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# For what value of $k$ do the following homogeneous system of equations posses a non-trivial.$x+ky+3z=0,3x+ky-2z=0,2x+3y-4z=0$

(A) $k=1$ (B) $k=3$ (C) $k=\large\frac{22}{2}$ (D) $k=\large\frac{33}{2}$

For non trivial solutions $D=0$
$\Rightarrow D=\begin{vmatrix}1 &k&3\\3 &k &-2\\2&3&-4\end{vmatrix}=0$
Apply $R_2-3R_1=R_2$
$R_3=2R_1$
$\begin{vmatrix}1 &k &3\\0 &-2k&-11\\0 &3-2k&-10\end{vmatrix}$
Evaluating by 1st column we get,
$-2k\times -10=-11(3-2k)$
$20k=-33+22k$
$20k-22k=-33$
$-2k=-33$
$k=\large\frac{33}{2}$
Hence (d) is the correct answer.
edited Jun 7, 2014