For the reaction 2 A + B $\to$Products, it is found that doubling the concentration of both reactants increases the rate by a factor of 8. But doubling the concentration of B alone, only doubles the rate. What is the order of the reaction w.r.t to A ?

When [B] is doubled keeping [A] constant ,rate doubles 

$\mathrm{r}\infty {\left[\mathrm{B}\right]}^{\mathrm{p}}$;$2\mathrm{r}\infty {\left[2\mathrm{B}\right]}^{\mathrm{p}}$

solving both equations, order with respect to "B", p = 1

when concentration of both are doubled rate increases by a factor of 8

$\mathrm{r}\infty {\left[\mathrm{A}\right]}^{\mathrm{q}}{\left[\mathrm{B}\right]}^{\mathrm{p}} 8\mathrm{r}\infty {\left[2\mathrm{A}\right]}^{\mathrm{q}}{\left[2\mathrm{B}\right]}^{\mathrm{p}}$

solving both reactions order with respect to "A" q =2

rate law =K[A]²[B]¹