Order of contact for a general tangent line of a cubic hypersurface
Solution 1:
Yes, this is true for all smooth cubic hypersurfaces.
A general $\mathbb P^2$ section of $X$ is a smooth plane cubic curve, and there are exactly $9$ points where tangent lines have contact order $3$ (also known as flex points).
However, if a general tangent line of $X$ has contact order $3$, then so is a general tangent line of $X$ contained in a general plane $\mathbb P^2\subset \mathbb P^N$. This says a smooth cubic curve has a dense subset of flex points. This is absurd.