By hit and trial, we put p = 5 and q = 1 so that (3p + 7q) = (3 x 5 + 7 x 1) = 22, which is divisible by 11.
(4p + 6q) = ( 4 x 5 + 6 x 1) = 26, which is not divisible by 11;
(p + q + 4 ) = (5 + 1 + 4) = 10, which is not divisible by 11;
(9p + 4q) = (9 x 5 + 4 x 1) = 49, which is not divisible by 11;
(4p - 9q) = (4 x 5 - 9 x 1) = 11, which is divisible by 11.