1) Is it permissible by Allah to keep a second wife not known by your family members?

2) Can I wear a wig instead of hijab to cover my hair?

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1- Yes a man can marry another wife without his wife’s knowledge. However, is that a wise thing to do? If it leads to problems then one should not.

2- Wearing a wig before men is not permissible because it is Zeena (adornment) and the Qur’an in Sura Nour verse 31 prohibits women frok displaying their adornments to men.