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 $z _{1}=x _{1}+i y _{1} z _{2}=x _{2}+i y _{2}$

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

Conjugate of $z$

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

Properties

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

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

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

Geometric representation

4.

Modulus and amplitude

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

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

Here $-\pi<\arg \mathrm{z} \leq \pi$

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

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

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

Properties of modulus

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

(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 $\left|\mathrm{z}^{\mathrm{n}}\right|=|\mathrm{z}|^{\mathrm{n}}$

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

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

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

Properties of argument

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

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

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

(iv) $\quad $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 $\arg (\mathrm{z})=\frac{\pi}{2} \Leftrightarrow \mathrm{z}$ is purely imaginary, $\operatorname{Im} z>0$

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

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

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

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

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

$\quad\arg z _{1}-\arg z _{2}=\pi, \frac{z _{1}}{z _{2}}<0$

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

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

$\quad\operatorname{argz _{1}-\operatorname {argz}} \mathrm{z} _{2}= \pm \frac{\pi}{2}$

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

De Moivres theorem

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

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

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

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

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

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

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

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

The $\mathbf{n}^{\text {th }}$ roots of unity

7. $ \mathrm{z}=\sqrt[n]{1}=1^{1 / \mathrm{n}}=\mathrm{e}^{\mathrm{i} 2 r \pi / \mathrm{n}}, \mathrm{r}=0,1,2, \ldots \ldots \ldots \ldots,(\mathrm{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 $\alpha=\mathrm{e}^{\mathrm{i} 2 \pi / \mathrm{n}}=\cos \frac{2 \pi}{\mathrm{n}}+\mathrm{i} \sin \frac{2 \pi}{\mathrm{n}}$

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 $e^{\mathrm{i} 2 \pi / \mathrm{n}}$

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

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

Cube roots of unity

8. $1^{1 / 3} =\cos \frac{2 \mathrm{r} \pi}{3}+i \sin \frac{2 \mathrm{r} \pi}{3} ; r=0,1,2 $

$ =1, \frac{-1+\mathrm{i} \sqrt{3}}{2}, \frac{-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 $\omega=\frac{-1+\mathrm{i} \sqrt{3}}{2}$, then $\omega^{2}=\frac{-1-\mathrm{i} \sqrt{3}}{2}$

$\therefore$ 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) $\quad \omega \& \omega^{2}$ 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) $\bar{\omega}=\frac{1}{\omega}=\omega^{2} ; \overline{\omega^{2}}=\frac{1}{\omega^{2}}=\omega$

(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 $\therefore$ 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 $\left|\frac{\text { real part }}{\text { imaginary part }}\right|=1: \sqrt{3}$ or $\sqrt{3}: 1$, can be expressed in terms of $\omega, \omega^{2} \& \mathrm{i}$

Square root of a complex number

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

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

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

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

Logarithm of a complex number

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

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

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

Expansions

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

(iii) $\cos ^{\mathrm{n}} \theta=\frac{1}{2^{\mathrm{n}-1}}\left\{{ }^{\mathrm{n}} \mathrm{C} _{0} \cos n \theta+{ }^{\mathrm{n}} \mathrm{C} _{1} \cos (\mathrm{n}-2) \theta+{ }^{+\mathrm{n}} \mathrm{C} _{2} \cos (\mathrm{n}-4)+………\right\}$

(iv) $\sin ^{\mathrm{n}} \theta=\frac{(-1)^{\mathrm{n} / 2}}{2^{\mathrm{n}-1}}\left\{\cos n{ }^{-\mathrm{n}} \mathrm{C} _{1} \cos (\mathrm{n}-2) \theta^{+\mathrm{n}} \mathrm{C} _{2} \cos (\mathrm{n}-4) \theta {+}…………\right\}$

$ \sin ^{\mathrm{n}} \theta=\frac{(-1)^{\frac{\mathrm{n}-1}{2}}}{2^{\mathrm{n}-1}}\left\{\sin n \theta^{-\mathrm{n}} \mathrm{C} _{1} \sin (\mathrm{n}-2) \theta^{+}{ }^{\mathrm{n}} \mathrm{C} _{2} \sin (\mathrm{n}-4) \theta +\ldots \ldots \ldots .\right\} $

Solved Examples

1. If $\omega$ is a complex cube root of unity, then the value of $\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}+\frac{a+b \omega+c \omega^{2}}{b+c \omega+a \omega^{2}}$ 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 $\left(x+\frac{1}{x}\right)^{2}+\left(x^{2}+\frac{1}{x^{2}}\right)^{2}+\ldots \ldots \ldots \ldots+\left(x^{27}+\frac{1}{x^{27}}\right)^{2}$ 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 $\frac{\alpha-1}{\beta-1}+\frac{\beta-1}{\gamma-1}+\frac{\gamma-1}{\alpha-1}$ 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) $1$

Practice questions

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

(a) $-1$

(b) $0$

(c) $-\mathrm{i}$

(d) $\mathrm{i}$

2. For positive integers $\mathrm{n} _{1}, \mathrm{n} _{2}$ 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) $n _{1}=n _{2}-1$

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

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

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 a cube root of unity, is

(a) $\sqrt{3}$

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

(c) $1$

(d) $0$

4. If $\mathrm{x}$ is a complex root of the equation

$ \left|\begin{array}{lll} 1& \mathrm{x} & \mathrm{x} \\ \mathrm{x} & 1 & \mathrm{x} \\ \mathrm{x} & \mathrm{x} & 1 \end{array}\right|+\left|\begin{array}{ccc} 1-\mathrm{x} & 1 & 1 \\ 1& 1-\mathrm{x} & 1 \\ 1& 1 & 1-\mathrm{x} \end{array}\right|=0 \text {, then } \mathrm{x}^{2005}+\frac{1}{\mathrm{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 $\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}=2 \omega^{2}$ and $\frac{1}{\mathrm{a}+\omega^{2}}+\frac{1}{\mathrm{~b}+\omega^{2}}+\frac{1}{\mathrm{c}+\omega^{2}}=2 \omega$, then $\frac{1}{\mathrm{a}+1}+\frac{1}{\mathrm{~b}+1}+\frac{1}{\mathrm{c}+1}=$

(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 $\mathrm{m}$ is

(a) $2$

(b) $5$

(c) $1$

(d) $3$

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

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

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

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

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

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

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

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

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

(d) None of these

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

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

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

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

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