In Sentence A we are given 'each of the six boys' i.e. here we are referring to each of the boy i.e. singular quantity and thus, instead of 'their' it should be 'his'.
In Sentence B we have 'One' as the subject. Thus, instead of the pronoun 'his' 'one's' should be used.
In Sentence C we have used the correlative conjunction 'either—or', the verb and the pronoun should agree with the subject nearer to the verb.
Thus, Since Shyam (a singular) is nearer to the verb, the word 'their'(a plural) can't be used instead a singular word should have been used(his)
Sentence D has no error.
Hence, option D is the correct answer.