2 questions from 2025
An audio transmitter (\( T \)) and a receiver (\( R \)) are hung vertically from two identical massless strings of length \( 8 \, \text{m} \) 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 \( 660 \, \text{Hz} \) and the speed of sound in air is \( 330 \, \text{m/s} \), the maximum variation in the frequency (in \( \text{Hz} \)) as measured by the receiver (Take the acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)) is ______. 
Consider a system of three connected strings, \( S_1, S_2 \) and \( S_3 \) with uniform linear mass densities \( \mu \, \text{kg} / \text{m} \), \( 4\mu \, \text{kg} / \text{m} \) and \( 16\mu \, \text{kg} / \text{m} \), respectively, as shown in the figure. \( S_1 \) and \( S_2 \) are connected at the point \( P \), whereas \( S_2 \) and \( S_3 \) are connected at the point \( Q \), and the other end of \( S_3 \) is connected to a wall. A wave generator \( O \) is connected to the free end of \( S_1 \). The wave from the generator is represented by \( y = y_0 \cos(\omega t - kx) \, \text{cm} \), where \( y_0 \), \( \omega \) and \( k \) are constants of appropriate dimensions.
Which of the following statements is/are correct:
