Question

For non zero solution $\left|\begin{array}{ccc}1& k& 3\\ 3& k& -2\\ 2& 4& -3\end{array}\right|$ = 0 
$\Rightarrow$ k = 11
Now equations
x + 11y + 3z = 0      .....(1)
3x + 11y - 2z = 0     .....(2)
2x + 4y - 3z = 0      .....(3)
on equation (1) + (3) we get 3x + 15y = 0
$\Rightarrow$ x = - 5y
Now put x = - 5y in equation (1)
we get - 5y + 11y + 3z = 0
$\Rightarrow$ z = - 2y
$\frac{xz}{y^2}=\frac{(-5y)(-2y)}{y^2}=10$