if a child had O type blood and the mother had B type blood and the father had A type blood, could a blood test eliminate the father

No, a blood test could not eliminate the father.

Explanation:

The mother had B type blood, so the possible genotypes are BB and BI.  The father had A type blood, so the possible genotypes are AA and AI.  The child had O type blood, so the only possible genotype is II.  If the mother has the genotype BI and the father has the genotype AI, then there is a 25% chance the child will have the blood type O, or II (I used a punnet square with the genotypes BI and AI to solve this.)  

high growth (apex)