4. none of these