An Exponential-Quadratic Mix

Let $f(x) = e^{x-a} + ax^2 - 3ax + 1$, where $a \in \mathbb{R}$.

1. When $a = 1$, find the equation of the tangent line to $y = f(x)$ at $x = 1$. 2. When $a > 1$, determine the number of zeros of $f(x)$ on $[1, +\infty)$, with justification. 3. If $f(x) \geqslant 0$ for all $x \geqslant 0$, find the range of possible values of $a$.