Solution:

(i) Given that $4 x-2 y=3$
$6 x-3 y=5$
We can write the above system of equations as
$\left[\begin{array}{ll}
4 & -2 \\
6 & -3
\end{array}\right]\left[\begin{array}{l}
\mathrm{X} \\
\mathrm{Y}
\end{array}\right]=\left[\begin{array}{l}
3 \\
5
\end{array}\right]_{\text {Or } \mathrm{AX}=\mathrm{B}}$
Where $A=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right]_{B}=\left[\begin{array}{l}3 \\ 5\end{array}\right]$ and $X=\left[\begin{array}{l}X \\ Y\end{array}\right]$
$|A|=-12+12=0$
Therefore, A is singular,
Now $X$ will be consistence if $(\operatorname{Adj} A) \times B=0$
$\begin{array}{l}
C_{11}=(-1)^{1+1}-3=-3 \\
C_{12}=(-1)^{1+2} 6=-6
\end{array}$
$\begin{array}{l}
C_{21}=(-1)^{2+1}-2=2 \\
C_{22}=(-1)^{2+2} 4=4
\end{array}$
Also, adj $A=\left[\begin{array}{cc}-3 & -2 \\ -6 & 4\end{array}\right]^{T}$ $=\left[\begin{array}{ll}-3 & 2 \\ -6 & 4\end{array}\right]$ $(\operatorname{Adj} A) \cdot B=\left[\begin{array}{ll}-3 & 2 \\ -6 & 4\end{array}\right]\left[\begin{array}{l}3 \\ 5\end{array}\right]$ $=\left[\begin{array}{c}-9+10 \\ -18+20\end{array}\right]=\left[\begin{array}{l}1 \\ 2\end{array}\right]$
As a result, the given system is inconsistent.

(ii) Given that $4 x-5 y-2 z=2$
$\begin{array}{l}
5 x-4 y+2 z=-2 \\
2 x+2 y+8 z=-1 \\
\quad\left[\begin{array}{ccc}
4 & -5 & -2 \\
5 & -4 & 2 \\
2 & 2 & 8
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
2 \\
-2 \\
-1
\end{array}\right] \\
|A|=4(-36)+5(36)-2(18) \\
|\mathrm{A}|=0
\end{array}$
Cofactors of A are:
$\begin{array}{l}
C_{11}=(-1)^{1+1}-32-4=-36 \\
C_{21}=(-1)^{2+1}-40+4=-36 \\
C_{31}=(-1)^{3+1}-10-8=-18 \\
C_{12}=(-1)^{1+2} 40-4=-36 \\
C_{22}=(-1)^{2+1} 32+4=36 \\
C_{32}=(-1)^{3+1} 8+10=-18 \\
C_{13}=(-1)^{1+2} 10+8=18 \\
C_{23}=(-1)^{2+1} 8+10=-18 \\
C_{33}=(-1)^{3+1}-16+25=9
\end{array}$
$\begin{array}{l}
\operatorname{Adj} A=\left[\begin{array}{ccc}
-36 & -34 & 18 \\
36 & 36 & -18 \\
-18 & -18 & 9
\end{array}\right]^{\mathrm{T}} \\
=\left[\begin{array}{ccc}
-36 & 36 & -18 \\
-36 & 36 & -18 \\
18 & -18 & 9
\end{array}\right] \\
\operatorname{Adj} \mathrm{A} \times \mathrm{B}=\left[\begin{array}{ccc}
-36 & 36 & -18 \\
-36 & 36 & -18 \\
18 & -18 & 9
\end{array}\right]\left[\begin{array}{c}
2 \\
-2 \\
-1
\end{array}\right] \\
=\left[\begin{array}{c}
-72-72+18 \\
-72-72+18 \\
36+36-9
\end{array}\right] \neq\left[\begin{array}{l}
0 \\
0 \\
0
\end{array}\right]
\end{array}$
As a result, the above system is inconsistent.