A ray parallel to principal axis is incident at 30° from normal on concave mirror having radius of curvature R. The point on principal axis where rays are crossing the axis is Q such that PQ is R(1-1/√3) which is not the nominal focus $R/2$. Why is that?
The paraxial approximation clearly doesn't hold ($30^\circ$ is a large angle), and so neither does the usual spherical mirror equation, because of spherical aberration. Forget that equation and focal length for a moment, take the mirror to be an exact circle and treat this as a geometry problem. You will need the law of reflection: $\angle QMC = 30^\circ$.
