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.
$\therefore \begin{vmatrix}1 & k & 3 \\ 3 & k & -2 \\ 2 & 4 & -3\end{vmatrix}=0$
$\Rightarrow 44-4 k=0$
$\therefore k=11$
Let $z=\lambda$
$\therefore x+11 y=-3 \lambda$
and $3 x+11 y=2 \lambda$
$\therefore x=\frac{5 \lambda}{2}, y=-\frac{\lambda}{2}, z=\lambda$
$\therefore \frac{x z}{y^{2}}=\frac{\frac{5 \lambda}{2} \cdot \lambda}{\left(-\frac{\lambda}{2}\right)^{2}}=10$