Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 2

At an intersection, $x^2=(x-1{)}^2=x^2-2x+1$, so $x=1/2$. Substitution into either parabola givesThe resulting linear equation for $x$ has only one solution, so this is the unique intersection.

The two derivatives are $2x$ and $2(x-1)$. At the intersection their slopes are $1$ and $-1$, so their direction vectors $(1,1)$ and $(1,-1)$ have zero dot product. Thus the tangent lines are perpendicular and their angle is