Bertini's theorem for $F$-rational $F$-pure singularities

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $X\subseteq {\mathbb{P}}_k^n$ be a quasi-projective variety that is $F$-rational and $F$-pure. We prove that if $H\subseteq {\mathbb{P}}_k^n$ is a general hyperplane, then $X\cap H$ is also $F$-rational and $F$-pure. Of related but independent interest, we present a relationship between the characteristic and index of a $\mathbb{Q}$-Gorenstein variety with isolated non-$F$-regular locus which is $F$-pure but not $F$-regular.