Is it permissible for a female engineer to invest her money with a contractor who has been selected in a national tender for projects affiliated with her directorate, if the contractor accepts this without coercion and she has no involvement with the project?

There is no impediment to the contractor participating with a sum of money in his project if there is no relation to the project in terms of allocation, supervision, or responsibility for receipt. The profit between the two of you shall be according to the agreement, and this is a permissible Mudarabah (profit-sharing partnership).