39. 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)^2 (qs) is even. 2. (q-s)q^2 s is even. 3. (q + r)^2 (p + s) is odd. Which of the statements given above are correct?

1. A. 1 and 2 only
2. B. 2 and 3 only
3. C. 1 and 3 only
4. D. 1, 2 and 3

Answer: D

Explanation

Given: p, q are odd integers; r, s are even integers.
1. (p – r)^2 (qs): (odd – even) = odd. So, (p – r)^2 = odd^2 = odd. (q × s) = (odd × even) = even. Therefore, (p – r)^2 (qs) = odd × even = even. Statement 1 is correct.
2. (q – s)q^2 s: (q – s) = (odd – even) = odd. q^2 = odd^2 = odd. s is even. Therefore, (q – s)q^2 s = odd × odd × even = even. Statement 2 is correct.
3. (q + r)^2 (p + s): (q + r) = (odd + even) = odd. So, (q + r)^2 = odd^2 = odd. (p + s) = (odd + even) = odd. Therefore, (q + r)^2 (p + s) = odd × odd = odd. Statement 3 is correct.
All three statements are correct. This question tests basic properties of odd and even numbers (parity) and their behavior under arithmetic operations.