How can we ask a question tag for the following questions? Negative or positive?
The boys problems are physical not mental,...?
They were warned not to climb the mountain in such bad weather, ...?
Is it correct to use negative question tag?
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For the first time you can ask
The boys problems are physical not mental, are they not?/aren't they?
you say it with a negative "aren't they" confirming if they really are or not, for the second one you can say
They were warned not to climb the mountain in such bad weather, were they not?/weren't they?
which basically points out that the asker makes it certain "They were warned about it, right?"
The purpose of a tag is to check for agreement. In the first case I simply wouldn't tag at the end but split into two sentences. Then there are two alternatives.
The boy's problems are physical, aren't they? They aren't mental.
The boy's problems are physical. They aren't mental, are they?
The second has a clear positive verb and so the tag would normally be negative. The negative verb in the infinitive clause is not being questioned.
They were warned not to climb the mountain in such bad weather, weren't they?
I've also stressed the verb "were", as would be natural, if this was in doubt.