If $\mathrm{z}=\cos \mathrm{\alpha }+\mathrm{isin}\mathrm{\alpha },0<\mathrm{\alpha }<\mathrm{\pi }/6$, then the principal argument of:

S.No.	COLUMN-I	S.No.	COLUMN-II
A)	$1+{\mathrm{z}}^3$	p)	$2\mathrm{\alpha }-\frac{\mathrm{\pi }}{2}$
B)	$1-{\mathrm{z}}^4$	q)	$\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\alpha }}{2}$
C)	$\frac{1+{\mathrm{z}}^3}{1-{\mathrm{z}}^4}$	r)	$\frac{3\mathrm{\alpha }}{2}$
D)	$\frac{{\mathrm{z}}^4-1}{{\mathrm{z}}^3+1}$	s)	$\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\alpha }}{2}$

$\text{ A) }1+{\mathrm{z}}^3=1+\cos 3\mathrm{\alpha }+\mathrm{isin}3\mathrm{\alpha }$
$=2{\cos }^2\frac{3\mathrm{\alpha }}{2}+2\mathrm{isin}\frac{3\mathrm{\alpha }}{2}\cos \frac{3\mathrm{\alpha }}{2}\Rightarrow \arg \left(1+{\mathrm{z}}^3\right)=3\mathrm{\alpha }/2$
$\text{ since }\cos \left(\frac{3\mathrm{\alpha }}{2}\right)>0$
$\text{ B) }1-{\mathrm{z}}^4=1-\cos 4\mathrm{\alpha }-\mathrm{isin}4\mathrm{\alpha }$
$\begin{array}{l}=2{\sin }^{2}2\mathrm{\alpha }-2\mathrm{icos}2\mathrm{\alpha sin}2\mathrm{\alpha }\\ \Rightarrow \arg \left(1-{\mathrm{z}}^4\right)=2\mathrm{\alpha }-\mathrm{\pi }/2\text{ since }\sin 2\mathrm{\alpha }>0\end{array}$
$\text{ C) }\frac{1+{\mathrm{z}}^3}{1-{\mathrm{z}}^4}=\frac{\cos (3\mathrm{\alpha }/2)}{\sin \left(2\mathrm{\alpha }\right)}\left[\cos \left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\alpha }}{2}\right)+\mathrm{isin}\left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\alpha }}{2}\right)\right]$
$\Rightarrow \arg \left(\frac{1+{\mathrm{z}}^3}{1-{\mathrm{z}}^4}\right)=\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\alpha }}{2}$
$\text{ D) }\frac{{\mathrm{z}}^4-1}{{\mathrm{z}}^3+1}=\frac{\sin \left(2\mathrm{\alpha }\right)}{\cos (3\mathrm{\alpha }/2)}\left[\cos \left(\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\alpha }}{2}\right)+\mathrm{isin}\left(\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\alpha }}{2}\right)\right]$
$\text{ Thus, }\arg \left(\frac{{\mathrm{z}}^4-1}{{\mathrm{z}}^3+1}\right)=\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\alpha }}{2}\text{. }$