A) \[k=\frac{5}{2},\,x=\frac{1}{5}\]
B) \[k=\frac{5}{2},\,x\ne \frac{1}{5}\]
C) \[k=\frac{1}{5},\,x=\frac{5}{2}\]
D) None of these
Correct Answer: A
Solution :
Let \[\Delta =\left| \begin{matrix} 4 & 2 & (1-x) \\ 5 & k & 1 \\ 6 & 3 & (1+x) \\ \end{matrix} \right|\] Applying \[{{R}_{1}}\to {{R}_{1}}+{{R}_{3}}\] \[\Rightarrow \] \[\Delta =\left| \begin{matrix} 10 & 5 & 2 \\ 5 & k & 1 \\ 6 & 3 & 1+x \\ \end{matrix} \right|\] Applying \[{{C}_{1}}\to {{C}_{1}}-2{{C}_{2}}\] \[\Rightarrow \] \[\Delta =\left| \begin{matrix} 0 & 5 & 2 \\ 5-2k & k & 1 \\ 0 & 3 & 1+x \\ \end{matrix} \right|\] \[\Rightarrow \] \[(5-2k)(5+5x-6)=0\] \[\Rightarrow \] \[(5-2k)(5+5x-6)=0\]
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