What is the ruling on objecting to the Mosque Imam regarding the joining of prayers due to rain?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for any of the congregants to object to the Imam in the mosque, whether the objection is to demand the joining of prayers or to oppose it; for the general principle is that the Imam possesses the religious knowledge (Al-Ilm al-Shar‘i) that qualifies him to make the decision in this matter. And Allah the Exalted knows best."
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
What is the ruling on one who vowed not to speak to a certain person, then forgot his vow and spoke to him?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Whoever swears an oath or makes a vow to abstain from something has a choice: if he wishes, he may keep to his vow, or if he wishes, he may offer the expiation of an oath (kaffārat yamīn). The Prophet (peace and blessings be upon him) said: “The expiation for a vow is the expiation for an oath.” Narrated by Muslim. This type of vow is called a nadhr al-lajāj (a vow made in stubbornness or anger).
If a person makes a vow and then forgets it, nothing is due from him, for forgetfulness is among the excuses that lift legal responsibility (taklīf) from a Muslim. This is based on the saying of the Prophet (peace and blessings be upon him): “Actions are judged only by intentions, and every person will have only what he intended.” [Narrated by al-Bukhārī]. There is no intention for the one who forgets or errs. It is also based on the report: “Indeed, Allah has pardoned for my Ummah mistakes, forgetfulness, and what they are compelled to do.” Narrated by Ibn Mājah and others, and graded authentic by Ibn Ḥibbān and al-Ḥākim.
It is stated in ʿUmdat al-Sālik (p. 243), one of the books of the Shāfiʿī school: “Whoever swears, [for example], ‘I will not enter the house,’ and then enters it forgetfully, ignorantly, under compulsion, or being carried, he does not break his oath.” And Allah Almighty knows best.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
