A long solid cylindrical conductor of radius a=10 cm carries a uniform current density J=5×10^5 A m^-2 along its axis. It has a long cylindrical hole…

Use superposition: the actual conductor = full cylinder with current density $+J$ plus a cylinder filling the hole with $-J$. For a cylinder, inside $\vec B=\tfrac12\mu_0\vec J\times\vec r$ (from its axis), outside $B=\dfrac{\mu_0J\rho^2}{2r}$. (A) Inside the hole $\vec B=\tfrac12\mu_0\vec J\times(\vec r_1-\vec r_2)=\tfrac12\mu_0\vec J\times\vec d$, uniform, magnitude $\tfrac12\mu_0Jd=\tfrac12(4\pi\times10^{-7})(5\times10^5)(0.04)=12.6\ \mathrm{mT}$: correct. (B) At the farthest surface point, $B_{full}=\tfrac12\mu_0Ja=31.4\ \mathrm{mT}$ and the hole contributes $-\dfrac{\mu_0Jb^2}{2(a+d)}=-0.22\ \mathrm{mT}$ (distance $a+d=14\ \mathrm{cm}$), giving $31.2\ \mathrm{mT}$: correct. (C) On the axis the full cylinder gives zero but the hole gives $\dfrac{\mu_0Jb^2}{2d}=0.79\ \mathrm{mT}$: incorrect. (D) $\vec J\times\vec d$ is perpendicular to $\vec d$, so the field in the hole is perpendicular to the line of axes: incorrect.