VAT

Respected sir,

Sir from my study text i understand that input VAT related to exempt supplies, is not recoverable. In relation to this statement I have some doubts.
One, why is an input VAT levied over goods on which output VAT can’t be levied? Any specific reason sir?
Two, is the reverse possible(no input VAT, but output VAT exists)? and output VAT not paid back to HMRC.
Three, could you give an example for the 2 above mentioned scenarios(if the latter, is possible also)?

Appreciate:)