Oxford University MAT 2021 Solved Papers

The tangent at $\mathit{a}$ is $\mathit{y}=\left(3{\mathit{a}}^2-3\right)\left(\mathit{x}-\mathit{a}\right)+\left({\mathit{a}}^3-3\mathit{a}\right)$, which passes through $\left(2,0\right)$ if and only if $0=\left(3{\mathit{a}}^2-3\right)\left(2-\mathit{a}\right)+\left({\mathit{a}}^3-3\mathit{a}\right)$. This simplifies to $2{\mathit{a}}^3-6{\mathit{a}}^2+6=0$. The left-hand side is a cubic in $\mathit{a}$ and we'd like to know how many roots it has. The turning points of $2{\mathit{a}}^3-6{\mathit{a}}^2+6$ are at $\mathit{a}=0$ and $\mathit{a}=2$, where the value of the cubic is 6 and $-2$ respectively. So this cubic starts negative, rises to a positive local maximum, then decreases to a negative local minimum before rising again. There are therefore three roots for this cubic, so three values of $\mathit{a}$ for which the tangent to the original cubic passes through the point $\left(2,0\right)$.