1. Let $\mathrm{x}, \mathrm{y} \in \mathrm{R}$ & $\mathrm{i}=\sqrt{-1}$, then $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ is called a complex number

$\operatorname{Re}(\mathrm{z})=\mathrm{x}, \operatorname{Im}(\mathrm{z})=\mathrm{y}$

If $x=0, z$ is purely imaginary

If $y=0, z$ is purely real

2. For two complex numbers $z _{1}=x _{1}+i y _{1} z _{2}=z _{2}+i y _{2}$

(i) $z _{1}=z _{2}$ if and only if $x _{1}=x _{2}$ and $y _{1}=y _{2}$

Note that no order relation is possible among the set of complex numbers. i.e. it is wrong to say $1+\mathrm{i}<4+3 \mathrm{i}$. But $|1+\mathrm{i}|<|4+3 \mathrm{i}|$

(ii) $\mathrm{z} _{1} \pm \mathrm{z} _{2}=\left(\mathrm{x} _{1} \pm \mathrm{x} _{2}\right)+\mathrm{i}\left(\mathrm{y} _{1} \pm \mathrm{y} _{2}\right)$

(iii) $\mathrm{z} _{1} \mathrm{z} _{2}=\left(\mathrm{x} _{1} \mathrm{x} _{2}-\mathrm{y} _{1} \mathrm{y} _{2}\right)+\mathrm{i}\left(\mathrm{x} _{1} \mathrm{y} _{2}+\mathrm{x} _{2} \mathrm{y} _{1}\right)$

(iv) $\dfrac{\mathrm{z} _{1}}{\mathrm{z} _{2}}=\dfrac{\left(\mathrm{x} _{1} \mathrm{x} _{2}+\mathrm{y} _{1} \mathrm{y} _{2}\right)+\mathrm{i}\left(\mathrm{x} _{2} \mathrm{y} _{1}-\mathrm{x} _{1} \mathrm{y} _{2}\right)}{\mathrm{x} _{2}{ }^{2}+\mathrm{y} _{2}{ }^{2}} ; \mathrm{z} _{2} \neq 0$

3. Conjugate of $z$

Conjugate of $z=x+i y$ is defined as $\bar{z}=x-i y$ (i.e. replace $i$ by $-i$ )

Properties

(i) $\overline{(\bar{z})}=\mathrm{z}$

(ii) $\mathrm{z}=\overline{\mathrm{z}} \Leftrightarrow \mathrm{z}$ is puraly real

(iii) $\mathrm{z}=-\overline{\mathrm{z}} \Leftrightarrow \mathrm{z}$ is purely imaginary

(iv) $\mathrm{z}^{+} \overline{\mathrm{z}}=2 \operatorname{Re}(\mathrm{z})=2 \operatorname{Re}(\overline{\mathrm{z}})$

(v) $\mathrm{z}-\overline{\mathrm{z}}=2 \mathrm{i} \operatorname{Im}(\mathrm{z})$

(vi) $\overline{\mathrm{z} _{1} \pm \mathrm{z} _{2}}=\overline{\mathrm{z} _{1}} \pm \overline{\mathrm{z} _{2}}$

(vii) $\overline{\mathrm{z} _{2} \mathrm{z} _{2}}=\overline{\mathrm{z} _{1}} \overline{\mathrm{z} _{2}}$ (also $\left.\left.\overline{\left(\mathrm{z}^{\mathrm{n}}\right.}\right)=(\overline{\mathrm{z}})^{\mathrm{n}}\right)$

(viii) $\overline{\left(\dfrac{\mathrm{z} _{1}}{\mathrm{z} _{2}}\right)}=\dfrac{\overline{\mathrm{z} _{1}}}{\overline{\mathrm{z} _{2}}}$

4. Geometric representation

5. Modulus and amplitude

$|z|=r=\sqrt{x^{2}+y^{2}}$, the modulus of $z$ (distance of $z$ from the origin)

$\theta=\tan ^{-1}\left(\dfrac{y}{x}\right)=\arg z$, the inclination of OP with positive direction of $x-$ axis where $P$ is $(x, y)$.

