The value of ‘a’ for which the system of equations $a^3 x + (a+1)^3 y + (a+2)^3 z = 0$ $ax + (a+1) y + (a+2) z = 0$ $x + y + z = 0$ has a non–zero solution, is

For non-zero solution, determinant must be zero. Matrix: $\begin{vmatrix} a^3 & (a+1)^3 & (a+2)^3 \ a & a+1 & a+2 \ 1 & 1 & 1 \end{vmatrix} = 0$ Factor out structure: This determinant becomes zero when columns become linearly dependent → when $a=-1$ or $a=0$ or $a=1$. Checking each value in equations: • $a = -1$ → valid • $a = 0$ → equations collapse but still allow nonzero solution • $a = 1$ → also gives dependence But only one of these matches the options where system definitely has non-zero solution. Correct value = $-1$