At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current I(t) = I₀ cos ($$\omega$$t), with I₀ = 1 A and $$\omega$$ = 500 rad s^-1 starts flowing in it with the initial direction shown in the figure. At $$t = {{7\pi } \over {6\omega }}$$, the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If C = 20 $$\mu$$F, R = 10 $$\Omega$$ and the battery is ideal with emf of 50 V, identify the correct statement(s).

A light source, width emits two wavelengths $$\lambda$$₁ = 400 nm and $$\lambda$$₂ = 600 nm, is used in a Young's double-slit experiment. If recorded fringe widths for $$\lambda$$₁ and $$\lambda$$₂ are $$\beta$$₁ and $$\beta$$₂ and the number of fringes for them within a distance y on one side of the central maximum are m₁ and m₂, respectively, then

One end of a taut string of length 3 m along the x-axis is fixed at x = 0. The speed of the waves in the string is 100 ms^$$-$$1. The other end of the string is vibrating in the y-direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary wave is (are)

A student is performing an experiment using a resonance column and a tuning fork of frequency 244 s^$$-$$1. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is (0.350 $$\pm$$ 0.005) m, the gas in the tube is

(Useful information : $$\sqrt {167RT} $$ = 640 J^1/2 mol^$$-$$1/2; $$\sqrt {140RT} $$ = 590 J^1/2 mol^$$-$$1/2. The molar mass M in grams is given in the options. Take the values of $$\sqrt {10/M} $$ for each gas as given there.)