How many ml of 1.00M HCl do I need to add to 10.0ml of 1.00M NaOH to get a solution with pH = 1?

Let V ml of HCl is added to get a solution with pH = 1.

Balanced equation for the reaction:
HCl + NaOH → NaCl + H₂O
Mole ratio HCl : NaOH = 1 : 1

No. of moles of NaOH = (1.00 mol/L) × (10.0/1000 L) = 0.01 mol
No. of moles of HCl added = (1.00 mol/L) × (V/1000 L) = 0.001V mol

No. of moles of HCl in the final solution = No. of moles of HCl left unreacted = (0.001V - 0.0100) mol
Volume of the final solution = (V + 10.0) ml = (V + 10)/1000 L

Concentration of H⁺ ions in the final solution:
(0.001V - 0.01) / [(V + 10)/1000] = 10^(-pH)
(V - 10) / (V + 10) = 10^(-1)
(V - 10) / (V + 10) = 0.1
V - 10 = 0.1V + 1
0.9V = 11
V = 12.2

Volume of HCl added = 12.2 ml