if harmonic mean of p and q is 3
then 2pq/(p+q)=3
2pq=3p+3q
2pq-3p=3q
p(2q-3)=3q
p=3q/(2q-3).......................(1)
and
[ (6p+1)/(7q-4) ] <(p/q)
6pq+q<7pq-4p
4p+q<pq
from (1)
{12q/(2q-3) + q }< 3q2/(2q-3)
(2q2+9q)/2q-3 <3q2/(2q-3)
{(2q2+9q)/2q-3} - 3q2/(2q-3)<0
(-q2+9q)/(2q-3)<0
(q2-9q)/(2q-3)>0
this is true if either both numerator and denominator is positive or both are negative
case 1) if (q2-9q)>0 and (2q-3)>0
ie q(q-9)>0 and (2q-3)>0
from here we get q > 9
case 2) if (q2-9q)<0 and (2q-3)<0
from here we get q belongs to (0,3/2)
so option b) is correct