MHTCET 10 May 2024 Shift 1 Paper

Concept:Beats occur when two sound waves of slightly different frequencies superpose.The beat frequency is the absolute difference of their frequencies.The maximum intensity depends on the sum of amplitudes, since intensity is proportional to the square of amplitude.Explanation:Compare each wave with the standard form $x = A \sin(2\pi f t)$.For $x_1 = 2 \sin(1000\pi t)$, the angular frequency is $1000\pi$.So, $f_1 = \frac{1000\pi}{2\pi} = 500\ \mathrm{Hz}$.For $x_2 = 3 \sin(1006\pi t)$, the angular frequency is $1006\pi$.So, $f_2 = \frac{1006\pi}{2\pi} = 503\ \mathrm{Hz}$.Beat frequency: $f_{\text{beat}} = |f_2 - f_1| = |503 - 500| = 3\ \mathrm{Hz}$.Maximum amplitude during constructive interference is the sum of individual amplitudes.So, $A_{\max} = 2 + 3 = 5$.Maximum intensity is proportional to the square of amplitude:$I_{\max} = (2+3)^2 = 5^2 = 25$ units.Thus, the waves produce 3 beats per second with maximum intensity 25 units.Answer:Option C: 3 beats/s with maximum intensity 25 units.