Moles of $CO _{2}=\frac{11}{44}$

$=0.25$

Moles of $N _{2}$ are,

Moles of $N _{2}=\frac{14}{28}$

$=0.50$

The value of $\gamma$ for the mixture of gases is given by the relation,

$\frac{n_{1}+n_{2}}{\gamma_{\operatorname{mix}}-1}=\frac{n_{1}}{\gamma_{1}-1}+\frac{n_{2}}{\gamma_{2}-1}$

Where, $n_{1}$ and $n_{2}$ are the number of moles of the two gases and $\gamma_{\text {mix }}$ is the specific heat ratio for mixture. Therefore,

$\frac{0.25+0.50}{\gamma_{\text {mix }}-1}=\frac{0.25}{1.29-1}+\frac{0.50}{1.4-1}$

$\frac{0.75}{\gamma_{\operatorname{mix}}-1}=0.862+1.25$

$2.112 \gamma_{\operatorname{mix}}=0.75+2.112$

$\gamma_{\operatorname{mix}} \approx 1.4$

$=\frac{7}{5}$