Answer is: pH is 1.92.
Balanced chemical reaction: Ba(OH)₂ + 2HNO₃ → Ba(NO₃)₂ + 2H₂O.
V(Ba(OH)₂) = 5.0 mL ÷ 1000 mL/L.
V(Ba(OH)₂) = 0.005 L; volume of barium hydroxide.
c(Ba(OH)₂) = 0.010 M; molarity of barium hydroxide.
n(Ba(OH)₂) = V(Ba(OH)₂) · c(Ba(OH)₂).
n(Ba(OH)₂) = 0.00005 mol; amount of barium hydroxide.
n(HNO₃) = 0.0004 mol; amoun of nitric acid.
From balanced chemical reaction: n(Ba(OH)₂) : n(HNO₃) = 1 : 2.
Δn(HNO₃) = 0.0004 mol - 0.0001 mol.
Δn(HNO₃) = 0.0003 mol; excess of nitri acid.
Δc(HNO₃) = 0.0003 mol ÷ 0.025 L.
Δc(HNO₃) = 0.012 M.
pH = -log(0.012 M).
pH = 1.92.