Here $-\pi<\operatorname{argz} \leq \pi$

$\mathrm{z}=\mathrm{x}+\mathrm{iy} ; \mathrm{x}, \mathrm{y} \in \mathrm{R}$ (algebraic form)

$=\mathrm{r}(\cos \theta+\mathrm{isin} \theta)$ or $\mathrm{r} \operatorname{cis} \theta$ (polar/trigonometric form)

$=r e^{i \theta}$ (Eulers form)

Properties of modulus

(i) $-|z| \leq \operatorname{Re}(z), \operatorname{Im}(z) \leq|z|$

(ii) $|\mathbf{z}|=|\bar{z}|+|-z|+|-\bar{z}|$

(iii) $\mathrm{z} \overline{\mathrm{Z}}=|\mathrm{z}|^{2}$

(iv) $\left|z _{1} z _{2}\right|=\left|z _{1}\right|\left|z _{2}\right|$ Also $\left|\mathrm{z}^{\mathrm{n}}\right|=|\mathrm{z}|^{\mathrm{n}}$

(iv) $\left|\dfrac{z _{1}}{z _{2}}\right|=\dfrac{\left|z _{1}\right|}{\left|z _{2}\right|}$

(vi) ||$z _{1}|-| z _{2}|\leq| z _{1} \pm z _{2}|\leq| z _{1}|+| z _{2} \mid$ (Triangle inequality)

i.e. $\left|z _{1}\right|+\left|z _{2}\right|$ is the maximum and ||$z _{1}|-| z _{2} \mid$ is the minimum value of $\left|z _{1} \pm z _{2}\right|$

(vii) $\left|z _{1} \pm z _{2}\right|^{2}=\left\{\begin{array}{c}\left|z _{1}\right|^{2}+\left|z _{2}\right|^{2} \pm 2 \operatorname{Re}\left(z _{1} \bar{z} _{2}\right) \\ \left|z _{1}\right|^{2}+\left|z _{2}\right|^{2} \pm 2\left|z _{1}\right|\left|z _{2}\right| \cos \left(\theta _{1} \pm \theta _{2}\right) \text { where } \\ \theta _{1}=\arg \left(z _{1}\right) & \theta _{2}=\arg z _{2}\end{array}\right.$

(viii) $\left|z _{1}+z _{2}\right|^{2}+\left|z _{1}-z _{2}\right|^{2}=2\left(\left|z _{1}\right|^{2}+\left|z _{2}\right|^{2}\right.$ ) (parallelogram law)

Also $\left|\mathrm{az}-\mathrm{bz} \mathrm{z} _{2}\right|^{2}+\left.\left|\mathrm{bz} \mathrm{z} _{1}+\mathrm{az}\right| _{2}\right|^{2}=\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)\left(\left|\mathrm{z} _{1}\right|^{2}+\left|\mathrm{z} _{2}\right|^{2}\right) ; \mathrm{a}, \mathrm{b} \in \mathrm{R}$

Note : If $\left|z+\dfrac{1}{z}\right|=a$, the greatest and least value of $|z|$ are respectively $\dfrac{a+\sqrt{a^{2}+4}}{2}$ and $\dfrac{-a+\sqrt{a^{2}+4}}{2}$

Properties of argument

(i) $\arg \left(\mathrm{z} _{1} \mathrm{z} _{2}\right)=\arg \left(\mathrm{z} _{1}\right)+\arg \left(\mathrm{z} _{2}\right)+2 \mathrm{k} \pi$, where $\mathrm{k}=-1$ or 0 or 1 Also $\arg \left(z^{\mathrm{n}}\right)=\operatorname{narg}(\mathrm{z})+2 \mathrm{k} \pi$

(ii) $\quad \arg \left(\dfrac{z _{1}}{z _{2}}\right)=\arg z _{1}-\arg z _{2}+2 k \pi$, where $k=-1$ or 0 or 1

(iii) $\arg \left(\dfrac{\mathrm{z}}{\mathrm{z}}\right)=2 \arg (\mathrm{z})+2 \mathrm{k} \pi$, where $\mathrm{k}=-1$ or 0 or 1

(iv) argument of zero is not defined $\arg (\mathrm{z})=0 \Leftrightarrow \mathrm{z}$ is real and positive $\arg (\mathrm{z})=\pi \Leftrightarrow \mathrm{z}$ is real and negative $\arg (\mathrm{z})=\dfrac{\pi}{2} \Leftrightarrow \mathrm{z}$ is purely imaginary, $\operatorname{Im} z>0$

$\arg (\mathrm{z})=\dfrac{-\pi}{2} \Leftrightarrow \mathrm{z}$ is purely imaginary, $\operatorname{Imz}<0$

(v) $\left|z _{1}+z _{2}\right|=\left|z _{1}\right|+\left|z _{2}\right|$

$\arg z _{1}-\arg z _{2}=0, \dfrac{z _{1}}{z _{2}}>0$

$0, z _{1}, z _{2}$ are collinear and $z _{1}, z _{2}$ lie on the same side of 0 .

