Question

If the system of linear equations
$x+ky+3z=0$
$3x+ky-2z=0$
$2x+4y-3z=0$
has a non-zero solution $(x,y,z)$ then $\frac{xz}{y^2}$ is equal to

• A. -10
• B. 10
• C. -30
• D. 30

Answer 1

Answer: (B)

$\because$ System of equation has non-zero solution.
$\therefore \left|\begin{array}{ccc}1& k& 3\\ 3& k& -2\\ 2& 4& -3\end{array}\right|=0$
$\Rightarrow 44-4k=0$
$\therefore k=11$
Let $z=\lambda$
$\therefore x+11y=-3\lambda$
and $3x+11y=2\lambda$
$\therefore x=\frac{5\lambda }{2},y=-\frac{\lambda }{2},z=\lambda$
$\therefore \frac{xz}{y^2}=\frac{\frac{5\lambda }{2}\cdot \lambda }{{\left(-\frac{\lambda }{2}\right)}^2}=10$