Solution:
The noun in the given statement, 'name' is described by the adjective 'French', and it is not specified which 'name' in particular, hence indefinite article 'a' is the correct option, since the noun and adjective both begin with a consonant sound.

Solution:
In the word "year" the letter Y makes a consonant sound. The rule of indefinite articles is that the word "a" goes before consonant sounds and the word "an" goes before vowel sounds.

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Solution:
Here, specific 'little information' is talked about. Hence, definite article 'the' will be used.

Solution:
The sentence suggests that the speaker believes that the listener already knows what he is referring to and the hint also indicates that. In this case, we use definite article, 'the' for both the nouns. Hence option C is the correct answer.

Solution:
Here, the sentence talks about the 'cycle' as general class of things. Since we are referring to general things, we need to use the definite article 'a'.

Solution:
As 'beauty' is an abstract noun, the second blank doesn't require an article the first blank requires article 'the' because there, beauty of something specific is talked about.