JEE Advanced 2021 — Set 2

One end of a horizontal uniform beam of weight 𝑊 and length 𝐿 is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height 𝐿 above the hinge at point O. A block of weight 𝛼𝑊 is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of (2√2)𝑊. 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 𝑓𝑠. 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 𝑓𝑠, will the sound reaching the pipe lead to a resonance?

For a prism of prism angle 𝜃= 60°, the refractive indices of the left half and the right half are, respectively, 𝑛1 and 𝑛2 (𝑛2 ≥𝑛1) as shown in the figure. The angle of incidence 𝑖 is chosen such that the incident light rays will have minimum deviation if 𝑛1 = 𝑛2 = 𝑛= 1.5. For the case of unequal refractive indices, 𝑛1 = 𝑛 and 𝑛2 = 𝑛+ ∆𝑛 (where ∆𝑛≪𝑛), the angle of emergence 𝑒= 𝑖+ ∆𝑒. Which of the following statement(s) is(are) correct?