Is it permissible to fast on the 11th of Muharram as a makeup for the 10th day, which my mother broke her fast on due to illness, knowing that she fasted the 9th day (Friday) individually?

Fasting on Friday alone is disliked if it is a voluntary fast, unless it coincides with a habitual fast such as Ashura or Arafah, or if one intended to fast the day after it but was unable to due to an excuse. However, if one does not fast it without an excuse, there is a difference of opinion regarding it. Since the mother was unable to fast Ashura due to illness, it is recommended for her to make it up, bearing in mind that Friday might be Ashura this year.