Question

If the system of equations $x-ky+3z=0,$ $2x+ky-2z=0$ and $3x-4y+2z=0$ has non-trivial solutions, then the value of $\frac{10 y}{x}$ is equal to

• A. $3$
• B. $-\frac{15}{2}$
• C. $\frac{5}{7}$
• D. $-\frac{5}{7}$

Answer 1

Answer: (B)

For non-trivial solutions,
$\begin{vmatrix} 1 & -k & 3 \\ 2 & k & -2 \\ 3 & -4 & 2 \end{vmatrix}=0\Rightarrow \left(2 k - 8\right)+k\left(10\right)+3\left(- 8 - 3 k\right)=0$
$\Rightarrow 3k-32=0\Rightarrow k=\frac{32}{3}$
Putting $z=\lambda $ , we get,
$3 x-32 y=-9 \lambda\, y = \frac{\lambda}{4}$
$3x - 4y = -2\lambda \,\, x = -\frac{\lambda}{3}$
i.e. $\frac{10 y}{x}=-\frac{15}{2}$