No, The 4 Pin connector and the 24 Pin connector Need to be attached.. (UNLESS your P1 COnnection is 20 Pins and not 24 Pins).
To test if the power supply is dead or not, you should dissconnect all connections but fans, then you take the 20/24 PIn connector, take an insulated paperclip, and connect a Black wire to a green wire.
IF the power supply turns on that means it is good and something else is wrong. IF it does not turn on, it means that the PSU is DOA and you should RMA it ASAP.