1. Tardigrade
  2. Question
  3. Mathematics
  4. Consider the system of equations x+k y=0, y+k z=0 and z+k x=0. The set of all real values of k for which the system has a unique solution, is

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Q. Consider the system of equations $x+k y=0, y+k z=0$ and $z+k x=0$. The set of all real values of $k$ for which the system has a unique solution, is

Determinants

Solution:

Given,
$ x + ky + 0.z = 0$
$ 0 + y + kz = 0$
$ kx + 0y + z = 0$
For unique solution, $ \Delta \neq 0 \Rightarrow \begin{vmatrix}1 & k & 0 \\ 0 & 1 & k \\ k & 0 & 1\end{vmatrix} \neq 0 \Rightarrow 1(1)- k \left(- k ^2\right) \neq 0$
$\Rightarrow k ^3+1 \neq 0 \Rightarrow k \in R -\{-1\}$