A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is (p² + q²) always a composite number, where p and q are different prime numbers?
Which one of the following is correct in respect of the above Question and the Statements?
- AThe Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
- BThe Question can be answered by using either Statement alone
- CThe Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- DThe Question cannot be answered even by using both the Statements together
Explanation
Statement I: p−q odd forces one of the primes to be 2 (the only even prime). Then p²+q²=4+q² with q an odd prime — but 2²+3²=13 is prime, so p²+q² is NOT always composite and the question is answered 'No' by statement I alone. Statement II: p+q odd likewise forces one prime to be 2, answering it the same way. Either statement alone suffices, option B. Note: UPSC's published provisional key lists this question as D, but independent solving confirms B (13 is prime), so it is flagged as disputed pending UPSC's final key.