A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $x_0$. Consider two cases: (i) when the block is at $x_0$; and (ii) when the block is at  $x=x_0+A$. In both the cases, a particle with mass m(<M)  is softly placed on the block after which they stick to each other. Which of the following statements(s) is (are) true about the motion after the mass m is placed on the mass M

When block is placed at a position  $x_0$
From conservation of momentum,

 $MV=\left(m+M\right)V^1$
$\Rightarrow V^1=\frac{MA\omega }{m+M}=\frac{MA}{m+M}\sqrt{\frac{K}{M}}$ 
$\Rightarrow A^1\sqrt{\frac{K}{M+m}}=\frac{MA}{m+M}\sqrt{\frac{K}{M}}$ 
$$\begin{gathered} \Rightarrow A^1=A\sqrt{\frac{M}{m+M}} \\ \Rightarrow \frac{A^1}{A}=\sqrt{\frac{M}{m+M}} \end{gathered}$$ 
When block is placed at extreme position amplitude does not change since at extreme position  M is at rest . So, initial and final momentums are not changed
B) $T_{initial}=2\pi \sqrt{\frac{M}{K}}$  
After block is placed  $T_{final}=2\pi \sqrt{\frac{m+M}{K}}$
C) Total energy   $E=\frac{1}{2}m{A}^2{w}^2=\frac{1}{2}m{V}_{mean}^2$
Before block is placed
At  $x_0:E_1=\frac{1}{2}M{V}^2$
At  $x_0+A : E_2=\frac{1}{2}M{V}^2=E_1\left[\text{energy constant}\right]$
After block is placed 
At  $x_0:E_3=\frac{1}{2}\left(m+M\right){\left(\frac{MV}{m+M}\right)}^2$
 $=\frac{1}{2}M{V}^2\left[\frac{M}{m+M}\right]$
At $x_0+A : E_4=E_3$ [energy constant]
As  $\frac{M}{m+M}<1$ so energy decreases
D) When block is placed at  $x_0$
$V^1=\frac{M}{m+M}V$ 
$V^1<V$ 
When block is placed at  $x_0+A$ velocity of system at its mean position 
$V^1=A{w}^1$  =   $A\sqrt{\frac{K}{M+m}}$    $V^1<V$