A three-digit number 4a3 is added to another three-digit numbers 984 to give the four digit number 13b7, which is divisible by 11. The value of |2a - b| is

The addition of 4a3 and 984 gives 13b7.

We shall refer to the divisibility rule of 11, which states that |sum of alternate digits from unit digit-sum of rest of the digits| shall be 0 or a multiple of 11. 

From the similar logic, 13b7 has 7+3 as the sum of alternate digits starting from unit digit, and 1+b as the remaining ones. 

It is clear that the difference cannot touch any higher value (starting from 11) as none of the two values (3+7 or 1+b) can be greater than 10, hence it has to be 0. 

(3+7)- (1+b) = 0

b=9

Now, we shall go back to the original equation: 4a3 + 984 = 1397

from this, 4a3= 413

Hence, a = 1

Substituting the values of a and b in |2a-b|, we get the final answer as 7.