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Q.
If the system of equations $2x + 3y+5 = 0$, $x + ky + 5 = 0$, $kx - 12y - 14 = 0$ has non-trivial solution, then the value of $k$ is
Determinants
Solution:
The homogeneous linear system of equations has non-trivial solution
$\therefore \quad\left|\begin{matrix}2&3&5\\ 1&k&5\\ k&-12&-14\end{matrix}\right|=0$
$\Rightarrow \quad2\left(-14k+60\right)-3\left(-14-5k\right)+5\left(-12-k^{2}\right)=0$
$\Rightarrow \quad5k^{2}+13k-102=0\,\Rightarrow \quad\left(5k-17\right)\left(k+6\right)=0$
$\Rightarrow \quad k=-6, \frac{17}{5}$