Am I sinful if my mother takes her money to a usurious bank to deposit the price of the chalet she bought in cash from a company that deals with this bank, and is she sinful for doing so?

The issue consists of two parts: First, buying the chalet. If it is for permissible personal use to avoid mixed-gender places, or for permissible investment by renting it out for lawful purposes without sins, then there is no harm. However, if the purchase is for use in forbidden activities such as mixing of genders and sins, then it is prohibited, and assisting in it is not permissible.

The second part: depositing the price of the chalet in a usurious bank. If buying the chalet is permissible, then there is no harm in depositing the price in a usurious bank to transfer it to the seller. However, if the purchase is forbidden, then paying the price is not permissible.

Generally, it is not permissible to open an account in a usurious bank unless there is a necessity or need, such as the absence of Islamic banks, and it should be limited to a current account. Your assistance to your mother depends on the ruling of the purchase: if it is permissible, then helping her is an act of righteousness; if it is forbidden, then helping her is prohibited.