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?

A physical quantity 𝑆⃗ is defined as 𝑆⃗= (𝐸⃗⃗× 𝐵⃗⃗)/𝜇0, where 𝐸⃗⃗ is electric field, 𝐵⃗⃗ is magnetic field and 𝜇0 is the permeability of free space. The dimensions of 𝑆⃗ are the same as the dimensions of which of the following quantity(ies) ?

A heavy nucleus 𝑁, at rest, undergoes fission 𝑁→𝑃+ 𝑄, where 𝑃 and 𝑄 are two lighter nuclei. Let 𝛿= 𝑀𝑁−𝑀𝑃−𝑀𝑄, where 𝑀𝑃, 𝑀𝑄 and 𝑀𝑁 are the masses of 𝑃, 𝑄 and 𝑁, respectively. 𝐸𝑃 and 𝐸𝑄 are the kinetic energies of 𝑃 and 𝑄, respectively. The speeds of 𝑃 and 𝑄 are 𝑣𝑃 and 𝑣𝑄, respectively. If 𝑐 is the speed of light, which of the following statement(s) is(are) correct?

Two concentric circular loops, one of radius 𝑅 and the other of radius 2𝑅, lie in the xy-plane with the origin as their common center, as shown in the figure. The smaller loop carries current 𝐼1 in the anti-clockwise direction and the larger loop carries current 𝐼2 in the clockwise direction, with 𝐼2 > 2𝐼1. 𝐵⃗⃗(𝑥, 𝑦) denotes the magnetic field at a point (𝑥, 𝑦) in the xy-plane. Which of the following statement(s) is(are) correct?

The value of 𝑋 is ___ .

The value of 𝑌 is ___. Question Stem for Question Nos. 9 and 10 Question Stem A pendulum consists of a bob of mass 𝑚= 0.1 kg and a massless inextensible string of length 𝐿= 1.0 m. It is suspended from a fixed point at height 𝐻= 0.9 m above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse 𝑃= 0.2 kg-m/s is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is 𝐽 kg-m2/s. The kinetic energy of the pendulum just after the lift- off is 𝐾 Joules.

The value of 𝐽 is ___ .

The value of 𝐾 is ___. Question Stem for Question Nos. 11 and 12 Question Stem In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C 𝜇𝐹 across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase- angle (in degrees) between the current and the supply voltage is 𝜑. Assume, 𝜋√3 ≈5.