Step 1: Analyze sentence (i)
\[
(i) \text{The number of candidates who appear for the GATE examination is staggering.}
\]
- Correct: The subject "number" is singular, so the verb "is" agrees with it.
Step 2: Analyze sentence (ii)
\[
(ii) \text{A number of candidates from my class are appearing for the GATE examination.}
\]
- Correct: "A number of" is a phrase that takes the plural verb "are."
Step 3: Analyze sentence (iii)
\[
(iii) \text{The number of candidates who appear for the GATE examination are staggering.}
\]
- Incorrect: The subject "number" is singular, so the verb should be "is" instead of "are."
Step 4: Analyze sentence (iv)
\[
(iv) \text{A number of candidates from my class is appearing for the GATE examination.}
\]
- Incorrect: "A number of" takes a plural verb, so it should be "are" instead of "is."
Conclusion:
Sentences (i) and (ii) are grammatically correct, while sentences (iii) and (iv) are incorrect. Thus, the correct answer is Option (A).
Final Answer: (A)