I think I solved my problem!
I noticed, when I connected, the ipad that the new "Trust this Computer" popup to Flash on and off very quickly (See http://support.apple.com/kb/HT5868 for info on purpose of the popup). It appeared for such a short time that I was never able to see it for long enough to respond to it.
Last night I switched off the lock password (the ipad was not connected to the mac when I did this). Today when I tried again to connect the Mac I heard a beep and saw the "Trust this Computer" popup. Instead of being presented for fractions of a second, as it had been previously, it was shown for a second or two before dissappearing and reappearing several seconds later etc.
When I realised that the "Trust this Computer! popup was being presented for long enough for me to push the Trust button, I attempted to press it. The popup closed and the ipad showed it was charging. The Mac presented a popup saying it had not been able to connect. I selected "OK", itunes loaded and started to sync to the ipad.
I subsequently conncted the ipad via a different cable. The problem started again (it is a genuine apple cable, supplied with the ipad). As soon as I added the cable that had worked all was OK again.
My wife's ipad had the same behaviour as I have described above.
I noticed that ios version 7.1.1 is now available. I have installed it. However this does not solve the problem as described above.
Conclusion: I now only have one out of 6 cables that I have that actually works!!! This cable is one of those "older types" that has "clips" in the side to "lock" it into place on the ipad. The ipad end is about twice the depth of the cable that was shipped with the ipad (possibly I got it with an old ipod) . What has Apple done to make so many cables unusable at overnight? I have not yet found any sources for this old type of cable.