Problem #1

[saturated]

I came across this problem in a book and I admit becoming a deer in the headlights 

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I remember looking at the answer and not liking it  though I do not recall

Initially I proposed that current can only flow in one direction and when the cathode end is negative then current will flow.

But the cathode end is also connected to ground  sooooo how's it gonna be negative 

I welcome as usual any advice hints speculation solutions discussion etc

The good news is I'm going to construct this circuit I have a few transformers and can put some low voltage like 6 to 12 volts on the secondary 

I'm still a bit hesitant to ground one terminal of the secondary 

But that's the good thing is i can simply build it and not take chances like have a fuse etc

And that's what I like is I don't have to stare at something for days trying to make sense of it or accept it when I have the components and test equipment to actually see what happens 

[g1]

The cathode is not grounded.  Think about how an inductor reacts to AC.
The question is a bit misleading as it implies there will be DC at the cathode, I would not call it exactly that.