Is it obligatory to fulfill the vow of cutting hair and praying two rak'ahs, which was conditioned upon the mother's recovery, given that she passed away before recovering?

The vow (nazr) that must be fulfilled is the vow of drawing closer to Allah (nazr al-qurbah), such as vowing to give charity, perform prayer, or fast. However, not cutting hair does not fall into this category. Vowing to perform two rak'ahs of prayer or similar acts of obedience must be fulfilled, based on the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." Tying a vow to the occurrence of something is disliked by most scholars and is called a conditional vow (nazr mu'allaq). It becomes obligatory to fulfill if the desired outcome occurs. Since your mother's recovery did not happen, there is no obligation to fulfill the vow. It is permissible for a woman to cut her hair for adornment for her husband, as long as it does not involve imitating disbelievers, prostitutes, or men.