Given: Moles of monoatomic gas $=\alpha$, moles of diatomic gas is $\beta$.

Solution:

The mixture is behaving as diatomic gas. If we neglect the vibrational mode of freedom then degree of freedom ( $\left.f_{\text {mix }}\right)$ of mixture is $3$ (for translational) $+2$ (for rotational $=5$.

We know that,

$f _{\text {mix }}=\frac{ f _1 n _1+ f _2 n _2}{ n _1+ n _2}$

$\Rightarrow 5=\frac{3 \alpha+6 \beta}{\alpha+\beta}$

$\Rightarrow 5 \alpha+5 \beta=3 \alpha+6 \beta$

$\therefore 2 \alpha=\beta$