S.No.	COLUMN - I	S.No.	COLUMN - II
A)	$\text{ If }{\mathrm{Z}}_1=1+\mathrm{i},{\mathrm{Z}}_2=\sqrt{3}+\mathrm{i},\text{ then }\arg {\left(\frac{{\mathrm{Z}}_1}{{\mathrm{Z}}_2}\right)}^4=$	p)	$\frac{3\mathrm{\pi }}{4}$
B)	$\text{ If }|\mathrm{Z}-\mathrm{i}|=\sqrt{2}.\text{ Then }\arg \left(\frac{\mathrm{Z}-1}{\mathrm{Z}+1}\right)=$	q)	$\pm \frac{\mathrm{\pi }}{2}$
C)	$\text{ If }\overline{\mathrm{Z}}+\mathrm{i}\overline{\mathrm{W}}=0,\arg (\mathrm{Z}.\mathrm{W})=\mathrm{\pi },\text{ then }\arg \left(\mathrm{Z}\right)=$	r)	$\frac{\mathrm{\pi }}{3}$
D)	$\text{ If }\left|{\mathrm{Z}}^2-1\right|=\left|{\mathrm{Z}}^2\right|+1,\text{ then }\arg \left(\mathrm{Z}\right)=$	s)	$\frac{\mathrm{\pi }}{4}$

$\text{ A) }4\arg \left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)=4\left(\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{6}\right)=\frac{\mathrm{\pi }}{3}$
$\text{ B) }\left|z-i\right|\text{=}\sqrt{2}\text{ represents a circle with centre C(0,1) and radius }\sqrt{2}$

$\mathrm{A}(1,0) \mathrm{and} \mathrm{B}(-1,0) \mathrm{lies} \mathrm{on} \mathrm{the} \mathrm{circle} \mathrm{and} \mathrm{AB} \mathrm{subtends} \mathrm{right} \mathrm{angle} \mathrm{at} \mathrm{the} \mathrm{centre}$

$\mathrm{So} \mathrm{angle} \mathrm{subtended} \mathrm{by} \mathrm{AB} \mathrm{at} \mathrm{a} \mathrm{point} \mathrm{P}\left(\mathrm{z}\right) \mathrm{on} \mathrm{the} \mathrm{circle} \mathrm{is} \frac{\mathrm{\pi }}{4}\Rightarrow \arg \left(\frac{\mathrm{Z}-1}{\mathrm{Z}+1}\right)= \frac{\mathrm{\pi }}{4}$
$$\begin{gathered} \text{ C) }\overline{\mathrm{z}}+\mathrm{i}\overline{\mathrm{\omega }}=0\Rightarrow \mathrm{z}-\mathrm{i\omega }=0\Rightarrow \mathrm{z}=\mathrm{i\omega }; \\ \arg \mathrm{z}+\arg \mathrm{w}=\mathrm{\pi } \\ \arg \mathrm{z}+\arg \frac{\mathrm{z}}{\mathrm{i}}=\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} \text{ or }\arg z+\arg z-\arg i=\pi \\ \text{ or }2\arg z-\frac{\pi }{2}=\pi \\ \text{ or }2\arg z=\frac{3\pi }{2} \\ \text{ or }\arg z=\frac{3\pi }{4} \end{gathered}$$
$\text{ D) }{\mathrm{z}}^2\text{ and }1\text{ have opposite directions }\arg \left({\mathrm{z}}^2\right)=\mathrm{\pi }$
etc…….