Argument of a Complex Number

Q. If $|\sin x+\cos x|=|\sin x|+|\cos x|$, where $\sin x\ne 0,\cos x\ne 0$, then in which quadrant does $x$ lie?

1. $\text{Quadrant I}$
2. $\text{Quadrant II}$
3. $\text{Quadrant III}$
4. $\text{Quadrant IV}$

Q. If $z_1,z_2$ are two complex numbers such that $|z_1|=|z_2|=2$ and $\arg z_1+\arg z_2=0$, then $z_1{z}_2=$

Q. Let $z,w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and $\arg zw=\pi$. Then $\arg z$ equals

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

Q. Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i|$ $\ge \sqrt{5}$. If the complex number $z_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set $\{\frac{1}{|z-1|}:z\in S\}$, then the principal argument of $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$ is

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

Q. Let $z_1,z_2,z_3,z_4$ are distinct complex numbers representing the vertices of a quadrilateral $ABCD$ taken in order. If $z_1-z_3=z_2-z_4$ and $\arg \left(\frac{z_4-z_1}{z_2-z_1}\right)=\frac{\pi }{2},$ then the quadrilateral is

1. rectangle
2. rhombus
3. square
4. trapezium

Q. Argument and modulus of $\frac{1+i}{1-i}$ are respectively
[RPET 1984; MP PET 1987; Karnataka CET 2001]

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

Q. If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then find the value of $\text{Im}(z_1+z_2)$. (where $\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

1. $\sin \frac{\pi }{3}$
2. $\text{cosec}\frac{\pi }{3}$
3. $\tan \frac{\pi }{3}$
4. $\cot \frac{\pi }{3}$

Q. If $\frac{3+i\sin \theta }{4-i\cos \theta },\theta \in [0,2\pi ]$, is a real number, then an argument of $\sin \theta +i\cos \theta$ is :

1. $\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$
2. $-{\tan }^{-1}\left(\frac{3}{4}\right)$
3. $\pi -{\tan }^{-1}\left(\frac{3}{4}\right)$
4. ${\tan }^{-1}\left(\frac{4}{3}\right)$

Q. If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then the value of $\text{Im}(z_1+z_2)$ is
($\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

1. $\sin \frac{\pi }{3}$
2. $\text{cosec}\frac{\pi }{3}$
3. $\tan \frac{\pi }{3}$
4. $\cot \frac{\pi }{3}$

Q. Let $z$ be a complex number such that $z+|z|=3+i$. Then which of the following is/are true?

1. ${\sin }^3\theta +{\cos }^3\theta =\frac{91}{125}$, where $\theta =\arg z$
2. $|z|=\frac{3}{5}$
3. $Re\left(\log z\right)=\log \left(\frac{5}{3}\right)$
4. $z-\overline{z}=-2i$

Q. If $\sqrt{5-12i}+\sqrt{-5-12i}=z$, then principal value of $\arg z$ can be

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