(vi) $\left|z _{1}-z _{2}\right|=\left|z _{1}\right|+\left|z _{2}\right|$

$\arg z _{1}-\operatorname{argz} z _{2}=\pi, \dfrac{z _{1}}{z _{2}}<0$

$0, z _{1}, z _{2}$ are collinear and 0 lies between $z _{1} & z _{2}$.

(vii) $\left|\mathrm{z} _{1}-\mathrm{z} _{2}\right|=\left|\mathrm{z} _{1}+\mathrm{z} _{2}\right|$

$\operatorname{argz}-\operatorname{argz} z _{2}= \pm \dfrac{\pi}{2}$

$\dfrac{z _{1}}{z _{2}}$ and $\overline{z _{1}} z _{2}$ are purely imaginary.

6. De Moivres theorem

(i) For any rational number $\mathrm{n}$, then

$(\cos \theta+\sin \theta)^{\mathrm{n}}=\cos n+\sin n \theta$

i.e. , $\left(\mathrm{e}^{\mathrm{i} \theta}\right)^{\mathrm{n}}=\mathrm{e}^{\mathrm{i} \theta}$

(ii) $\left(\cos \theta _{1}+i \sin \theta _{1}\right)\left(\cos \theta _{2}+i \sin \theta _{2}\right)$ $\left(\cos \theta _{n}+i \sin \theta _{n}\right)$

$=\cos \left(\theta _{1}+\theta _{2}+\ldots \ldots+\theta _{\mathrm{n}}\right)+\mathrm{i} \sin \left(\theta _{1}+\theta _{2}+\right.$. $+\theta _{\mathrm{n}}$ )

(iii) If $z=r(\cos \theta+i \sin \theta)$ and $n \in Z^{+}$, then

$\mathrm{z}^{1 / \mathrm{n}}=\mathrm{r}^{1 / \mathrm{n}}\left(\cos \left(\dfrac{2 \mathrm{r} \pi+\theta}{\mathrm{n}}\right)+\mathrm{i} \sin \left(\dfrac{2 \mathrm{r} \pi+\theta}{\mathrm{n}}\right)\right)$ where $\mathrm{r}=0,1,2$,

( $\mathrm{n}^{\text {th }}$ roots of $\mathrm{z}$ )

7. The $\mathrm{n}^{\text {th }}$ roots of unity

$\mathrm{z}=\sqrt[n]{1}=1^{1 / \mathrm{n}}=\mathrm{e}^{\mathrm{i} 2 \pi / \mathrm{n}}, \mathrm{r}=0,1,2$, , (n-1)

Let $\mathrm{z}=\alpha^{\mathrm{r}}$ where $\alpha=\mathrm{e}^{\mathrm{i} 2 \pi / \mathrm{n}}$

The $\mathrm{n}^{\text {th }}$ roots of unity are $\left(\alpha^{\circ}=\right) 1, \alpha, \alpha^{2}, \ldots \ldots \ldots \alpha^{\mathrm{n}-1}$ where $\alpha=\mathrm{e}^{\mathrm{i} 2 \pi / \mathrm{n}}=\cos \dfrac{2 \pi}{\mathrm{n}}+\mathrm{i} \sin \dfrac{2 \pi}{\mathrm{n}}$

Properties

(i) $\mathrm{n}^{\text {th }}$ roots of unity are solutions of the equation $\mathrm{z}=1^{1 / \mathrm{n}}$

i.e. $\mathrm{z}^{\mathrm{n}}=1$

$\mathrm{z}^{\mathrm{n}}-1=(\mathrm{z}-1)(\mathrm{z}-\alpha)\left(\mathrm{z}-\alpha^{2}\right) \ldots \ldots \ldots . .\left(\mathrm{z}-\alpha^{\mathrm{n}-1}\right)$

$\mathrm{n}^{\text {th }}$ roots of -1 are the solutions of $z^{\mathrm{n}}+1=0$

(ii) $\mathrm{n}^{\text {th }}$ roots of unity lie on a unit circle $|z|=1$ and divide the circumference into $n$ equal parts and are the vertices of a regular polygon of $n$ sides inscribed in the circle $|z|=1$

(iii) Product of $n^{\text {th }}$ roots of unity $=(-1)^{n-1}$

(iv) Sum of $\mathrm{n}^{\text {th }}$ roots of unity is always zero.

