Is this solution wrong?

I suppose the alpha and beta particles are travelling at RIGHT ANGLE to the electric field lines. In this situation, the "deflection" that you mentioned in your question will be along the field lines, i.e. perpendicular to the original direction of travel of these two particles. Hence, the original travelling velocities, 0.3c and 0.1c, respectively of beta and alpha do not affect their "lateral displacement", as the two (original travelling direction and displacement direction) are perpendicular to each other.

For a given time interval, the force that acts on alpha is two times that on the beta, because of the charge on alpha (two protons) is two times that on beta (a single electron). But the mass of an alpha (2 protons and 2 neutrons) is thousands of times heavier than that of a beta (an electron), the acceleration on the alpha is clearly much much lower than that on the beta. As a result, the lateral displacement of the alpha is less than that of the beta.

1. We mearsure the radius of curvature. Irrelevant with the travelling distance.
2. The distances your mentioned here are 2 different things and 2 different directions, The displacement toward the plate is perpendicular to the travelling distance. Displacement toward the plate is related to deflection but travelling distance is not. you just first make it clear.