There is a mutation in the operator of the lac operon in a cell such that the lac repressor always stays bound to the operator. If lactose is added to the cell, what will happen?

Yes, if there is a mutation in the operator of the lac operon in a cell such that the lac repressor always stays bound to the operator. If lactose is added to the cell, the lac repressor loses its ability to bind DNA. It clear off the operator, clearing the way for RNA polymerase to transcribe the operon.