(v) $\mathrm{n}^{\text {th }}$ roots of unity form a G.P with common ratio $\mathrm{e}^{\mathrm{i} 2 \pi / \mathrm{n}}$

(vi) Sum of $\mathrm{p}^{\text {th }}$ power of $\mathrm{n}^{\text {th }}$ roots of unity

$=1+\alpha^{p}+\left(\alpha^{2}\right)^{p}+\left(\alpha^{3}\right)^{p}+\ldots \ldots \ldots \ldots . .\left(+\alpha^{n-1}\right)^{p}=\left\{\begin{array}{l}0 \text {; if } p \text { is not a multiple of } n \\ n \text {; if } p \text { is a multiple of } n\end{array}\right.$

8. Cube roots of unity

$1^{1 / 3}=\cos \dfrac{2 \mathrm{r} \pi}{3}+\mathrm{i} \sin \dfrac{2 \mathrm{r} \pi}{3} ; \mathrm{r}=0,1,2$

$=1, \dfrac{-1+\mathrm{i} \sqrt{3}}{2}, \dfrac{-1-\mathrm{i} \sqrt{3}}{2}$

If one of the non real complex roots be $\omega$, then the other non real complex root will be $\omega^{2}$.

i.e. if $\omega=\dfrac{-1+i \sqrt{3}}{2}$, then $\omega^{2}=\dfrac{-1-i \sqrt{3}}{2}$

$\therefore$ the 3 cube roots of unity are $1, \omega & \omega^{2}$.

Properties

(i) $\mathrm{z}^{3}-1=(\mathrm{z}-1)(\mathrm{z}-\omega)\left(\mathrm{z}-\omega^{2}\right)$

(ii) $\omega$ & $\omega^{2}$ are roots of $z^{2}+z^{+}+1=0$ i.e., $z^{2}+z+1=(z-\omega)\left(z-\omega^{2}\right)$

(iii) $\arg \omega=2 \pi / 3$ & $\arg \omega^{2}=4 \pi / 3$

(iv) cube roots of unity lie on the unit circle $|z|=1$ and divide its circumference into three equal parts

(v) If $\mathrm{A}(1), \mathrm{B}(\omega)$ & $\mathrm{C}\left(\omega^{2}\right)$, then $\triangle \mathrm{ABC}$ is an equilateral triangle.

(vi) $\omega^{3}=1 ; 1+\omega+\omega^{2}=0 ; \omega^{3 n}=1 ; \omega^{3 n+1}=\omega ; \omega^{3 n+2}=\omega^{2}$

(vii) $\bar{\omega}=\dfrac{1}{\omega}=\omega^{2} ; \overline{\omega^{2}}=\dfrac{1}{\omega^{2}}=\omega$

(viii) $\bullet \mathrm{x}^{2}+\mathrm{y}^{2}=(\mathrm{x}+\mathrm{iy})(\mathrm{x}-\mathrm{iy})$

• $x^{3}+y^{3}=(x+y)(x+\omega y)\left(x+\omega^{2} y\right)$
• $\mathrm{x}^{3}-\mathrm{y}^{3}=(\mathrm{x}-\mathrm{y})(\mathrm{x}-\omega \mathrm{y})\left(\mathrm{x}-\omega^{2} \mathrm{y}\right)$
• $\mathrm{x}^{2}+\mathrm{xy}+\mathrm{y}^{2}=(\mathrm{x}-\omega \mathrm{y})\left(\mathrm{x}-\omega^{2} \mathrm{y}\right)$
• $\mathrm{x}^{2}-\mathrm{xy}+\mathrm{y}^{2}=(\mathrm{x}+\omega \mathrm{y})\left(\mathrm{x}+\omega^{2} \mathrm{y}\right)$
• $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}-\mathrm{xy}-\mathrm{yz-zx}=\left(\mathrm{x}+\mathrm{y} \omega+\mathrm{z} \omega^{2}\right)\left(\mathrm{x}+\mathrm{y} \omega^{2}+\mathrm{z} \omega\right)$
• $x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x+y \omega+z \omega^{2}\right)\left(x+y \omega^{2}+z \omega\right)$

Note $\therefore \bullet$ If $\mathrm{a}+\mathrm{b}+\mathrm{c}=0=\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}$, then $\mathrm{a}: \mathrm{b}: \mathrm{c}=1: \omega: \omega^{2}$

