JEE (Advanced) 2021 Paper-2
For the following questions answer them individually
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 $$f_0$$ 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?
For a prism of prism angle $$\theta = 60^\circ$$, the refractive indices of the left half and the right half are, respectively, $$n_1$$ and $$n_2 (n_2 \geq n_1)$$ as shown in the figure. The angle of incidence 𝑖 is chosen such that the incident light rays will have minimum deviation if $$n_1 = n_2 = n = 1.5$$. For the case of unequal refractive indices, $$n_1 = n$$ and $$n_2 = n + \triangle n$$ (where $$\triangle n \ll n$$), the angle of emergence $$e = i + \triangle e$$. Which of the following statement(s) is(are) correct?
A physical quantity $$\overrightarrow{S}$$ is defined as $$\overrightarrow{S} = \frac{\left(\overrightarrow{E} \times \overrightarrow{B}\right)}{\mu_0}$$, where $$\overrightarrow{E}$$ is electric field, $$\overrightarrow{B}$$ is magnetic field and $$\mu_0$$ is the permeability of free space. The dimensions of $$\overrightarrow{S}$$ are the same as the dimensions of which of the following quantity(ies) ?
A heavy nucleus N, at rest, undergoes fission $$N \rightarrow P + Q$$, where P and Q are two lighter nuclei. Let $$\delta = M_N − M_P − M_Q$$, where $$M_P$$, $$M_Q$$ and $$M_N$$ 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 $$v_P$$ and $$v_Q$$, 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 $$I_1$$ in the anti-clockwise direction and the larger loop carries current $$I_2$$ in the clockwise direction, with $$I_2 > 2I_1. \overrightarrow{B}(x, y)$$ denotes the magnetic field at a point (x, y) in the xy-plane. Which of the following statement(s) is(are) correct?
Question Stem
A soft plastic bottle, filled with water of density 1 gm/cc, carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of 5 gm, and it is made of a thick glass of density 2.5 gm/cc. Initially the bottle is sealed at atmospheric pressure $$p_0 = 10^5$$ Pa so that the volume of the trapped air is $$v_0 = 3.3 cc$$ When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure $$p_0 + \triangle p$$ without changing its orientation. At this pressure, the volume of the trapped air is $$v_0 - \triangle v$$.
Let $$\triangle v = X cc$$ and $$\triangle p = Y \times 10^3$$ Pa.
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-m$$^2$$/s. The kinetic energy of the pendulum just after the lift-off is 𝐾 Joules.
Question Stem
In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C $$\mu F$$ 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 $$\psi$$. Assume, $$\pi \sqrt{3} \approx 5$$.
