Question

# The constant $$k$$ is such that the following system of equations posses a non-trivial(i.e., not all zero) solution over the set of rationals $$Q$$  $$x+ky+3z=0,\quad 3x+ky-2z=0\quad 2x+3y-4z=0$$Then $$\displaystyle \frac{2k}{11}$$ is equal to

Solution

## For non-trivial solution $$\Delta = \begin{vmatrix} 1&k&3 \\ 3 & k & -2 \\2&3&-4 \end{vmatrix}=0$$ Expanding  along first column,$$1(-4k+6)-3(-4k-9)+2(-2k-3k)=0\Rightarrow -2k+33=0 \Rightarrow k = 33/2$$ Mathematics

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