From the given set of equations,augmented matrix=
$\begin{bmatrix} 1&1 &1 &1 \\ 1& 2 & 4 & a\\ 1& 4& 10& a^{2} \end{bmatrix}$
$R_2$$\rightarrow$$R_2$-$R_1$
$R_3$$\rightarrow$$R_3$-$R_1$
$\begin{bmatrix} 1 & 1 & 1& 1\\ 0& 1 & 3 & a-1\\ 0& 3&9 & a^{2}-1 \end{bmatrix}$
$R_3$$\rightarrow$$R_3$-$3R_2$
$\begin{bmatrix} 1 & 1 & 1& 1\\ 0& 1 & 3 & a-1\\ 0& 0&0 & a^{2}-3a+2 \end{bmatrix}$
For the system of equations to have solution,Rank of augmented matrix must be equal to rank of coefficient matrix
Rank of coefficient matrix=2
so for the rank of augmented matrix to be equal to 2, $a^{2}-3a+2$ must be equal to 0.
Therefore,a=1,2