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(B) At the extreme position,the velocity of the object of mass $m$ is zero. When an object of mass $2m$ is placed gently on it,the velocity of the com

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Only $a, c, d$

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(B) At the extreme position,the velocity of the object of mass $m$ is zero. When an object of mass $2m$ is placed gently on it,the velocity of the combined system $(m + 2m = 3m)$ remains zero.$(a)$ Since the object is at the extreme position,its displacement $x = A$ is the new amplitude. The potential energy stored in the spring is $U = \frac{1}{2}kA^2$. Since the position $x = A$ does not change and the spring constant $k$ is the same,the amplitude $A$ remains unchanged. Thus,$(a)$ is correct.$(b)$ The time period of a spring-mass system is given by $T = 2\pi \sqrt{\frac{M}{k}}$. Initially,$M = m$,so $T_1 = 2\pi \sqrt{\frac{m}{k}}$. After adding the mass,$M' = 3m$,so $T_2 = 2\pi \sqrt{\frac{3m}{k}}$. Since $T_2 
eq T_1$,the time period changes. Thus,$(b)$ is incorrect.$(c)$ The total mechanical energy is $E = \frac{1}{2}kA^2$. Since $k$ and $A$ remain unchanged,the total mechanical energy remains unchanged. Thus,$(c)$ is correct.$(d)$ The maximum speed is given by $v_{max} = A\omega = A \sqrt{\frac{k}{M}}$. Since $M$ increases to $3m$,$\omega$ decreases,so $v_{max}$ changes. Thus,$(d)$ is correct.Therefore,statements $(a), (c),$ and $(d)$ are correct.

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