Q. Let p, q, r and s be distinct positive integers.
Let p, q be odd and r, s be even. Consider the following statements:
1. (p – r) (qs) is even.
2. (q – s) q s is even.
3. (q + r)² (p + s) is odd.
Which of the statements given above are correct?
a) 1 and 2 only
b) 2 and 3 only
c) 1 and 3 only
d) 1, 2 and 3
Correct answer: d) 1, 2 and 3
Question from UPSC Prelims 2024 CSAT
Explanation :
Let’s analyze each statement given the conditions:
Statement 1: (p – r)(qs) is even.
– (p – r): Since p is odd and r is even, p – r is odd.
– qs: q is odd and s is even, so qs is even.
– Therefore, (p – r)(qs) = odd × even = even.
– Conclusion: Statement 1 is correct.
Statement 2: (q – s) q s is even.
– (q – s): q is odd and s is even, so q – s is odd.
– q × s: q is odd and s is even, making qs even.
– Thus, (q – s)qs = odd × even = even.
– Conclusion: Statement 2 is correct.
Statement 3: (q + r)^2 (p + s) is odd.
– (q + r): q is odd and r is even, so q + r is odd. Squaring it, (q + r)^2, remains odd.
– (p + s): p is odd and s is even, so p + s is odd.
– Therefore, (q + r)^2 (p + s) = odd × odd = odd.
– Conclusion: Statement 3 is correct.
All three statements are correct.
Answer: d) 1, 2 and 3