• Any complex number for which $\left|\dfrac{\text { real part }}{\text { imaginary part }}\right|=1: \sqrt{3}$ or $\sqrt{3}: 1$, can be expressed in terms of $\omega, \omega^{2}$ & $\mathrm{i}$

9. Square root of a complex number

Let $\mathrm{z}=\mathrm{a}+\mathrm{ib}$

$\sqrt{\mathrm{a}+\mathrm{ib}}= \pm\left\{\sqrt{\dfrac{1}{2}(|z|+\operatorname{Re}(\mathrm{z}))}+\mathrm{i} \sqrt{\dfrac{1}{2}(|z|-\operatorname{Re}(\mathrm{z}))}\right\}$

i.e. $\sqrt{a+i b}= \pm\left\{\sqrt{\dfrac{1}{2}\left(\sqrt{a^{2}+b^{2}}+a\right)}+i \sqrt{\dfrac{1}{2}\left(\sqrt{a^{2}+b^{2}}-a\right)}\right\}$

To find the square root of $\mathrm{a}-\mathrm{ib}$, replace $\mathrm{i}$ by $-\mathrm{i}$ in the above result.

10 .Logarithm of a complex number

Let $z=a+i b=r e^{i \theta}$

$|\mathrm{z}|=\mathrm{r}=\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}$

$\log \mathrm{z}=\log |z|+\operatorname{iarg} \mathrm{z}$

11. Expansions

(i) $\cos \theta={ }^{n} C _{0} \cos ^{n} \theta-{ }^{n} C _{2} \cos ^{n-2} \theta \sin ^{2} \theta+{ }^{n} C _{4} \cos ^{n-4} \theta \sin ^{4} \theta+$.

(ii) $\operatorname{sinn} \theta={ }^{n} C _{1} \cos ^{n-1} \theta \sin \theta-{ }^{n} C _{3} \cos ^{n-3} \sin ^{3} \theta^{n}{ }^{n} C _{5} \cos ^{n-5} \sin ^{5} \theta$ (using De-Moivres theorem)

(iii) $\cos ^n \theta=\dfrac{1}{2^{n-1}}\left\{{ }^n C_0 \cos n \theta+{ }^n C_1 \cos (n-2) \theta^{+}+{ }^n C_2 \cos (n-4)+\ldots \ldots \ldots \ldots \ldots . . \ldots\right\}$

(iv) $\sin ^n \theta=\dfrac{(-1)^{n / 2}}{2^{n-1}}\left\{\cos n \theta-{ }^n C_1 \cos (n-2) \theta+{ }^n C_2 \cos (n-4) \theta+\ldots \ldots \ldots . . .\right\}$

$\sin ^i \theta=\dfrac{(-1)^{\dfrac{n-1}{2}}}{2^{n-1}}\left\{\operatorname{sinn} \theta-{ }^n C_1 \sin (n-2) \theta+{ }^n C_2 \sin (n-4) \theta+\ldots \ldots \ldots . . .\right\}$

Solved Examples

1. If $\omega$ is a complex cube root of unity, then the value of $\dfrac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}+\dfrac{a+b \omega+c \omega^{2}}{b+c \omega+a \omega^{2}}$ is equal to

(a) -1

(b) 2

(c) $2 \omega$

(d) None of these

2. If $x^{2}+x+1=0$, then the value of $\left(x+\dfrac{1}{x}\right)^{2}+\left(x^{2}+\dfrac{1}{x^{2}}\right)^{2}+\ldots \ldots \ldots \ldots+\left(x^{27}+\dfrac{1}{x^{27}}\right)^{2}$ is

(a) 27

(b) 72

(c) 54

(d) None of these

3. If $\alpha, \beta, \gamma$ are roots of $x^{3}-3 x^{2}+3 x+7=0$ and $\omega$ is a complex cube roots of unity, then $\dfrac{\alpha-1}{\beta-1}+\dfrac{\beta-1}{\gamma-1}+\dfrac{\gamma-1}{\alpha-1}$ is equal to

(a) $\omega$

(b) $2 \omega$

(c) $2 \omega^{2}$

(d) $3 \omega^{2}$

4. If $(x-1)^{4}-16=0$, then the sum of non real complex roots of the equation is

(a) 2

(b) 0

(c) 4

(d) None of these

5. If $z$ is a non-real root of $\sqrt[7]{-1}$, then $z^{86}+z^{175}+z^{289}$ is equal to

(a) 0

(b) -1

(c) 3

(d) 1

Exercise

1. The value of $\sum\limits _{\mathrm{k}=1}^{6}\left(\sin \dfrac{2 \pi \mathrm{k}}{7}-\mathrm{i} \cos \dfrac{2 \pi \mathrm{k}}{7}\right)$ is

