Properties of nth Root of a Complex Number

Q.

If 1, $\alpha$, ${\alpha }^2$, ${\alpha }^3$, ............, ${\alpha }^{n-1}$ are roots of unity.What is the value of If 1 $\times$ $\alpha$ $\times$ ${\alpha }^2$ $\times$ ${\alpha }^3$, $\times$ ............, ${\alpha }^{n-1}$.

1. $(-1{)}^n$

2. $(-1{)}^{n-1}$

3. $(-1{)}^{n+1}$

4. $(-1{)}^{n+1}$

Q.

Find the value of 'a' so that $x^2-11x+a=0and{x}^2-14x+2a=0$ have a common root.

1. 0

2. 12

3. 24

4. 48

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

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. If $|z-3+2i|=4,$ then the difference between the greatest value and the least value of $|z|$ is :

1. $2\sqrt{13}$
2. $4+\sqrt{13}$
3. $\sqrt{13}$
4. $8$

Q. Let $\alpha ,\beta$ be the roots of $x^2-x+p=0$ and $\gamma ,\delta$ be the roots of $x^2-4x+q=0$. If $\alpha ,\beta ,\gamma ,\delta$ are in G.P., the integral values of p, q are respectively

1. – 2, 3
2. – 2, – 32
3. – 6, – 32
4. – 6, 3

Q.

If $\alpha {=}^m{C}_2$, Then ${}^{\alpha }{C}_2$ is equal to ..........

1. 

2. 

3. 

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. ${\alpha }_1,{\alpha }_2,{\alpha }_3,........{\alpha }_{100}$ 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. 
4. none of these

Q. Prove the following identity:
$1-{\sin }^4\alpha -{\cos }^4\alpha =\frac{1}{2}{\sin }^{2}2\alpha$

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.

Find the value of ${\alpha }^2+{\beta }^2$ if $\alpha ,\beta$ are the roots of $x^2+5x+2=0$

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Q. If $\alpha ,\beta$ are the roots of the equation $2x^2+4x-5=0$, the equation whose roots are the reciprocals of $2\alpha -3$ and $2\beta -3$ is

1. $x^2+10x-11=0$
2. $x^2+10x+11=0$
3. $11x^2+10x+1=0$
4. $11x^2-10x+1=0$

Q. Which of the following statements are true and which are false? In each case give a valid reason for saying so
(i) p : Each radius of a circle is a chord of the circle
(ii) q : The centre of a circle bisects each chord of the circle
(iii) r : Circle is a particular case of an ellipse
(iv) s : If x and y are integers such that $x>y$ then $-x<-y$
(v) t : $\sqrt{11}$ is a rational number

Q. If $\cos \alpha +2\cos \beta +3\cos \gamma =\sin \alpha +2\sin \beta +3\sin \gamma =0,$ then

1. $\cos 3\alpha +8\cos 3\beta +27\cos 3\gamma =18\cos (\alpha +\beta +\gamma )$
2. $\cos 3\alpha +8\cos 3\beta +27\cos 3\gamma =18\sin (\alpha +\beta +\gamma )$
3. $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma =18\sin (\alpha +\beta +\gamma )$
4. $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma =18\cos (\alpha +\beta +\gamma )$

Q.

Find the value of

$\sum {}_{k=1}^6$ $(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})$.

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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 $2\cos \alpha =x+\frac{1}{x},2\cos \beta =y+\frac{1}{y}$ 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. $\pm i\sin (10\alpha -12\beta )$