Is it permissible to buy a house with the intention of renting it out and then repay its value from the resulting rental income, especially if the loan is from one's mother and provided that the house is bought for personal residence, and does the prohibition against borrowing apply to this case?

It is permissible to purchase a house with a non-interest-bearing benevolent loan (qard hasan), because a benevolent loan is one on which no interest accrues to the lender. Once the borrower receives the money, it becomes his property, and he may dispose of it as he wishes. If the mother stipulates that the house is for her son's residence, then her condition must be adhered to, and the house may not be rented out except with her permission, because "Muslims are bound by their conditions." However, if the son no longer needs the house afterwards, or needs to rent it out, there is no harm in that, as long as it is not a circumvention of the mother's condition. But if the intention behind purchasing the house is to rent it out, the son must clarify this to his mother.