Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $\mathrm{O}$, and tied at two fixed ends $\mathrm{P}$ and $\mathrm{Q}$, 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 source, approaching with speed u towards the open end of a stationary pipe of length L, is emitting a sound of frequency f_s. The farther end of the pipe is closed. The speed of sound in air is v and f₀ is the fundamental frequency of the pipe. For which of the following combination(s) of u and f_s, will the sound reaching the pipe lead to a resonance?

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $$500$$ $$Hz$$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $$50.7$$ $$cm$$ and $$83.9$$ $$cm.$$ Which of the following statements is (are) true?

A block $$M$$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to fixed rigid support at $$O.$$ A transverse wave pulse (Pulse 1) of wavelength $${\lambda _0}$$ is produced at point $$O$$ on the rope. The pulse takes time $${T_{OA}}$$ to reach point $$A.$$ If the wave pulse of wavelength $${\lambda _0}$$ is produced at point $$A$$ (Pulse 2) without disturbing the position of $$M$$ it takes time $${T_{AO}}$$ to reach point $$O.$$ which of the following options is/are correct?