, 
, has a non-zero solution is
A) -1
B) 0
C) 1
D) None of these
Correct Answer: A
Solution :
The system will have a non-zero solution, if \[\Delta \,\,\,\equiv \,\,\,\left| \begin{matrix} {{a}^{3}} & {{(a+1)}^{3}} & {{(a+2)}^{3}} \\ a & a+1 & a+2 \\ 1 & 1 & 1 \\ \end{matrix} \right|=0\] \[\Rightarrow \,\,\,\left| \begin{matrix} {{a}^{3}} & 3{{a}^{2}}+3a+1 & 3{{(a+1)}^{2}}+3(a+1)+1 \\ {{a}^{2}}1 & 1 & 1 \\ 1 & 0 & 0 \\ \end{matrix} \right|=0\] \[\left[ \begin{align} & {{C}_{2}}\to {{C}_{2}}\to {{C}_{1}} \\ & {{C}_{3}}\to {{C}_{3}}\to {{C}_{2}} \\ \end{align} \right]\] \[\Rightarrow \,\,\,3{{a}^{2}}+3a+1-\{3{{(a+1)}^{2}}+3(a+1)+1\}=0\] [Expanding along \[{{R}_{3}}\]] \[\Rightarrow \,\,\,-6\left( a+1 \right)\,\,=0\,\,\,\Rightarrow \,\,\,a=-1.\]
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