The value of a for which the system of equations

$a^3x + (a+1)^3y+(a+2)^3x=0$

$ax+(a+1)y + (a=2) z=0$

$x+ y + z= 0 $

has a non-zero solution, is

-1

The correct answer is option (3) : -1

The given system of equations will have a non-zero solution, if

$\begin{vmatrix}a^3 & (a+1)^3 & (a=2)^3\\a & (a+1) & (a+2) \\ 1 & 1 & 1\end{vmatrix} = 0 $

$⇒\begin{vmatrix}a^3-(a+2)^3 & (a+1)^3-(a+2)^3 & (a+2)^3\\-2 & -1 & a+2 \\ 0& 0 & 1\end{vmatrix}=0$

$\begin{vmatrix} Applying\, C_1→C_1-C_3\\and\, C_2→C_2-C_3 \end{vmatrix}$

$⇒-a^3+(a+2)^3 +2(a+1)^3 -2(a+2)^3 = 0 $

$⇒(a+2)^3-2(a+1)^3+a^3=0$

$⇒(6a^2+12a+8) -2(3a^2 +3a+1) = 0 $

$⇒6a + 6 = 0 $

$⇒a = - 1$