How does increase in volume change the speed of reaction in production of NO2?
I'm not an expert but this is how I would do it.


Determine the reaction rate formula from the reaction equation

$$r = K[\ce{NO}]^2[\ce{O2}]$$
Changing the volume by a factor $2$ means the concentration of $\ce{NO}$ and $\ce{O2}$ will become half

$$r' = K\cdot\frac{[\ce{NO}]}{2}^2\cdot\frac{[\ce{O2}]}{2}$$
Compare the reaction rates

$$r' = K[\ce{NO}]^2\cdot\frac{1}{4}\cdot [\ce{O2}]\cdot\frac{1}{2}$$

$$\frac{r}{r'} = \frac{K[\ce{NO}]^2[\ce{O2}]}{K[\ce{NO}]^2[\ce{O2}]\cdot\frac{1}{8}} = 8$$


Conclusion: increasing the volume by $2$ will decrease the concentration (or pressure) of $\ce{NO}$ and $\ce{O2}.$ This causes the reaction rate to decrease by a factor of $8$.