A particle is observed from two frames $S_1$ and$S_2$. The frame $S_2$ moves with respect to $S_1$ with an acceleration a. Let $F_1$ and $F_2$ be the pseudo forces on the particle when seen from $S_1$ and $S_2$respectively. Which of the following are not possible ?

If both $F_1$ and $F_2$ are zero, then the accelerations of the body observed from different frames will be zero. And consecutively relative acceleration between the frames will also be zero. Since there is some relative acceleration between the frames, this scenario is not possible.