Question

Consider the system of equations
$x-2y+3z=-1$
$-x+y-2z=k$
$z-3y+4z=1$
Statement $1:$ The system of equations has no solution for $k\ne 3.$
Statement $2:$ The determinant $\left|\begin{array}{ccc}1& 3& -1\\ -1& -2& k\\ 1& 4& 1\end{array}\right|\ne 0$ for $k\ne 3$

• A. Statement $1$ is True, Statement $2$ is True and Statement $2$ is a correct explanation for Statement $1$
• B.
  Statement $1$ is True, Statement $2$ is True and Statement $2$ is NOT a correct explanantion for Statement $1$
• C. Statement $1$ is True, Statement $2$ is False.
• D. Statement $1$ is False, Statement $2$ is True

Answer 1

The correct option is A Statement $1$ is True, Statement $2$ is True and Statement $2$ is a correct explanation for Statement $1$

$D=\left|\begin{array}{ccc}1& -2& 3\\ -1& 1& -2\\ 1& -3& 4\end{array}\right|$

Here, $D=0$
For no solution, at least one of $D_x,D_y,D_z\ne 0$

$D_x=\left|\begin{array}{ccc}-1& -2& 3\\ k& 1& -2\\ 1& -3& 4\end{array}\right|=3-k$

$D_y=\left|\begin{array}{ccc}1& -1& 3\\ -1& k& -2\\ 1& 1& 4\end{array}\right|=k-3$

$D_z=\left|\begin{array}{ccc}1& -2& -1\\ -1& 1& k\\ 1& -3& 1\end{array}\right|=k-3$
Clearly, for $k\ne 3$, we have $D_x,D_y,D_z\ne 0$. For $k=3,$ we have infinite solutions.

Statement $1$ is True, Statement $2$ is True and Statement $2$ is a correct explanation for Statement $1.$