Consider the system of equations $x - 2y + 3z = -1; -x + y - 2z = k; x - 3y + 4z = 1$ \ Statement-I : The system of equations has no solution for $k \neq 3$ and \ Statement-II : The determinant $\begin{vmatrix} 1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1 \end{vmatrix} \neq 0$, for $k \neq 3$.

$x-2y+3z=-1; \quad -x+y-2z=k; \quad x-3y+4z=1$ $\text{S-I: no solution for } k\ne 3$ $\text{S-II: } \begin{vmatrix} 1 & 3 & -1 \\ -1 & -2 & k \\ 1 & 4 & 1 \end{vmatrix}\ne 0 \text{ for } k\ne 3$ $\text{Check consistency of the system, then the determinant}$ $\text{Choose the correct option (1)--(4)}$ $1.\ \det A \text{ and the matrix form } AX=B$ $2.\ \text{consistency for each } k$ $3.\ \text{the determinant of Statement-II}$ $A=\begin{bmatrix} 1 & -2 & 3 \\ -1 & 1 & -2 \\ 1 & -3 & 4 \end{bmatrix}, \quad B=\begin{bmatrix} -1 \\ k \\ 1 \end{bmatrix}$ $\det A=1(4-6)+2(-4+2)+3(3-1)$ $=-2-4+6=0$ $P_3-P_1:\ -y+z=2 \Rightarrow z=y+2$ $x=-1+2y-3z=-y-7$ $-x+y-2z=-(-y-7)+y-2(y+2)=3$ $3=k$ $\text{Consistent} \iff k=3 \Rightarrow \text{no solution for } k\ne3$ $\text{Statement-I: TRUE}$ $\Delta=\begin{vmatrix}1&3&-1\\-1&-2&k\\1&4&1\end{vmatrix}$ $=1(-2-4k)-3(-1-k)+(-1)(-4+2)$ $=(-2-4k)+(3+3k)+2$ $\Delta=3-k$ $\Delta=3-k\ne0 \iff k\ne3$ $\text{Statement-II: TRUE}$ $\det A=0,\ \Delta\ne0 \Rightarrow \text{inconsistent}$ $\Rightarrow \text{S-II explains S-I}$ $\text{(2) wrong: II is the explanation}$ $\text{(3) wrong: S-II is true}; \quad \text{(4) wrong: S-I is true}$ $\text{Both true; II explains I}$ $\boxed{\text{Answer: option (1)}}$