A block of mass M=3 kg executes simple harmonic motion of amplitude A on a smooth horizontal surface, attached to a light spring of force constant k.…

Case I: the putty has no horizontal velocity, so horizontal momentum is conserved: $Mv_0=(M+m)v'$ with $v_0=\omega A$. The new speed at the mean position is $\tfrac34 v_0$, the new angular frequency is $\omega'=\omega\sqrt{3/4}$, so $A'=v'/\omega'=\tfrac34 A\sqrt{4/3}=\tfrac{\sqrt3}{2}A$ and (A) is correct. The energy is $\tfrac12 kA'^2=\tfrac34$ of the original, i.e. a $25\%$ loss, not one half, so (C) is wrong. Case II: the block is momentarily at rest, so nothing changes in the energy or displacement; the amplitude stays $A$ and (B) is correct, while the energy stays $\tfrac12 kA^2$, so the claimed $25\%$ reduction in (D) is wrong. Answer: A, B.