$\Rightarrow$$\frac{v}{{2{l_0}}} = \frac{{3v}}{{4{l_c}}}$
$\Rightarrow$${l_c} = \frac{{3 \times 2 \times 0.5}}{4} = 0.75m$
$(A)$ With a node at $O$, the minimum frequency of vibration of the composite string is $v_0$
$(B)$ With an antinode at $O$, the minimum frequency of vibration of the composite string is $2 v_0$
$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$, it has $6$ nodes, including the end nodes
$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string
$(A)$ a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is open.
$(B)$ a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is open.
$(C)$ a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.
$(D)$ a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.