Existence of primitive pairs with prescribed traces over finite fields

Let F = Fqm, m > 7, n a positive integer, and f = p(1)/p(2) with p(1), p(2) coprime irreducible polynomials in F[x] and deg(p(1)) + deg(p(2)) = n. We obtain a sufficient condition on (q, m), which guarantees, for any prescribed a, b in E = F-q, the existence of primitive pair (alpha, f (alpha )) in F such that Tr-F/E(alpha) = a and Tr-F/E(a(-1)) = b: Further, for every positive integer n, such a pair definitely exists for large enough m. The case n = 2 is dealt separately and proved that such a pair exists for all (q, m) apart from at most 64 choices.