$A$ pulley fixed to the ceiling carries a string with blocks of mass $m$ and $3m$ attached to its ends. The masses of the string and pulley are negligible. When the system is released,what is the acceleration of its centre of mass?

In the arrangement shown in the figure,the pulley is smooth and massless,and all the strings are light. Let $F_1$ be the force exerted on the pulley in case $(i)$ and $F_2$ be the force in case $(ii)$. Then:

Consider a body of mass $3\,kg$ at rest on a smooth horizontal table. This body is connected by a light string,which passes over a smooth pulley at the edge of the table,to another body of mass $2\,kg$ hanging freely. This $2\,kg$ mass is released from rest. Now consider the following statements:
$(A)$ The masses remain at rest.
$(B)$ The $3\,kg$ mass moves uniformly while the $2\,kg$ mass moves with acceleration $\frac{2}{5}g\,m/s^2$.
$(C)$ Both bodies move with acceleration $\frac{2}{5}g\,m/s^2$.
$(D)$ The tension in the string near the first body is more than that near the second body.
$(E)$ The tension in the string is $\frac{6g}{5}\,N$.
Then the correct statements are:

Tension in the rope at the rigid support is ............ $N$ $\left(g=10 \,m / s ^2\right)$

$A$ lift of mass $1000\,kg$ is moving with an acceleration of $1\,m/s^2$ in the upward direction. The tension developed in the cable connected to the lift is ........... $N$ $(g = 9.8\,m/s^2)$.

$A$ light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (where $m_2 > m_1$). If the acceleration of the system is $\frac{g}{\sqrt{2}}$,then the ratio of the masses $\frac{m_1}{m_2}$ is: