Geared and Ungeared cost of Equity

Dear Sir,
can you please confirm if my understanding is correct?

in APV to calculate the base present value we use the Cost of equity ungeared (so or the cost of equity is given and we need to ungeared it or we need to calculate it with the asset beta if the cost of equity is not given).

if the entity is fully finance through equity, the asset beta = beta equity so the cost of equity is ungeared (because we do not have debt)

if we need to use the CAPM to discount a project, we use the cost of equity geared to calculate the WACC.

did i say anything wrong?
thanks a lot