Question
The number of values of $$k$$ for which the linear equations $$4x + ky + 2z = 0, kx + 4y + z = 0$$ and $$2x + 2y + z = 0$$ possess a non-zero solution is
A.
2
B.
1
C.
zero
D.
3
Answer :
2
Solution :
\[\Delta = 0\,\, \Rightarrow \,\,\left| \begin{array}{l}
4\,\,\,\,\,\,k\,\,\,\,\,\,2\\
k\,\,\,\,\,\,4\,\,\,\,\,\,1\\
2\,\,\,\,\,\,2\,\,\,\,\,\,1
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \,\,4\left( {4 - 2} \right) - k\left( {k - 2} \right) + 2\left( {2k - 8} \right) = 0 \cr
& \Rightarrow \,\,8 - {k^2} + 2k + 4k - 16 = 0 \cr
& \Rightarrow \,\,{k^2} - 6k + 8 = 0 \cr
& \Rightarrow \,\,\left( {k - 4} \right)\left( {k - 2} \right) = 0,\,\,k = 4,2 \cr} $$