Question

If the system of linear equations,
$\begin{array}{c}x+y+z=6\\ x+2y+3z=10\\ 3x+2y+\lambda z=\mu \end{array}$
has more than two solutions, then $\mu -{\lambda }^2$ is equal to

Answer 1

The system of equations has more than two solutions.
This means the system of equations has infinite solutions.
$\therefore D=D_3=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 3& 2& \lambda \end{array}\right|=0$

$\Rightarrow 2\lambda -6-\lambda +9+2-6=0$
$\Rightarrow \lambda =1$

$\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 3& 2& \mu \end{array}\right|=0$

$\Rightarrow 2\mu -20-\mu +30-24=0$
$\Rightarrow \mu =14$
So, $\mu -{\lambda }^2=13$