$A$ wave equation is given as $y = \cos(500t - 70x)$,where $y$ is in $mm$,$x$ is in $m$,and $t$ is in $s$. Which of the following statements is correct?

Two uniform strings of mass per unit length $\mu$ and $4 \mu$,and length $L$ and $2 L$,respectively,are joined at point $O$,and tied at two fixed ends $P$ and $Q$,as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$,which of the following statement$(s)$ is(are) correct?
$(A)$ 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

$A$ rod of length $L$ and negligible mass is suspended by two identical strings $AB$ and $CD$ as shown in the figure. $A$ mass $M$ is suspended from point $O$ which is at a distance $x$ from $B$. If the frequency of the first harmonic of $AB$ is equal to the frequency of the second harmonic of $CD$, then the value of $x$ is

$A$ wave is travelling along a string. At an instant,the shape of the string is as shown in the figure. At this instant,point $A$ is moving upwards. Which of the following statements is/are correct?

$A$ man standing in front of a mountain beats a drum at regular intervals. The rate of drumming is gradually increased and he finds that the echo is not heard distinctly when the rate becomes $40$ per minute. He then moves nearer to the mountain by $90 \ m$ and finds that the echo is again not heard when the drumming rate becomes $60$ per minute. The distance between the mountain and the initial position of the man is .... $m$

$Assertion :$ The pitch of wind instruments rises and that of string instruments falls as an orchestra warms up.
$Reason :$ When temperature rises,the speed of sound increases,but the speed of a wave in a string fixed at both ends decreases.