You use an when the next word starts with a vowel sound. There is no hard rule as to what letter follows a/an in writing - it's dependent on pronunciation:
an hour ("h" is silent, so the word starts with the sound /a/)
a hit ("h" is not silent in this case, so the word starts with /h/).
an N.Y.C. ("N.Y.C." is pronounced as ...
Because you have "that" acting as a determiner, you don't need an article as well.
after that first year.
Consider as an example:
Look at the car.
Look at that car.
Both "the" and "that" in these two examples serve to highlight which particular car you mean.
In this usage, the article is correct. The sense is 'an appearance of puzzlement which seemed strange'.
Compare 'He accepted the compliments with a becoming modesty.'
'She displayed her work with a shy pride'.
Many (perhaps most) uncountable words can be used as countable when the reference is to a particular instance or type.
I will drink some wine (uncountable) with That is a good wine (a type, so countable).
I feel love (uncountable) with I have a love of music (a type, so countable).
We hate injustice (uncountable) with An injustice which must be ...
You can not omit the article.
The first thing is while there is only one light, you are referring to one time when it was lit -- that is a single alarm/warning. The other ones may be nothing more than possible, but they will happen at different times.
You can make clear which time you are referring to, by context or the like, but then the second is that ...
This is an odd case where either works, but the meanings are slightly different.
If we think of “water gas” as a general idea of what happens when you mix two gases, then it’s “a” mixture, whereas the contents of a specific container with those gases would be “the” mixture.
If we think of it as a specific mixture as opposed to other mixtures, then it is “the”...
“An exchange of money” refers to a singular exchange of money.
“The exchange of money” refers to the process itself generally (for this reason, A1 may be considered slightly incorrect, though it would be considered acceptable). It may also refer to some specific exchange of money that you’re already talking about or have alluded to. E.g.:
The exchange of ...
'The' is necessary and required, because there is only one start of a project, and unique items are preceded by the definite article. You may find websites such as Grammar Monster useful:
The definite article is the word "the." It is used before a noun to
define it as something specific (e.g., something previously mentioned
or known, something ...
A proper noun is a particular name of a person, place or thing. They are written with a capital letter. Articles can be part the proper noun:
I visited the United States last year. (the is part of the proper noun)
I visited Canada last year ("the" is not part of the name of Canada)
It is rare to use "a" with proper nouns, but ...
The difference is to do with whether you are talking about specific things or things in general.
In the case of memes if you are talking about memes in general you would omit the article as I have just done, and you can refer to a subset of memes without the article as well (you can talk about 'cat memes' for example without using the article) but if you ...
Since the idea is specifically related to making money, a definite article must be used. The definite article (the) is used before a noun to indicate that the identity of the noun is known to the reader.
Do you like the idea of making some money on the side?
This statement indicates that you have already an idea proposed to you by someone or an idea made ...