MacBook Air 2012 Mini DisplayPort not working with HDTV
I have exactly same set up down to the MBA 2012 + Adapter and Cable. 

It has been working flawlessly since.

So far I used it on 3 different TV models.

In your case I would say it is the adapter, they tend to be flaky (what do you expect for $6).

But, you can take a look in the Console and see what the "Windoserver" is doing when you plug in the cable.

If you live in the USA, go to store like Best buy or Walmart, get another adapter and try it. If it works you have your proof. If it does not just return it to the store for full refund.