What is the numerical value of the rate constant?

Let the order of reaction are a and b with respect to A and B respectively. 

r = k[A]^a[B]^b ....(1)

According to the question,

2r = k[A]^a[2B]^b....(2)

and 

4r = k[2A]^a[B]^b ....(3)

3/1, we have 2^2 = 2^a or, a = 2 

2/1, we have 2^1 = 2^b or, b = 1. 

So, rate law : r = k[A]^2[B]^1