Geometrical Representation of Argument and Modulus

Q. If $z=-3+4i$ and $zw=-14+2i$, then

1. $\arg w=\frac{\pi }{4}$
2. $\arg w=-\frac{\pi }{4}$
3. $|w|=2\sqrt{2}$
4. $|w|=4$

Q. If $z$ is a complex number such that $|3z-2|+|3z+2|=4$, then the locus of $z$ is

1. a line
2. a circle
3. a point
4. a line segment

Q. If $z$ be a complex number satisfying $|Re\left(z\right)|+|Im\left(z\right)|=4$, then $|z|$ cannot be:

1. $\sqrt{7}$
2. $\sqrt{\frac{17}{2}}$
3. $\sqrt{10}$
4. $\sqrt{8}$

Q. If $\left|\frac{z_1}{z_2}\right|=1$ and $\arg \left(z_1{z}_2\right)=0$, then

1. $z_1=z_2$
2. $|z_2{|}^2=z_1{z}_2$
3. $z_1{z}_2=1$
4. $z_1=2z_2$

Q. For a complex number $z,$ if $|z-1+i|+|z+i|=1,$ then the range of the principle argument of $z$ is
$($ Here, principle argument $\in (-\pi ,\pi ])$

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

Q. Polar form of $z=\frac{(1+7i)}{(2-i{)}^2}$ is

1. $\sqrt{2}(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
2. $2(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
3. $\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$
4. $2(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$

Q. The number of complex numbers $z$ that satisfies both the equation $|z-1-i|=\sqrt{2}$ and $|z+1+i|=2$, is

1. $1$
2. $2$
3. $4$
4. infinite

Q. If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\arg \left(z\right)-\arg \left(\omega \right)=\frac{3\pi }{2},$ then $\arg \left(\frac{1-2\overline{z}\omega }{1+3\overline{z}\omega }\right)$ is
(Here $\arg \left(z\right)$ denotes the principal argument of complex number $z$ )

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