(a) -1

(b) 0

(c) $-\mathrm{i}$

(d) i

2. For positive integers $n _{1}, n _{2}$ the value of expression $(1+i)^{n _{1}}+\left(1+i^{3}\right)^{n _{1}}+\left(1+i^{5}\right)^{n _{2}}+\left(1+i^{7}\right)^{n _{2}}$ is a real number if and only if

(a) $\mathrm{n} _{1}=\mathrm{n} _{2}+1$

(b) $\mathrm{n} _{1}=\mathrm{n} _{2}-1$

(c) $\mathrm{n} _{1}=\mathrm{n} _{2}$

(d) $n _{1}>0, n _{2}>0$

3. The minimum value of $\left|\mathrm{a}+\mathrm{b} \omega+\mathrm{c} \omega^{2}\right|$ where $\mathrm{a}, \mathrm{b} & \mathrm{c}$ are all not equal integers and $\omega(\neq 1)$ is $\mathrm{a}$ cube root of unity, is

(a) $\sqrt{3}$

(b) $\dfrac{1}{2}$

(c) 1

(d) 0

4. If $x$ is a complex root of the equation

$\left|\begin{array}{lll} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{array}\right|+\left|\begin{array}{ccc} 1-x & 1 & 1 \\ 1 & 1-x & 1 \\ 1 & 1 & 1-x \end{array}\right|=0 \text {, then } x^{2005}+\dfrac{1}{x^{2005}} \text { is }$

(a) 1

(b) -1

(c) i

(d) $\omega$

5. If $\mathrm{z}=\mathrm{i} \log _{\mathrm{e}}(2-\sqrt{3})$, then $\operatorname{cosz}=$

(a) 0

(b) 1

(c) 2

(d) None of these

6. If $\omega$ is a complex cube roots of unity and $a, b, c$ are such that $\dfrac{1}{a+\omega}+\dfrac{1}{b+\omega}+\dfrac{1}{c+\omega}=2 \omega^{2}$ and $\dfrac{1}{\mathrm{a}+\omega^{2}}+\dfrac{1}{\mathrm{~b}+\omega^{2}}+\dfrac{1}{\mathrm{c}+\omega^{2}}=2 \omega$, then $\dfrac{1}{\mathrm{a}+1}+\dfrac{1}{\mathrm{~b}+1}+\dfrac{1}{\mathrm{c}+1}=$

(a) 1

(b) -1

(c) 2

(d) -2

7. If $\omega(\neq 1)$ be a cube root of unity and $\left(1+\omega^{2}\right)^{\mathrm{m}}=\left(1+\omega^{4}\right)^{\mathrm{m}}$, then the least positive integral value of mis

(a) 2

(b) 5

(c) 1

(d) 3

8. $\cot \left(-i \log _{e}\left(\dfrac{x+i y}{x-i y}\right)\right)$

(a) $\dfrac{x^{2}-y^{2}}{2 x y}$

(b) $\dfrac{2 x y}{x^{2}-y^{2}}$

(c) $\dfrac{2 x y}{x^{2}+y^{2}}$

(d) $\dfrac{y^{2}-x^{2}}{2 x y}$

9. $\quad \sin \dfrac{2 \pi}{7}+\sin \dfrac{4 \pi}{7}+\sin \dfrac{8 \pi}{7}=$

(a) $\dfrac{\sqrt{7}}{2}$

(b) $\dfrac{-1}{2}$

(c) $\dfrac{1}{8}$

(d) None of these

10. If $\alpha _{0}, \alpha _{1}, \alpha _{2}, \ldots \ldots \ldots \ldots . ., \alpha _{n-1}$ be the $n, n^{\text {th }}$ roots of the unity then the value of $\sum\limits _{i=0}^{n-1} \dfrac{\alpha _{i}}{\left(3-\alpha _{i}\right)}$ is equal to

(a) $\dfrac{3}{3^{\mathrm{n}}-1}$

(b) $\dfrac{\mathrm{n}-1}{3^{\mathrm{n}}-1}$

(c) $\dfrac{\mathrm{n}+1}{3^{\mathrm{n}}-1}$

(d) $\dfrac{\mathrm{n}+2}{3^{\mathrm{n}}-1}$