How to determine diode will conduct or not
Note: whenever you have a resistor in series with a current source you can ignore the resistor for your analysis. 

I.e. you can ignore the right resistor and this makes the problem easy:

Hint: Sum of currents into (or out of) any node is 0 (KCL).
So what happens if \$I_{from-left} = V_{src}/R\$ is less than or larger than \$I_{from-right} = -I_{src} = -1A\$?