1. Find the 5 th roots of $\frac{(-1-\sqrt{3} i)^{2021}}{(- \sqrt{3}+i)^{3}(-2+2 i)^{12}}$ and plot them in the complex plane 2. Consider the linear system $$ \left(\begin{array}{1111} 1 & 1 & 1 & 2 2 & 3 & 3 & 5 4 & 2 & a&7 3 & 4 & 4 & a \end{array} ight]\left(\begin{array}{1} x_{1} 1 x_{2} x_{3} x_{4} \end{array} ight)=\left[\begin{array}{1} 11 4 V 011 \end{array} ight) $$ (a) Find the conditions satisfied by $a$ and $b$ such that the system has i. no solutions; ii. infinitely many solutions; iii. a unique solution. (b) Consider the vectors $\boldsymbol{v}_{1}=\left\begin{array}{1}1 1 2 4 3\end{array} ight], \boldsymbol{v}_{2}=\left(\begin{array}{1}1 \\ 3 12 4\end{array} ight), \boldsymbol{v}_{3}=\left(\begin{array}{1}1 3 Vall 4\end{array} ight], \boldsymbol{v}_{4}=\left(\begin{array}{1}2 15171a\end{array} ight]$ and $\boldsymbol{v}_{5}=\left[\begin{array}{1}1 1 4 10 b\end{array} ight]$ in $\mathbb{R}_{4}$ i. Find the values of $a$ and $b$ such that $\boldsymbol{v}_{3}, \boldsymbol{v}_{4}$ and $\boldsymbol{v}_{5} $ are linearly dependent and write $\boldsymbol{v}_{3}$ as a linear combination of $\boldsymbol{v}_{4}$ and $\boldsymbol{v}_{5}$, if possible. ii. Find the values of $a$ and $b$ such that $$ operatornamespan}\left\{\boldsymbol{v}_{1}, \boldsymbol{v}_{2}, \boldsymbol{v}_{3}, \boldsymbol{v}_{4}, \boldsymbol{v}_{5} ight\} eg \mathbb{R}_{4} $$ CS.SD. 109