Properties of Argument

Q.

Let $z$ and $\omega$ be two non zero complex numbers such that $\left|z\right|=\left|\omega \right|$ and $Argz+Arg\omega =\pi$, then $z$ equals:

1. $\omega$

2. $-\omega$

3. $\overline{\omega }$

4. $-\overline{\omega }$

Q. If complex numbers $z_1,z_2$ are such that $|z_1|=\sqrt{2},|z_2|=\sqrt{3}\text{and}|z_1+z_2|=\sqrt{5-2\sqrt{3}}$, then the value of $|\arg \left(z_1\right)-\arg \left(z_2\right)|$ is

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{3\pi }{4}$

Q. If $arg\left(z\right)<0$, then $arg(-z)-arg\left(z\right)$ equals

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

Q. If ${\theta }_1$ and ${\theta }_2$ are arguments of two non-zero complex numbers $z_1$ and $z_2$ respectively such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$, then

1. $|z|=\sqrt{\frac{5}{2}}$
2. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1-{\theta }_2)$
3. $|z|=\sqrt{\frac{5}{4}}$
4. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1+{\theta }_2)$

Q. If z and $\omega$ are two non-zero complex numbers such that $|z\omega |=1$ and arg $\left(z\right)-arg\left(\omega \right)=\frac{\pi }{2},$ then $\overline{z}\omega$ is equal to

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

Q. Let $z$ and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and $argz\omega =\pi$. Then $argz$ equals

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

Q. For two complex numbers $z_1$ and $z_2;(a{z}_1+b\overline{z_1})(c{z}_2+d\overline{z_2})=(c{z}_1+d\overline{z_1})(a{z}_2+b\overline{z_2})$ $b\ne 0,d\ne 0$ if

1. $\frac{a}{b}=\frac{c}{d}$
2. $\frac{a}{d}=\frac{b}{c}$
3. $|z_1|=|z_2|$
4. $\arg \left(z_1\right)=\arg \left(z_2\right)$

Q. If $z=\frac{(1+i)(1+2i)(1+3i)\dots \dots \dots (1+ni)}{(1-i)(2-i)(3-i)\dots \dots \dots (n-i)}$, where $i=\sqrt{-1},n\in N$, then principal argument of $z$ can be -

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

Q. If complex numbers $z_1,z_2$ are such that $|z_1|=\sqrt{2},|z_2|=\sqrt{3}\text{and}|z_1+z_2|=\sqrt{5-2\sqrt{3}}$, then the value of $|\arg \left(z_1\right)-\arg \left(z_2\right)|$ is

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $\frac{3\pi }{4}$

Q. If $x=9^{1/3}\cdot 9^{1/9}\cdot 9^{1/27}\cdots \infty ;y=4^{1/3}\cdot 4^{-1/9}\cdot 4^{1/27}\cdots \infty$ and $z=\sum _{r=1}^{\infty }(1+i{)}^{-r}$ and the principal argument of the complex number $p=x+yz$ is $-{\tan }^{-1}\frac{\sqrt{a}}{b}$, then the value of $a^2+b^2$ is ($a$ and $b$ are coprime natural numbers)

Q. Let $z$ and $w$ be two non- zero complex numbers such that $|z|=|w|$ and $\arg \left(z\right)+\arg \left(w\right)=\pi$. Then the value of $(z+\overline{w}{)}^{10}$ is

Q. If $\arg \left(z_1\right)={170}^{\circ }$ and $\arg \left(z_2\right)={70}^{\circ }$, then the principal argument of $z_1{z}_2$ is

1. ${120}^{\circ }$
2. $-{120}^{\circ }$
3. ${240}^{\circ }$
4. $-{240}^{\circ }$

Q. If $z=\sqrt{3}+i$ and $w=3i$, then $\arg \left(zw\right)$ and $\arg \frac{w}{z}$ is

1. $\arg \left(zw\right)=\frac{2\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{3}$
2. $\arg \left(zw\right)=\frac{2\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{2\pi }{3}$
3. $\arg \left(zw\right)=\frac{\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{3}$
4. $\arg \left(zw\right)=\frac{\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{5}$

Q. If $a=\cos \alpha +i\sin \alpha ,b=\cos \beta +i\sin \beta ,$
$c=\cos \gamma +i\sin \gamma$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=1,$ then

1. $\cos (\alpha -\beta )+\cos (\beta -\gamma )+\cos (\gamma -\alpha )=0$
2. $\sin (\alpha -\beta )+\sin (\beta -\gamma )+\sin (\gamma -\alpha )=0$
3. $\cos (\alpha -\beta )+\cos (\beta -\gamma )+\cos (\gamma -\alpha )=1$
4. $\sin (\alpha -\beta )+\sin (\beta -\gamma )+\sin (\gamma -\alpha )=1$

Q. If $\arg \left(z_1\right)={170}^{\circ }$ and $\arg \left(z_2\right)={70}^{\circ }$, then the principal argument of $z_1{z}_2$ is

1. ${120}^{\circ }$
2. $-{120}^{\circ }$
3. ${240}^{\circ }$
4. $-{240}^{\circ }$

Q. If $z=\frac{(1+i)(1+2i)(1+3i)\dots \dots \dots (1+ni)}{(1-i)(2-i)(3-i)\dots \dots \dots (n-i)}$, where $i=\sqrt{-1},n\in N$, then principal argument of $z$ can be -

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

Q. For a non-zero complex number $z$, let $\arg \left(z\right)$ denote the principal argument with $-\pi <\arg \left(z\right)\le \pi$. Then which of the following statement(s) is (are) $\text{FALSE}$?

1. $\arg (-1-i)=\frac{\pi }{4},$ where $i=\sqrt{-1}$
2. The function $f:R\to (-\pi ,\pi ]$, defined by $f\left(t\right)=\arg (-1+it)$ for all $t\in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$
3. For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)$ is an integer multiple of $2\pi$.
4. For any three given distinct complex numbers $z_1,z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$,
   lies on a straight line.

Q. Let $z$ and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and $argz\omega =\pi$. Then $argz$ equals

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

Q. Let $z_1$ and $z_2$be nth roots of unity which subtend a right angle at the origin, then n must be of the form (where, k is an integer)

1. 4k+1
2. 4k+2
3. 4k+3
4. 4k