The answer is “$-3$”.

Recall that $\left[\begin{array}{c}a\\ 1\\ 2\end{array}\right]$ in $N(A)$ means $$\left[\begin{array}{ccc}1& -1& 2\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}a\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right].$$ Multiplying out the left-hand side gives $$\left[\begin{array}{c}a+3\\ a+3\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right].$$ Hence, $a=-3$.