Determining if a diode conducts or is cutoff?
The model you are using is:


If \$V_{anode} &lt; V_{cathode}\$, \$I_D = 0\$ (cutoff)
If \$I_D &gt; 0\$, \$V_{anode} = V_{cathode}\$ (conducting)


The general procedure for solving systems with diodes is:


Make an assumption about the state of every diode (either cutoff or conducting).
Try to solve the circuit with those assumptions.
If you have a solution, check to make sure that solution is consistent, i.e. all of your original assumptions were correct. If there is no valid solution, your original assumptions were probably wrong. Pick a new set of assumptions and repeat.


For example, take circuit a. Let's assume D1 and D2 are both conducting. By definition then, the node labeled \$V\$ must be at 1V and at 2V at the same time, which is nonsense. So we know that both diodes can't be conducting at the same time. Now pick new assumptions and try again (left as an exercise for the reader).