Evaluate the determinant
$\left(i\right)$ $\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& 0\end{array}\right|$
$\left(ii\right)$ $\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$
$\left(iii\right)$ $\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$
$\left(iv\right)$ $\left|\begin{array}{ccc}2& -1& -2\\ 0& 2& -1\\ 3& -5& 0\end{array}\right|$

$\left(i\right)$ Given: $\left|\begin{array}{ccc}3& -1& -2\\ 0& 0& -1\\ 3& -5& 0\end{array}\right|$
$=3\left|\begin{array}{cc}0& -1\\ -5& 0\end{array}\right|-(-1)\left|\begin{array}{cc}0& -1\\ 3& 0\end{array}\right|+(-2)\left|\begin{array}{cc}0& 0\\ 3& -5\end{array}\right|$
$=3\left(0\right(0)-(-5\left)\right(-1)+1(0\left(0\right)-3(-1))-2(0(-5)-3\left(0\right))$
$=3(0-5)+1(0+3)-2(0-0)$
$=3(-5)+1\left(3\right)+0$
$=-15+3$
$=-12$

$\left(ii\right)$ Given: $\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$
$=3\left|\begin{array}{cc}1& -2\\ 3& 1\end{array}\right|-(-4)\left|\begin{array}{cc}1& -2\\ 2& 1\end{array}\right|+5\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|$
$=3\left(1\right(1)-3(-2\left)\right)+4\left(1\right(1)-(2\left)\right(-2\left)\right)+5\left(1\right(3)-2(1\left)\right)$
$=3(1+6)+4(1+4)+5(3-2)$
$=3\left(7\right)+4\left(5\right)+5\left(1\right)$
$=21+20+5$
$=46$

$\left(iii\right)$ Given: $\left|\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right|$
$=0\left|\begin{array}{cc}0& -3\\ 3& 0\end{array}\right|-1\left|\begin{array}{cc}-1& -3\\ -2& 0\end{array}\right|+2\left|\begin{array}{cc}-1& 0\\ -2& 3\end{array}\right|$
$=0\left(0\right(0)-3(-3\left)\right)-1(-1(0)-(-2\left)\right(-3\left)\right)+2(-1(3)-(-2\left)0\right)$
$=0(0+9)-1(0-6)+2(-3-0)$
$=-1(-6)+2(-3)$
$=6-6$
$=0$

$\left(iv\right)$ Given: $\left|\begin{array}{ccc}2& -1& -2\\ 0& 2& -1\\ 3& -5& 0\end{array}\right|$
$=2\left|\begin{array}{cc}2& -1\\ -5& 0\end{array}\right|-(-1)\left|\begin{array}{cc}0& -1\\ 3& 0\end{array}\right|+(-2)\left|\begin{array}{cc}0& 2\\ 3& -5\end{array}\right|$
$=2\left(2\right(0)-(-5\left)\right(-1\left)\right)+1\left(0\right(0)-3(-1\left)\right)-2\left(0\right(-5)-3(2\left)\right)$
$=2(0-5)+1(0+3)-2(0-6)$
$=2(-5)+1\left(3\right)-2(-6)$
$=-10+3+12$
$=5$