JEE Main 2008 — Waves & Oscillations

A wave travelling along the $x$-axis is described by the equation $y(x, t)=0.005 \cos (\alpha x-\beta t)$. If the wavelength and the time period of the wave are $0.08 \mathrm{~m}$ and $2.0 \mathrm{~s}$, respectively, then $\alpha$ and $\beta$ in appropriate units are

The speed of sound in oxygen $\left(\mathrm{O}_2\right)$ at a certain temperature is $460 \mathrm{~ms}^{-1}$. The speed of sound in helium (He) at the same temperature will be (assumed both gases to be ideal)

While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of $18 \mathrm{~cm}$ during winter. Repeating the same experiment during summer, she measures the column length to be $\mathrm{xcm}$ for the second resonance. Then