$A$ car $P$ travelling at $20\,ms^{-1}$ sounds its horn at a frequency of $400\,Hz$. Another car $Q$ is travelling behind the first car in the same direction with a velocity $40\,ms^{-1}$. The frequency heard by the passenger of the car $Q$ is approximately $.......\,Hz$. [Take,velocity of sound $= 360\,ms^{-1}$]

An observer is moving towards a stationary source of sound. In this case:

$A$ whistle sends out $256$ waves in a second. If the whistle approaches the observer with a velocity equal to $\frac{1}{3}$ of the velocity of sound in air,calculate the number of waves per second the observer will receive.

$A$ person driving a car at a constant speed of $15\,m/s$ is approaching a vertical wall. The person notices a change of $40\,Hz$ in the frequency of his car's horn upon reflection from the wall. The frequency of the horn is ............ $Hz$. (Given: Speed of sound: $330\,m/s$)

An audio transmitter $(T)$ and a receiver $(R)$ are hung vertically from two identical massless strings of length $8 \ m$ with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0 = \cos^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is $660 \ Hz$ and the speed of sound in air is $330 \ m/s$,the maximum variation in the frequency (in $Hz$) as measured by the receiver (Take the acceleration due to gravity $g = 10 \ m/s^2$) is:

$A$ train,standing at the outer signal of a railway station,blows a whistle of frequency $400\; Hz$ in still air.
$(i)$ What is the frequency of the whistle for a platform observer when the train $(a)$ approaches the platform with a speed of $10\; m s^{-1}$,$(b)$ recedes from the platform with a speed of $10\; m s^{-1}$?
$(ii)$ What is the speed of sound in each case? The speed of sound in still air can be taken as $340\; m s^{-1}$.