$A$ wave pulse on a string has the dimensions shown in the figure. The wave speed is $v = 1 \, cm/s$. If point $O$ is a free end,what is the shape of the wave at time $t = 3 \, s$?

$A$ hollow pipe of length $0.8 \ m$ is closed at one end. At its open end,a $0.8 \ m$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the string is $50 \ N$ and the speed of sound in air is $320 \ m/s$,the mass of the string is: (in $g$)

$Assertion :$ 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.

Match Column-$1$ with Column-$2$.

Column-$1$	Column-$2$
$(a)$ Light waves	$(i)$ Mechanical and transverse.
$(b)$ Sound waves	$(ii)$ Mechanical and longitudinal.
$(c)$ Earthquake waves	$(iii)$ Non-mechanical and transverse.
$(d)$ Waves on a tense string	$(iv)$ Mechanical,transverse and longitudinal.

$A$ 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

$A$ vibrating string of length $\ell$ under a tension $T$ resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length $75 \,cm$ inside a tube closed at one end. The string also generates $4$ beats per second when excited along with a tuning fork of frequency $n$. Now, when the tension of the string is slightly increased, the number of beats reduces to $2$ per second. Assuming the velocity of sound in air to be $340 \,m/s$, the frequency $n$ of the tuning fork in $Hz$ is: