The adjective ‘no’ means ‘not a’ and ‘not any’. A singular or plural noun can follow it depending on the meaning you want to convey. ‘No cars’ in the 1st sentence means ‘not any cars’. ‘Alternative’ could mean ‘a choice between 2 things’. In this sense, there’s only one other choice “to what he had done.” You therefore have the singular ‘alternative’ in "There was no alternative to what he had done." Because of the word origin, some writers consider ‘alternative’ to mean 1 of 2 (not more than 2) possibilities. It is commonly used to mean one (and limited to one only) of several possibilities.
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